Simple writing DUE TOMORROW (THURSDAY)

Please respond to the following paragraphs casually and meet the word count. I have provided materials if needed, but most information can be found through a basic Google search. YOU DO NOT NEED PREVIOUS KNOWLEDGE.

150 words

1.
How is dividing a polynomial by a binomial similar to or different from the long division you learned in elementary school?
Can understanding how to do one kind of division help you with understanding the other kind? What are some examples from real life in which you might use polynomial division?

200 words

What are the four elements of financial management? Why do we need financial and managerial accounting? What are the differences between the two? List references if you use a source other than the text. (200 words)

P A R T

Healthcar

Finan

O v e r v i e w

Introduction to
Healthcare Finance 1

C H A P T E R

THE HISTORY

Financial management has a long and distinguished his-
tory

Consider, for example, that Socrates wrote abou

the universal function of management in human en-
deavors in 400 B.C. and that Plato developed the concept
of specialization for efficiency in 350 B.C. Evidence of so-
phisticated financial management exists for much earlier
times: the Chinese produced a planning and control sys-
tem in 1100 B.C., a minimum-wage system was developed
by Hammurabi in 1800 B.C., and the Egyptians and
Sumerians developed planning and record-keeping sys-
tems in 4000 B.C.1

Many managers in early history discovered and redis-
covered managerial principles while attempting to rea

their goals. Because the idea of management thought a

a discipline had not yet evolved, they formulated princi-
ples of management because certain goals had to be ac-
complished. As management thought became codified
over time, however, the building of techniques for man-
agement became more organized. Management as a dis-
cipline for educational purposes began in the United
States in 1881. In that year, Joseph Wharton created the
Wharton School, offering college courses in business
management at the University of Pennsylvania. It was the
only such school until 1898, when the Universities of
Chicago and California established their business
schools. Thirteen years later, in 1911, 30 such schools
were in operation in the United States.2

Over the long span of history, managers have al

sought how to make organizations work more effectively.
Financial management is a vital part of organizational

After completing this chapter,
you should be able to

1. Discuss the three viewpoints of
managers in organizations.

2. Identify the four elements of
financial management.

3. Understand the differences
between the two types of
accounting.

4. Identify the types of
organizations.

5. Understand the compositio

and purpose of an organization
chart.

P r o g r e s s N o t e s

effectiveness. This book’s goal is to provide the keys to unlock the secrets of financial man-
agement for nonfinancial managers.

THE CONCEPT

A Method of Getting Money in and out of the Business

One of our colleagues, a nurse, talks about the area of healthcare finance as “a method of
getting money in and out of the business.” It is not a bad description. As we shall see, rev-
enues represent inflow and expenses represent outflow. Thus, “getting money in” repre-
sents the inflow (revenues), whereas “getting money out” (expenses) represents the
outflow. The successful manager, through planning, organizing, controlling, and decision
making, is able to adjust the inflow and outflow to achieve the most beneficial outcome for
the organization.

HOW DOES FINANCE WORK IN THE HEALTHCARE BUSINESS?

The purpose of this book is to show how the various elements of finance fit together: in
other words, how finance works in the healthcare business. The real key to understanding
finance is understanding the various pieces and their relationship to each other. If you, the
manager, truly see how the elements work, then they are yours. They become your tools to
achieve management success.

The healthcare industry is a service industry. It is not in the business of manufacturing,
say, widgets. Instead, its essential business is the delivery of healthcare services. It may have
inventories of medical supplies and drugs, but those inventories are necessary to service de-
livery, not to manufacturing functions. Because the business of health care is service, the
explanations and illustrations within this book focus on the practice of financial manage-
ment in the service industries.

VIEWPOINTS

The managers within a healthcare organization will generally have one of three views: (1) fi-
nancial, (2) process, or (3) clinical. The way they manage will be influenced by which view
they hold.

1. The financial view. These managers generally work with finance on a daily basis. The
reporting function is part of their responsibility. They usually perform much of the
strategic planning for the organization.

2. The process view. These managers generally work with the system of the organization.
They may be responsible for data accumulation. They are often affiliated with the in-
formation system hierarchy in the organization.

3. The clinical view. These managers generally are responsible for service delivery. They
have direct interaction with the patients and are responsible for clinical outcomes of
the organization.

4 CHAPTER 1 Introduction to Healthcare Finance

The Elements of Financial Management 5

Managers must, of necessity, interact with
one another. Thus, managers holding differ-
ent views will be required to work together.
Their concerns will intersect to some degree,
as illustrated by Figure 1-1. The nonfinancial
manager who understands healthcare fi-
nance will be able to interpret and negotiate
successfully such interactions between and
among viewpoints.

In summary, financial management is a
discipline with a long and respected history.
Healthcare service delivery is a business,
and the concept of financial management
assists in balancing the inflows and outflows
that are a part of the business.

WHY MANAGE?

Business does not run itself. It requires a variety of management activities in order to oper-
ate properly.

THE ELEMENTS OF FINANCIAL MANAGEMENT

There are four recognized elements of financial management: (1) planning, (2) control-
ling, (3) organizing and directing, and (4) decision making. The four divisions are based
on the purpose of each task. Some authorities stress only three elements (planning, con-
trolling, and decision making) and consider organizing and directing as a part of the con-
trolling element. This text recognizes organizing and directing as a separate element of
financial management, primarily because such a large proportion of a manager’s time is
taken up with performing these duties.

1. Planning. The financial manager identifies the steps that must be taken to accom-
plish the organization’s objectives. Thus, the purpose is to identify objectives and
then to identify the steps required for accomplishing these objectives.

2. Controlling. The financial manager makes sure that each area of the organization is
following the plans that have been established. One way to do this is to study current
reports and compare them with reports from earlier periods. This comparison often
shows where the organization may need attention because that area is not effective.
The reports that the manager uses for this purpose are often called feedback. The
purpose of controlling is to ensure that plans are being followed.

3. Organizing and directing. When organizing, the financial manager decides how to
use the resources of the organization to most effectively carry out the plans that have
been established. When directing, the manager works on a day-to-day basis to keep
the results of the organizing running efficiently. The purpose is to ensure effective re-
source use and provide daily supervision.

Financial

Clinical

Process

Figure 1–1 3 Views of Mgmt within an Organization.

6 CHAPTER 1 Introduction to Healthcare Finance

4. Decision making. The financial manager makes choices among available alternatives.
Decision making actually occurs parallel to planning, organizing, and controlling. All
types of decision making rely on information, and the primary tasks are analysis and
evaluation. Thus, the purpose is to make informed choices.

THE ORGANIZATION’S STRUCTURE

The structure of an organization is an important factor in management.

Organization Types

Organizations fall into one of two basic types: profit oriented or nonprofit oriented. In the
United States, these designations follow the taxable status of the organizations. The profit-
oriented entities, also known as proprietary organizations, are responsible for paying in-
come taxes. Proprietary subgroups include individuals, partnerships, and corporations.
The nonprofit organizations do not pay income taxes.

There are two subgroups of nonprofit entities: voluntary and government. Voluntary
nonprofits have sought tax-exempt status. In general, voluntary nonprofits are associated
with churches, private schools, or foundations. Government nonprofits, on the other hand,
do not pay taxes because they are government entities. Government nonprofits can be
(1) federal, (2) state, (3) county, (4) city, (5) a combination of city and county, (6) a hos-

pital taxing district (with the power to raise
revenues through taxes), or (7) a state uni-
versity (perhaps with a teaching hospital
affiliated with the university). The organiza-
tion’s type may affect its structure. Exhibit
1-1 summarizes the subgroups of both pro-
prietary and nonprofit organizations.

Organization Charts

In a small organization, top management
will be able to see what is happening. Exten-
sive measures and indicators are not neces-
sary because management can view overall
operations. But in a large organization, top
management must use the management
control system to understand what is going
on. In other words, to view operations, man-
agement must use measures and indicators
because he or she cannot get a firsthand
overall picture of the total organization.

As a rule of thumb, an informal manage-
ment control system is acceptable only if the

Exhibit 1–1 Types of Organizations

Profit Oriented—Proprietary
Individual
Partnership
Corporation
Other

Nonprofit—Voluntary
Church Associated
Private School Associated
Foundation Associated
Other

Nonprofit—Government
Federal
State
County
City
City-County
Hospital District
State University
Other

manager can stay in close contact with all aspects of the operation. Otherwise, a formal sys-
tem is required. In the context of health care, therefore, a one-physician practice (Figure
1-2) could use an informal method, but a hospital system (Figure 1-3) must use a formal
method of management control.

The structure of the organization will affect its financial management. Organization
charts are often used to illustrate the structure of the organization. Each box on an orga-
nization chart represents a particular area of management responsibility. The lines between
the boxes are lines of authority.

In the health system organization chart illustrated in Figure 1-3, the president/chief ex-
ecutive officer oversees seven senior vice presidents. Each senior vice president has vice
presidents reporting to him or her in each particular area of responsibility designated on
the chart. These vice presidents, in turn, have an array of other managers reporting to them
at varying levels of managerial responsibility.

The organization chart also shows the degree of decentralization within the organiza-
tion. Decentralization indicates the delegating of authority for decision making. The chart
thus illustrates the pattern of how managers are allowed—or required—to make key deci-
sions within the particular organization.

The purpose of an organization chart, then, is to indicate how responsibility is assigned
to managers and to indicate the formal lines of communication and reporting.

TWO TYPES OF ACCOUNTING

Financial

Financial accounting is generally for outside, or third party, use. Thus, financial accounting
emphasizes external reporting. External reporting to third parties in health care includes,
for example, government entities (Medicare, Medicaid, and other government programs)
and health plan payers. In addition, proprietary organizations may have to report to stock-
holders, taxing district hospitals have to report to taxpayers, and so on.

Two Types of Accounting 7

Reception
Scheduling

Billing
Accounting

Front
Office

Physician’s
Assistant

Registered
Nurse

Clinical
Services

Physician

Figure 1–2 Physicians Office Organization Chart.
Source: Courtesy of Resource Group, Ltd., Dallas, Texas.

8 CHAPTER 1 Introduction to Healthcare Finance
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Key Terms 9

Financial reporting for external purposes must be in accordance with generally accepted
accounting principles. Financial reporting is usually concerned with transactions that have
already occurred: that is, it is retrospective.

Managerial

Managerial accounting is generally for inside, or internal, use. Managerial accounting, as
its title implies, is used by managers. The planning and control of operations and related
performance measures are common day-by-day uses of managerial accounting. Likewise,
the reporting of profitability of services and the pricing of services are other common on-
going uses of managerial accounting. Strategic planning and other intermediate and long-
term decision making represent an additional use of managerial accounting.3

Managerial accounting intended for internal use is not bound by generally accepted ac-
counting principles. Managerial accounting deals with transactions that have already oc-
curred, but it is also concerned with the future, in the form of projecting outcomes and
preparing budgets. Thus, managerial accounting is prospective as well as retrospective.

INFORMATION CHECKPOINT

What Is Needed? Reports for management purposes.
Where Is It Found? With your supervisor.
How Is It Used? To manage better.
What Is Needed? Organization chart.
Where Is It Found? With your supervisor or in the administrative offices.
How Is It Used? To better understand the structure and lines of authority in

your organization.

KEY TERMS

Controlling
Decision Making
Financial Accounting
Managerial Accounting
Nonprofit Organization (also see Voluntary Organization)
Organization Chart
Organizing
Planning
Proprietary Organization (also see Profit-Oriented Organization)

10 CHAPTER 1 Introduction to Healthcare Finance

DISCUSSION QUESTIONS

1. What element of financial management do you perform most often in your job?
2. Do you perform all four elements? If not, why not?
3. Of the organization types described in this chapter, what type is the one you work

for?
4. Have you ever seen your company’s organization chart? If so, how decentralized is

it?
5. If you receive reports in the course of your work, do you believe that they are pre-

pared for outside (third party) use or for internal (management) use? What leads
you to believe this?

What Does the
Healthcare

Manager Need to
K n o w ?

2
C H A P T E R

HOW THE SYSTEM WORKS IN HEALTH CARE

The information that you, as a manager, work with is only
one part of an overall system. To understand financial
management, it is essential to recognize the overall sys-
tem in which your organization operates. An order exists
within the system, and it is generally up to you to find
that order. Watch for how the information fits together.
The four segments that make a healthcare financial sys-
tem work are (1) the original records, (2) the informa-
tion system, (3) the accounting system, (4) and the
reporting system. Generally speaking, the original
records provide evidence that some event has occurred;
the information system gathers this evidence; the ac-
counting system records the evidence, and the reporting
system produces reports of the effect. The healthcare
manager needs to know that these separate elements
exist and that they work together for an end result.

THE INFORMATION FLOW

Structure of the Information System

Information systems can be simplistic or highly com-
plex. They can be fully automated or semiautomated.
Occasionally—even today—they can still be generated
by hand and not by computer. (This last instance is be-
coming rare and can happen today only in certain small
and relatively isolated healthcare organizations that are
not yet required to electronically submit their billings.)

We will examine a particular information system
and point out the basics that a manager should be able to

After completing this chapter,
you should be able to

1. Understand that four segments
make a financial management
system work.

2. Follow an information flow.
3. Recognize the basic system

elements.
4. Follow the annual management

cycle.

P r o g r e s s N o t e s

recognize. Figure 2-1 shows information system components for an ambulatory care setting.
This complex system uses a clinical and financial data repository; in other words, both clin-
ical and financial data are fed into the same system. An automated medical record is also
linked to the system. These are basic facts that a manager should recognize about this am-
bulatory information system.

In addition, the financial information, both outpatient and any relevant inpatient, is fed
into the data repository. Scheduling-system data also enter the data repository, along with
any relevant inpatient care plan and nursing information. Again, all of these are basic facts
that a manager should recognize about this ambulatory care information system.

These items have all been inputs. One output from the clinical and financial data repos-
itory (also shown in Figure 2-1) is insurance verification for patients through an electronic
data information (EDI) link to insurance company databases. Insurance verification is daily
operating information. Another output is decision-making information for managed care
strategic planning, including support for demand, utilization, enrollment, and eligibility,
plus some statistical support. The manager does not have to understand the specifics of all
the inputs and outputs of this complex system, but he or she should recognize that these
outputs occur when this ambulatory system is activated.

Function of Flowsheets

Flowsheets illustrate, as in this case, the flow of activities that capture information.1 Flow-
sheets are useful because they portray who is responsible for what piece of information as it

12 CHAPTER 2 What Does the Healthcare Manager Need to Know?

Managed Care Systems
–Enrollment/eligibility
–Utilization management
–Demand management
–Algorithmic scheduling
support

Insurance

Verification

EDI
Link

Enterprise-Wide
Master Patient Index

Interface Engine

Clinical and Financial
Data Repository

Practice Management
System Scheduling

Ancillary Scheduling

Patient Accounting System
Financial (OP and IP)

OR Scheduling

Other Provider
Inpatient System

–Clinical order entry
–IP care plans
–IP nursing

Chart
Tracking

(IP and OP)

A
u
t
o
m
a
t
e
d

M
e
d
i
c
a
l

R
e
c
o
r
d

Figure 2–1 Information System Components for an Ambulatory Care Setting; OP, Outpatient; IP, Inpa-
tient; OR, Operating Room.

The Information Flow 13

enters the system. The manager needs to realize the significance of such information. We
give, as an example, obtaining confirmation of a patient’s correct address. The manager
should know that a correct address for a patient is vital to the smooth operation of the sys-
tem. An incorrect address will, for example, cause the billing to be rejected. Understand-
ing this connection between deficient data (e.g., a bad address) and the consequences (the
bill will be rejected by the payer and thus not be paid) illustrates the essence of good fi-
nancial management knowledge.

We can examine two examples of patient information flows. The first, shown in Figure
2-2, is a physician’s office flowsheet for address confirmation. Four different personnel are
involved, in addition to the patient. This physician has computed the cost of a bad address
as $12.30 to track down each address correction. He pays close attention to the handling of
this information because he knows there is a direct financial management consequence in
his operation.

The second example, shown in Figure 2-3, is a health system flowsheet for verification of
patient information. This flowsheet illustrates the process for a home care system. In this
case, the flow begins not with a receptionist, as in the physician office example, but with a
central database. This central database downloads the information and generates a sum-
mary report to be reviewed the next day. Appropriate verification is then made in a series of
steps, and any necessary corrections are made before the form goes to the billing depart-
ment. The object of the flow is the same in both examples: that is, the billing must have a

Initiates
Call

Mark
Superbill
if Change

Patient

Enter
Corrected
Address in
Computer

Copy
Insurance

Card

Instruct
Patient

to
Correct

Financial
Data

Check
Address

Insurance

Review
Patient

Records

Record
Message

Receive
Call

Phone Intake
Entry/Exit

Receptionist

Ask
Address
Change

Medical
Assistant

Ask
Address
Change

Doctor

Type
Chart
Label

Coder

Ask
Insurance
Change

Figure 2–2 Physician’s Office Flowsheet for Address Confirmation.

14 CHAPTER 2 What Does the Healthcare Manager Need to Know?

correct address to receive payment. But the flow is different within two different systems. A
manager must understand how the system works to understand the consequences—then
good financial management can prevail.

BASIC SYSTEM ELEMENTS

To understand financial management, it is essential to decipher the reports provided to the
manager. To comprehend these reports, it is helpful to understand certain basic system el-
ements that are used to create the information contained in the reports.

Chart of Accounts—The Map

The chart of accounts is a map. It outlines the elements of your company in an organized
manner. The chart of accounts maps out account titles with a method of numeric coding. It
is designed to compile financial data in a uniform manner that the user can decode.

The groupings of accounts in the chart of accounts should match the groupings of the
organization. In other words, the classification on the organization chart (as discussed in
the previous chapter) should be compatible with the groupings on the chart of accounts.
Thus, if there is a human resources department on your facility’s organization chart, and if

Central intake enters
demographics at time

referral received

Patient Accounts
Clerk generates

Patient Information
Summary next day

Data downloads to
CDB overnight

Information on Patient
Information Summary

verified by
care manager at next

visit

Correct information
written in by staff

Information
Correct?

Form is placed in
appropriate

care manager’s
mailbox

Form turned in to
Patient Accounts
Representative

Data in CDB’s central
intake updated

by Patient Accounts
Representative

Form placed in
billing folder

New labels generated
if necessary

YesNo

Figure 2–3 Health System Flowsheet for Verification of Patient Information.

expenses are grouped by department in your facility, then we would expect to find a human
resources grouping in the chart of accounts.

The manager who is working with financial data needs to be able to read and compre-
hend how the dollars are laid out and how they are gathered together, or assembled. This
assembly happens through the guidance of the chart of accounts. That is why we compare
it to a map.

Basic guidance for healthcare charts of accounts is set out in publications such as that of
Seawell’s Chart of Accounts for Hospitals.2 However, generic guides are just that—generic.
Every organization exhibits differences in its own chart of accounts that express the unique
aspects of its structure. We examine three examples to illustrate these differences. Remem-
ber, we are spending time on the chart of accounts because your comprehension of de-
tailed financial data may well depend on whether you can decipher your facility’s own chart
of accounts mapping in the information forwarded for your use.

The first format, shown in Exhibit 2-1, is a basic use, probably for a smaller organization.
The exhibit is in two horizontal segments, “Structure” and “Example.” There are three
parts to the account number. The first part is one digit and indicates the financial statement
element. Thus, our example shows “1,” which is for “Asset.” The second part is two digits
and is the primary subclassification. Our example shows “10,” which stands for “Current
Asset” in this case. The third and final part is also two digits and is the secondary subclassi-
fication. Our example shows “11,” which stands for “Petty Cash—Front Office” in this case.
On a report, this account number would probably appear as 1-10-11.

The second format, shown in Exhibit 2-2, is full use and would be for a large organiza-
tion. The exhibit is again in two horizontal segments, “Structure” and “Example,” and there

Basic System Elements 15

Exhibit 2–1 Chart of Accounts, Format 1

Structure
X XX XX
Financial Primary Secondary
Statement Subclassification Subclassification

Element

Example
1 10 11
Asset Current Petty Cash—

Asset Front Office
(Financial (Primary (Secondary
Statement Subclassification) Subclassification)

Element)

16 CHAPTER 2 What Does the Healthcare Manager Need to Know?

are now two line items appearing in the Example section. This full-use example has five
parts to the account number. The first part is two digits and indicates the entity designator
number. Thus, we conclude that there is more than one entity within this system. Our ex-
ample shows “10,” which stands for “Hospital A.” The second part is two digits and indicates
the fund designator number. Thus, we conclude that there is more than one fund within
this system. Our example shows “10,” which stands for “General Fund.”

The third part of Exhibit 2-2 is one digit and indicates the financial statement element.
Thus, the first line of our example shows “4,” which is for “Revenue,” and the second line of
our example shows “6,” which is for “Expense.” (The third part of this example is the first
part of the simpler example shown in Exhibit 2-1.) The fourth part is four digits and is the
primary subclassification. Our example shows 3125, which stands for “Lab—Microbiology.”
The number 3125 appears on both lines of this example, indicating that both the revenue
and the expense belong to Lab—Microbiology. (The fourth part of this example is the sec-
ond part of the simpler example shown in Exhibit 2-1. The simpler example used only two
digits for this part, but this full-use example uses four digits.) The fifth and final part is two
digits and is the secondary subclassification. Our example shows “03” on the first line, the
revenue line, which stands for “Payer: XYZ HMO” and indicates the source of the revenue.
On the second line, the expense line, our example shows “10,” which stands for “Clerical
Salaries.” Therefore, we understand that these are the clerical salaries belonging to Lab—

Exhibit 2–2 Chart of Accounts, Format 2

Structure
XX XX X XXXX XX
Entity Fund Financial Primary Secondary
Designator Designator Statement Subclassification Subclassification

Element

Example
10 10 4 3125 03
Hospital General Revenue Lab—Microbiology Payer: XYZ HMO
A Fund

10 10 6 3125 10
Hospital General Expense Lab—Microbiology Clerical Salaries
A Fund

(Entity (Fund (Financial (Primary (Secondary
Designator) Designator) Statement Subclassification) Subclassification)

Element)

Basic System Elements 17

Microbiology in Hospital A. (The fifth part of this example is the third and final part of the
simpler example shown in Exhibit 2-1.) On a report, these account numbers might appear
as 10-10-4-3125-03 and 10-10-6-3125-10. Another optional use that is easier to read at a
glance is 10104-3125-03 and 10106-3125-10.

Because every organization is unique and because the chart of accounts reflects that
uniqueness, the third format, shown in Exhibit 2-3, illustrates a customized use of the chart
of accounts. This example is adapted from a large hospital system. There are four parts to
its chart of accounts number. The first part is an entity designator and designates a com-
pany within the hospital system. The fund designator two-digit part, as traditionally used
(see Exhibit 2-2), is missing here. The financial statement element one-digit part, as tradi-
tionally used (see Exhibit 2-2), is also missing here. Instead, the second part of Exhibit 2-3
represents the primary classification, which is shown as an expense category (“Payroll”) in
the example line. The third part of Exhibit 2-3 is the secondary subclassification, repre-
senting a labor subaccount expense designation (“Regular per-Visit RN”). The fourth and
final part of Exhibit 2-3 is another subclassification that indicates the department within the
company (“Home Health”). On a report for this organization, therefore, the account num-
ber 21-7000-2200-7151 would indicate the home care services company’s payroll for regu-
lar per-visit registered nurses (RNs) in the home health department. Finally, remember that
time spent understanding your own facility’s chart of accounts will be time well spent.

Exhibit 2–3 Chart of Accounts, Format 3

Structure
XX XXXX XXXX XXXX
Company Expense Subaccount Department

Category

(Entity (Primary (Secondary (Additional
Designator) Classification) Subclassification) Subclassification)

Example
21 7000 2200 7151
Home Payroll Regular Home Health
Care per-Visit RN
Services

(Company) (Expense (Subaccount) (Department)
Category)

18 CHAPTER 2 What Does the Healthcare Manager Need to Know?

Books and Records—Capture Transactions

The books and records of the financial information system for the organization ser ve to
capture transactions. Figure 2-4 illustrates the relationship of the books and records to
each other. As a single transaction occurs, the process begins. The individual transaction is
recorded in the appropriate subsidiar y journal. Similar such transactions are then
grouped and balanced within the subsidiary journal. At periodic intervals, the groups of
transactions are gathered, summarized, and entered in the general ledger. Within the gen-
eral ledger, the transaction groups are reviewed and adjusted. After such review and ad-
justment, the transactions for the period within the general ledger are balanced. A
document known as the trial balance is used for this purpose. The final step in the process
is to create statements that reflect the transactions for the period. The trial balance is used
to produce the statements.

All transactions for the period reside in the general ledger. The subsidiary journals are so
named because they are “subsidiary” to the general ledger: in other words, they serve to
support the general ledger. Figure 2-5 illustrates this concept. Another way to think of the
subsidiary journals is to picture them as feeding the general ledger. The important point
here is to understand the source and the flow of information as it is recorded.

Individual Transaction
[begins process]

Individual Transaction
Is Recorded

Similar Transactions
Are Grouped and Balanced

Groups of Transactions
Are Gathered and Summarized

Summarized Groups of Transactions
Are Reviewed and Adjusted

Adjusted and Reviewed Transactions
for Period Are Balanced

Statements Reflecting Transactions Are Created
[ends process]

Creates
Original Record

Into
Subsidiary Journals

Within the
Subsidiary Journals

Into the
General Ledger

Within the
General Ledger

Trial Balance Is Produced
from the General Ledger
for This Purpose

Statements Are Produced
from the Trial Balance

Figure 2–4 The Progress of a Transaction.
Source: Courtesy of Resource Group, Ltd., Dallas, Texas.

Reports—The Product

Reports are more fully treated in a subsequent chapter of this text (see Chapter 10). It is
sufficient at this point to recognize that reports are the final product of a process that com-
mences with an original transaction.

THE ANNUAL MANAGEMENT CYCLE

The annual management cycle affects the type and status of information that the manager
is expected to use. Some operating information is “raw”—that is, unadjusted. When the
same information has passed further through the system and has been verified, adjusted,
and balanced, it will usually vary from the initial raw data. These differences are a part of
the process just described.

Daily and Weekly Operating Reports

The daily and weekly operating reports generally contain raw data, as discussed in the pre-
ceding paragraph. The purpose of such daily and weekly reports is to provide immediate
operating information to use for day-by-day management purposes.

Quarterly Reports and Statistics

The quarterly reports and statistics generally have been verified, adjusted, and balanced.
They are called interim reports because they have been generated some time during the
reporting period of the organization and not at the end of that period. Managers often use
quarterly reports as milestones. A common milestone is the quarterly budget review.

The Annual Management Cycle 19

Payroll

Journal

Accounts
Receivable

Journal

Cash
Receipts
Journal

Cash
Disbursements

Journal

Accounts
Payable
Journal

GENERAL LEDGER

SUBSIDIARY JOURNALS

THE BOOKS

Figure 2–5 Recording Information: Relationship of Subsidiary Journals to the General Ledger.
Source: Courtesy of Resource Group, Ltd., Dallas, Texas.

20 CHAPTER 2 What Does the Healthcare Manager Need to Know?

Annual Year-End Reports

Most organizations have a 12-month reporting period known as a fiscal year. A fiscal year,
therefore, covers a period from the first day of a particular month (e.g., January 1) through
the last day of a month that is one year, or 12 months, in the future (e.g., December 31). If
we see a heading that reads, “For the year ended June 30,” we know that the fiscal year
began on July 1 of the previous year. Anything less than a full 12-month year is called a “stub
period” and is fully spelled out in the heading. If, therefore, a company is reporting for a
three-month stub period ending on December 31, the heading on the report will read, “For
the three-month period ended December 31.” An alternative treatment uses a heading that
reads, “For the period October 1 to December 31.”

Annual year-end reports cover the full 12-month reporting period or the fiscal year. Such
annual year-end reports are not primarily intended for managers’ use. Their primary pur-
pose is for reporting the operations of the organization for the period to outsiders, or third
parties.

Annual year-end reports represent the closing out of the information system for a spe-
cific reporting period. The recording and reporting of operations will now begin a new
cycle with a new year.

COMMUNICATING FINANCIAL INFORMATION TO OTHERS

The ability to communicate financial information effectively to others is a valuable skill. It
is important to

• Create a report as your method of communication.
• Use accepted terminology.
• Use standard formats that are accepted in the accounting profession.
• Begin with an executive summary.
• Organize the body of the report in a logical flow.
• Place extensive detail into an appendix.

The rest of this book will help you learn how to create such a report. Our book will also
sharpen your communication skills by helping you better understand how heathcare fi-
nance works.

INFORMATION CHECKPOINT

What Is Needed? An explanation of how the information flow works in your
unit.

Where Is It Found? Probably with the information system staff; perhaps in the
administrative offices.

How Is It Used? Study the flow and relate it to the paperwork that you
handle.

KEY TERMS

Accounting System
Chart of Accounts
General Ledger
Information System
Original Records
Reporting System
Subsidiary Journals
Trial Balance

DISCUSSION QUESTIONS

1. Have you ever been informed of the information flow in your unit or division?
2. If so, did you receive the information in a formal seminar or in an informal manner,

one-on-one with another individual? Do you think this was the best way? Why?
3. Do you know about the chart of accounts in your organization as it pertains to infor-

mation you receive?
4. If so, is it similar to one of the three formats illustrated in this chapter? If not, how is

it different?
5. Do you work with daily or weekly operating reports? With quarterly reports and sta-

tistics?
6. If so, do these reports give you useful information? How do you think they could be

improved?

Discussion Questions 21

Source: NASA

5.1 Rules for Exponents
5.2 Addition and Subtrac-

tion of Polynomials
5.3 Multiplication of

Polynomials
5.4 Special Products
5.5 Integer Exponents and

the Quotient Rule
5.6 Division of Polynomials

Digital images were first sent between New York and London by cable in the
early 1920s. Unfortunately, the transmission time was 3 hours and the quality was poor.
Digital photography was developed further by NASA in the 1960s because ordinary pic-
tures were subject to interference when transmitted through space. Today, digital pic-
tures remain crystal clear even if they travel millions of miles. The following digital
picture shows the planet Mars.

Whether they are taken with a webcam, with a smartphone, or by the Mars rover,
digital images comprise tiny units called pixels, which are represented by numbers. As a
result, mathematics plays an important role in digital images. In this chapter we illustrate
some of the ways mathematics is used to describe digital pictures (see Example 4 and
Exercise 80 in Section 5.4). We also discuss how mathematics is used to model things
such as heart rate, computer sales, motion of the planets, and interest on money.

5 Polynomials and Exponents

If you want to do
something, do it!

—PLAUTUS

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298 CHAPTER 5 POLYNOMIALS AND EXPONENTS

When evaluating expressions, evaluate exponents before performing addition, subtraction,
multiplication, division, or negation.

5.1 Rules for Exponents
Review of Bases and Exponents ● Zero Exponents ● The Product Rule ● Power Rules

A LOOK INTO MATH N Electronic devices such as tablet computers and smartphones store information as bits.
A bit is either a 0 or a 1, and a string of 8 bits is called a byte. In the 1970s, IBM devel-
oped punch cards made out of paper that could hold up to 120 bits of information. Today,
many computer hard drives can hold more than 1 terabyte of information; that’s more than
8,000,000,000,000 bits! In mathematics, we often use exponents to express such large
numbers. In this section, we discuss the rules for exponents.

Review of Bases and Exponents
The expression 53 is an exponential expression with base 5 and exponent 3. Its value is

5 # 5 # 5 = 125.
In general, b n is an exponential expression with base b and exponent n. If n is a natural
number, it indicates the number of times the base b is to be multiplied with itself.

Exponent
T

b n = b # b # b # g #

Base c

n times

STUDY TI

Exponents occur throughout
mathematics. Because expo-
nents are so important, this
section is essential for your
success in mathematics. It
takes practice, so set aside
some extra time.

EVALUATING EXPRESSIONS

When evaluating expressions, use the following order of operations.

1. Evaluate exponents.
2. Perform negation.
3. Do multiplication and division from left to right.
4. Do addition and subtraction from left to right.

EXAMPLE 1 Evaluating exponential expressions

Evaluate each expression.

(a)

1 +

4
(b) 3a

b2 (c) – 24 (d) ( – 2)4

Solution

(a) Evaluate the exponent first.

4 factors

1 +

4
= 1 +

2 # 2 # 2 # 2
4

= 1 +
1

4
= 1 + 4 = 5

∂

2 factors

(b) 3a1
3
b2 = 3a1

3
#

3
b = 3 # 1

9
=

3
9
=
1
3

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RULES FOR EXPONENTS

4 factors

– 24 = – (2 # 2 # 2 # 2) = – 16.∂

4 factors

(d) (�2)4 = (�2)(�2)(�2)(�2) =

Now Try Exercises 9, 11, 15, 17

NOTE: Parts (c) and (d) of Example 1 appear to be very similar. However, the negation
sign is inside the parentheses in part (d), which means that the base for the exponential
expression is – 2. In part (c), no parentheses are used, indicating that the base of the expo-
nential expression is 2.

READING CHECK

• Explain how to tell the difference between a negative number raised to a power and the
opposite of a positive number raised to a power.

TECHNOLOGY NOTE

Evaluating Exponents
Exponents can often be evaluated on calculators by using the ^ key. The four expressions from
Example 1 are evaluated with a calculator and the results are shown in the following two figures.
When evaluating the last two expressions on your calculator, remember to use the negation key
rather than the subtraction key.

1�2^4/4
5

3(1/3) 2�Frac
1/3

�2^4
�16

(�2)^4
16

CALCULATOR HELP
To evaluate exponents, see Appendix
A (page AP-1).

Zero Exponents
So far we have discussed natural number exponents. What if an exponent is 0? What does
20 equal? To answer these questions, consider Table 5.1, which shows values for decreas-
ing powers of 2. Note that each time the power of 2 decreases by 1, the resulting value is
divided by 2. For this pattern to continue, we need to define 20 to be 1 because dividing 2
by 2 results in 1.

This discussion suggests that 20 = 1, and is generalized as follows.

TABLE 5.1 Powers of 2

Power of 2 Value

23 8

22 4

21 2

20 ?
ZERO EXPONENT

For any nonzero real number b,

b 0 = 1.

The expression 00 is undefined.

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EXAMPLE 2 Evaluating zero exponents

Evaluate each expression. Assume that all variables represent nonzero numbers.

(a) 70 (b) 3a4
9
b0 (c) a

2y5

Solution
(a) 70 = 1
(b) 314920 = 3(1) = 3. (Note that the exponent 0 does not apply to 3.)
(c) All variables are nonzero, so the expression inside the parentheses is also nonzero.

Thus 1x 2y53z 20 = 1.
Now Try Exercises 13, 41, 67

THE PRODUCT RULE

For any real number a and natural numbers m and n,

am # an = am + n.

READING CHECK

• State the product rule in
your own words.

The expression 43 # 42 has 3 + 2 = 5 factors of 4, so the result is 43 + 2 = 45. To multiply
exponential expressions with the same base, add exponents and keep the base.

5 factors

The Product Rule
We can use a special rule to calculate products of exponential expressions provided their
bases are the same. For example,

43 # 42 = (4 # 4 # 4) # (4 # 4) = 45.v

NOTE: The product 24 # 35 cannot be simplified by using the product rule because the
exponential expressions have different bases: 2 and 3.

EXAMPLE 3 Using the product rule

Multiply and simplify.
(a) 23 # 22 (b) x 4x 5 (c) 2 x 2 # 5x 6 (d) x 3(2 x + 3x 2)
Solution
(a) 23 # 22 = 23 + 2 = 25 = 32
(b) x 4x 5 = x 4 + 5 = x 9
(c) Begin by applying the commutative property of multiplication to write the product in a

more convenient order.

2 x 2 # 5x 6 = 2 # 5 # x 2 # x 6 = 10x 2 + 6 = 10x 8
(d) To simplify this expression, first apply the distributive property.

3�1 T T
3�2

x 3(2 x + 3x 2) = x 3 # 2 x + x 3 # 3x 2 = 2 x 4 + 3

x 5

c Exponent is 1.

Now Try Exercises 21, 23, 27, 71

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3015.1 RULES FOR EXPONENTS

EXAMPLE 4 Applying the product rule

Multiply and simplify.
(a) x # x 3 (b) (a + b)(a + b)4

Solution
(a) Begin by writing x as x 1. Then x 1 # x 3 = x 1 + 3 = x 4.
(b) First write (a + b) as (a + b)1. Then

(a � b)1 # (a � b)4 = (a � b)1 + 4 = (a � b)5.
Now Try Exercises 19, 63

Power Rules
How should (43)2 be evaluated? To answer this question, consider how the product rule can
be used in evaluating

(43)2 = 43 # 43 = 43 + 3 = 46.

Product rule

5 3 + 3 = 3 # 2
Similarly,

5 + 5 + 5 = 5
#

$%&
(a5)3 = a5 # a5 # a5 = a5 + 5 + 5 = a15.

Product rule

This discussion suggests that to raise a power to a power, we multiply the exponents.

RAISING A POWER TO A POWER

For any real number a and natural numbers m and n,

(am)n = amn.

EXAMPLE 5 Raising a power to a power

Simplify each expression.

(a) (32)4 (b) (a3)2

Solution
(a) (32)4 = 32

# 4 = 38 (b) (a3)2 = a3 # 2 = a6

Now Try Exercises 31, 33

To decide how to simplify the expression (2 x)3, consider

(2 x)3 = 2 x # 2 x # 2 x = (2 # 2 # 2) # (x # x # x) = 23x 3.
3 factors 3 factors 3 factors

∂ ¶ ¶

To raise a product to a power, we raise each factor to the power.

NOTE: If an exponent does not appear in an expression, it is assumed to be 1. For example,
x can be written as x 1 and (x + y) can be written as (x + y)1.

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302 CHAPTER 5 POLYNOMIALS AND EXPONENTS

READING CHECK

• State the rule for raising a
product to a power in your
own words.

RAISING A PRODUCT TO A POWER

For any real numbers a and b and natural number n,

(ab)n = anb n.

EXAMPLE 6 Raising a product to a power

Simplify each expression.
(a) (3z)2 (b) ( – 2 x 2)3 (c) 4(x 2y3)5 (d) ( – 22a5)3

Solution
(a) (3z)2 = 32z2 = 9z2
(b) ( – 2 x 2)3 = ( – 2)3(x 2)3 = – 8

x 6

Now Try Exercises 37, 39, 43, 45

The following equation illustrates another power rule.

3
b4 = 2

3
# 2

3
# 2
3
# 2

3
=

2 # 2 # 2 # 2
3 # 3 # 3 # 3 =

4 factors

To raise a quotient to a power, raise both the numerator and the denominator to the power.

RAISING A QUOTIENT TO A POWER

For any real numbers a and b and natural number n,

b
bn =

b n

. b � 0

EXAMPLE 7 Raising a quotient to a power

Simplify each expression.

(a) a2
3
b3 (b) aa

b
b9 (c) a

a + b

Solution

(a) a2
3
b3 =

33
=

(b) aa

b
b9 =

b 9

(c) Because the numerator is an expression with more than one term, we must place paren-
theses around it before raising it to the power 2.

aa + b
5
b2 =

(a + b)2

52
=

(a + b)2
25

Now Try Exercises 51, 53, 55

3035.1 RULES FOR EXPONENTS

The five rules for exponents discussed in this section are summarized as follows.

MAKING CONNECTIONS

Raising a Sum or Difference to a Power

Although there are power rules for products and quotients, there are not similar rules for sums
and differences. In general, (a + b)n 3 an + b n and (a – b)n 3 an – b n. For example,
(3 + 4)2 = 72 = 49 but 32 + 42 = 9 + 16 = 25. Similarly, (4 – 1)3 = 33 = 27 but
43 – 13 = 64 – 1 = 63.

RULES FOR EXPONENTS

The following rules hold for real numbers a and b, and natural numbers m and n.

Description Rule Example

Zero Exponent b 0 = 1, for b � 0 ( – 13)0 = 1

The Product Rule am # an = am + n 54 # 53 = 54 + 3 =

Power to a Power (am)n = am # n ( y2)5 = y2 # 5 = y10

Product to a Power (ab)n = anb n ( pq)7 = p7q 7

Quotient to a Power aa
b
bn = an

b n
, for b � 0 a

b3 =

x 3

, for y � 0

Simplification of some expressions may require the application of more than one rule
of exponents. This is demonstrated in the next example.

EXAMPLE 8 Combining rules for exponents

Simplify each expression.

(a) (2a)2(3a)3 (b) aa2b 3
c
b4 (c) (2 x 3y)2( – 4 x 2y3)3

Solution
(a) (2a)2(3a)3 = 22a2 # 33a3 Raising a product to a power
= 4 # 27 # a2 # a3 Evaluate powers; commutative property
= 108a5 Product rule

(b) aa2b 3
c
b4 = (a2)4(b 3)4

c 4
Raising a quotient to a power; raising

a product to a power

=
a8b

c 4
Raising a power to a power

(c) (2 x 3y)2( – 4 x 2y3)3 = 22(x 3)2y2( – 4)3(x 2)3( y3)3 Raising a product to a power
= 4 x 6y2( – 64)x 6y9 Raising a power to a power
= 4( – 64)x 6x 6y2y9 Commutative property
= – 256 x 12y11 Product rule

Now Try Exercises 47, 49, 61

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304 CHAPTER 5 POLYNOMIALS AND EXPONENTS

N REAL-WORLD CONNECTION Exponents occur frequently in calculations involving yearly
percent increases, such as the increase in property value illustrated in the next example.

EXAMPLE 9 Calculating growth in property value

If a parcel of property increases in value by about 11% each year for 20 years, then its value
will double three times.
(a) Write an exponential expression that represents “doubling three times.”
(b) If the property is initially worth $25,000, how much will it be worth after it doubles

3 times?

Solution
(a) Doubling three times is represented by 23.
(b) 23(25,000) = 8(25,000) = $200,000

Now Try Exercise 85

5.1 Putting It All Together

Bases and Exponents In the expression b n, b is the base
and n is the exponent. If n is a natural
number, then

b n = b # b # g # b.
5
n times

23 has base 2 and exponent 3.

91 = 9,
32 = 3 # 3 = 9,
43 = 4 # 4 # 4 = 64, and
– 62 = – (6 # 6) = – 36

Zero Exponents For any nonzero number b, b 0 = 1. 50 = 1, x 0 = 1, and (xy3)0 = 1

The Product Rule

For any real number a and natural
numbers m and n,

am # an = am + n.

24 # 23 = 24 + 3 = 27,
x # x 2 # x 6 = x 1 + 2 + 6 = x 9, and
(x + 1) # (x + 1)2 = (x + 1)3

CONCEPT EXPLANATION EXAMPLES

Raising a Power to a
Power

For any real number a and natural
numbers m and n,
(am)n = amn.

(24)2 = 24
# 2 = 28,

(x 2)5 =

x 2

# 5 = x 10, and

(a4)3 = a4
# 3 = a12

Raising a Product to a
Power

For any real numbers a and b and
natural number n,

(ab)n = anb n.

(3x)3 = 33x 3 = 27x 3,
(x 2y)4 = (x 2)4y4 = x 8y4, and
( – xy)6 = ( – x)6y6 = x 6

Raising a Quotient to a
Power

For any real numbers a and b and
natural number n,
aa
b
bn = an
b n
. b � 0

ax
y
b5 = x 5

and

aa2b
d 3
b4 = (a2)4b 4

(d 3)4
=

a8b 4

d 12

3055.1 RULES FOR EXPONENTS

5.1 Exercises

CONCEPTS AND VOCABULARY

1. In the expression b n, b is the _____ and n is the _____.

2. The expression b 0 = for any nonzero number b.

3. am # an = 4. (am)n =
5. (ab)n = 6. aa

b
bn =

PROPERTIES OF EXPONENTS

Exercises 7–18: Evaluate the expression.

7. 82 8. 43

9. ( – 2)3 10. ( – 3)4

11. – 23 12. – 34

13. 60 14. ( – 0.5)0

15. 3 +

2
16. 6 – a – 4

2
b2

17. 4a1
2
b3 18. 16a1

4
b2

Exercises 19–74: Simplify the expression. Assume that all
variables represent nonzero numbers.

19. 3 # 32 20. 53 # 53
21. 42 # 46 22. 104 #

103

23. x 3x 6 24. a5a2

25. x 2x 2x 2 26. y7y3y0

27. 4 x 2 # 5x 5 28. – 2y6 # 5y2
29. 3( – xy3)(x 2y) 30. (a2b 3)( – ab 2)

31. (23)2 32. (103)4

33. (n3)4 34. (z7)3

35. x(x 3)2 36. (z3)2(5z5)

37. ( – 7b)2 38. ( – 4z)3

39. (ab)3 40. (xy)8

41. (2 x 2)0 42. (3a2)4

43. ( – 4b 2)3 44. ( – 3r 4t 3)2

45. (x 2y3)7 46. (rt 2)5

47. ( y3)2(x 4y)3 48. (ab 3)2(ab)3

49. (a2b)2(a2b 2)3 50. (x 3y)(x 2y4)2

51. a1
3
b3 52.

2
b2

53. aa
b
b5 54. ax

2
b4

55. ax – y
3
b3 56.

a 4

x + y
b2

57. a 5
a + b

b2 58. aa – b
2
b3

59. a2 x
5
b3 60. a

2
b4

61. a3x 2
5y4
b3 62. aa2b 3

3
b5

63. (x + y)(x + y)3 64. (a – b)2(a – b)

65. (a + b)2(a + b)3 66. (x – y)5(x – y)4

67. 6(x 4y6)0 68. axy
z2
b0

69. a(a2 + 2b 2) 70. x 3(3x – 5y4)

71. 3a3(4a2 + 2b) 72. 2 x 2(5 – 4y3)

73. (r + t)(rt)

74. (x – y)(x 2y3)

75. Thinking Generally Students sometimes mistakenly
apply the “rule” am # b n � (ab)m + n. In general, this
equation is not true. Find values for a, b, m, and n with
a � b and m � n that will make this equation true.

76. Thinking Generally Students sometimes mistakenly

apply the “rule” (a + b)n � an + b n. In general,
this equation is not true. Find values for a, b, and n
with a � b that will make this equation true.

APPLICATIONS

Exercises 77–80: Write a simplified expression for the
area of the given figure.

77.
2

5×2

78.

2ab

79.

3×2

80.

7y3

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306 CHAPTER 5 POLYNOMIALS AND EXPONENTS

Exercises 81 and 82: Write a simplified expression for the
volume of the given figure.

81.

82.

3a2

3a2
3a2

83. Compound Interest If P dollars are deposited in an
account that pays 5% annual interest, then the amount
of money in the account after 3 years is P(1 + 0.05)3.
Find the amount when P = $1000.

84. Compound Interest If P dollars are deposited in an
account that pays 9% annual interest, then the amount
of money in the account after 4 years is P(1 + 0.09)4.
Find the amount when P = $500.

85. Investment Growth If an investment increases in
value by about 10% each year for 22 years, then its
value will triple two times.

(a) Write an exponential expression that represents
“tripling two times.”

(b) If the investment has an initial value of $8000,
how much will it be worth if it triples two times?

86. Stock Value If a stock decreases in value by about
23% each year for 9 years, then its value will be
halved three times.
(a) Write an exponential expression that represents

“halved three times.”
(b) If the stock is initially worth $88 per share, how

much will it be worth if it is halved three times?

WRITING ABOUT MATHEMATICS

87. Are the expressions (4 x)2 and 4 x 2 equal in value?
Explain your answer.

88. Are the expressions 33 # 23 and 66 equal in value?
Explain your answer.

5.2 Addition and Subtraction of Polynomials
Monomials and Polynomials ● Addition of Polynomials ● Subtraction of Polynomials ●
Evaluating Polynomial Expressions

A LOOK INTO MATH N If you have ever exercised strenuously and then taken your pulse immediately afterward,
you may have discovered that your pulse slowed quickly at first and then gradually leveled
off. A typical scatterplot of this phenomenon is shown in Figure 5.1(a). These data
points cannot be modeled accurately with a line, so a new expression, called a polynomial,
is needed to model them. A graph of a polynomial that models these data is shown in
Figure 5.1(b) and discussed in Exercise 71. (Source: V. Thomas, Science and Sport.)

NEW VOCABULARY

n Monomia

n Degree of a monomial
n Coefficient of a monomial
n Polynomial
n Polynomial in one variable
n Binomial
n Trinomial
n Degree of a polynomial
n Like terms

Figure 5.1 Heart Rate After Exercising

10 2 3 4 5 6 7

100

125

150

175

200

t
P

Time (minute

(a)

P
ul

se
(

be
at

s
pe

in
ut

Heart Rate

10 2 3 4 5 6 7
25
50
75
100
125
150
175
200
t
P
Time (minutes)

(b)

P
ul
se
(
be
at
s
pe
r
m
in
ut
e)

Polynomial Model

3075.2 ADDITION AND

SUBTRACTION OF POLYNOMIALS

Monomials and Polynomials
A monomial is a number, a variable, or a product of numbers and variables raised to natural
number powers. Examples of monomials include

– 3, xy2, 5a2, – z 3, and –
1

2
xy3.

A monomial may contain more than one variable, but monomials do not contain division by
variables. For example, the expression 3z is not a monomial. If an expression contains addi-
tion or subtraction signs, it is not a monomial.

The degree of a monomial is the sum of the exponents of the variables. If the mono-
mial has only one variable, its degree is the exponent of that variable. Remember, when
a variable does not have a written exponent, the exponent is implied to be 1. A nonzero
number has degree 0, and the number 0 has undefined degree. The number in a monomial
is called the coefficient of the monomial. Table 5.2 contains the degree and coefficient of
several monomials.

READING CHECK

• How do you determine the
degree of a monomial?

A polynomial is a monomial or the sum of two or more monomials. Each monomial is
called a term of the polynomial. Addition or subtraction signs separate terms. The expres-
sion 2 x 2 – 3x + 5 is a polynomial in one variable with three terms. Examples of polyno-
mials in one variable include

– 2 x, 3x + 1, 4y2 – y + 7, and x 5 – 3x 3 + x – 7.

These polynomials have 1, 2, 3, and 4 terms, respectively. A polynomial with two terms is
called a binomial, and a polynomial with three terms is called a trinomial.

A polynomial can have more than one variable, as in

x 2y2, 2 xy2 + 5x 2y – 1, and a2 + 2ab + b 2.

Note that all variables in a polynomial are raised to natural number powers. The degree of a
polynomial is the degree of the term (or monomial) with greatest degree.

READING CHECK

• How do you determine the
degree of a polyomial?

TABLE 5.2 Properties of Monomials

Monomial – 5 6a3b – xy 7y3

Degree 0 4 2 3

Coe fficient – 5 6 – 1 7

EXAMPLE 1 Identifying properties of polynomials

Determine whether the expression is a polynomial. If it is, state how many terms and vari-
ables the polynomial contains and give its degree.

(a) 7x 2 – 3x + 1 (b) 5x 3 – 3x 2y3 + xy2 – 2y3 (c) 4 x 2 +
5

x + 1

Solution
(a) The expression 7x 2 – 3x + 1 is a polynomial with three terms and one variable. The

first term 7x 2 has degree 2 because the exponent on the variable is 2. The second term
– 3x has degree 1 because the exponent on the variable is implied to be 1. The third
term 1 has degree 0 because it is a nonzero number. The term with greatest degree is
7x 2, so the polynomial has degree 2.

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308 CHAPTER 5 POLYNOMIALS AND EXPONENTS

(b) The expression 5x 3 – 3x 2y3 + xy2 – 2y3 is a polynomial with four terms and two
variables, x and y. The first term has degree 3 because the exponent on the variable is 3.
The second term has degree 5 because the sum of the exponents on the variables x and
y is 5. Likewise, the third term has degree 3 and the fourth term has degree 3. The term
with greatest degree is – 3x 2y3, so the polynomial has degree 2 + 3 = 5.

Now Try Exercises 21, 23, 27

Figure 5.2 Adding lw + lw

l lw

+ l lw

= l 2lw

If two monomials contain the same variables raised to the same powers, we call
them like terms. We can add or subtract (combine) like terms but cannot combine unlike
terms. The terms lw and 2lw are like terms and can be combined geometrically, as shown
in Figure 5.3. If we joined one of the small rectangles with area lw and a larger rect-
angle with area 2 lw, then the total area is 3 lw.

The distributive property justifies combining like terms.

1lw + 2lw = (1 + 2)lw = 3lw

The rectangles shown in Figure 5.4 have areas of ab and xy. Together, their area is the sum,
ab + xy. However, because these monomials are unlike terms, they cannot be combined
into one term.

Figure 5.3 Adding lw + 2lw

l lw 3lw

+ l 2lw = l

2w 3w

STUDY TIP

Do you want to know what
material will be covered on
your next exam? Often, the
best place to look is on previ-
ously completed assignments
and quizzes. If a topic is not
discussed in class, is not
found on the syllabus, and is
not part of your assignments,
then your time may be better
spent studying other topics.

Figure 5.4 Unlike terms: ab + xy

a ab xy

b
ab
b

+ x

xy x

= a

Addition of Polynomials
Suppose that we have 2 identical rectangles with length l and width w, as illustrated in
Figure 5.2. Then the area of one rectangle is lw and the total area is

lw + lw.

This area is equivalent to 2 times lw, which can be expressed as 2lw, or

lw + lw = 2lw.

3095.2 ADDITION AND SUBTRACTION OF POLYNOMIALS

EXAMPLE 2 Adding like terms

State whether each pair of expressions contains like terms or unlike terms. If they are like
terms, add them.

(a) 5x 2, – x 2 (b) 7a2b, 10ab 2 (c) 4rt 2,
1

2
rt 2

Solution
(a) The terms 5x 2 and – x 2 have the same variable raised to the same power, so they are like

terms. To add like terms, add their coefficients. Note that the coefficient of – x 2 is – 1.

5x 2 + ( – x 2) = (5 + ( – 1))x 2 Distributive property
= 4 x 2 Add.

(b) The terms 7a2b and 10 ab 2 have the same variables, but these variables are not raised to
the same powers. They are unlike terms and cannot be added.

like terms. We add them as follows.

4rt 2 +
1
2

rt 2 = a4 + 1
2
brt 2 Distributive property

=
9

2
rt 2 Add.

Now Try Exercises 29, 31, 33

EXAMPLE 3 Adding polynomials

Add by combining like terms.
(a) (3x + 4) + ( – 4 x + 2)
(b) ( y2 – 2y + 1) + (3y2 + y + 11)

Solution
(a) (3x + 4) + (�4 x + 2) = 3x + (�4 x) + 4 + 2
= (3 – 4)x + (4 + 2)
= –

x + 6

(b) ( y2 – 2y + 1) + (3y2 + y + 11) = y2 + 3y2 – 2y + y + 1 + 11
= (1 + 3)y2 + ( – 2 + 1)y + (1 + 11)
= 4y2 – y + 12

Now Try Exercises 37, 39

Recall that the commutative and associative properties of addition allow us to rearrange
a sum in any order. For example, if we write each subtraction in 2 x – 5 – 4 x + 10 as
addition of the opposite, we have

2 x � 5 � 4 x + 10 = 2 x + (�5) + (�4 x) + 10,

and the terms can be rearranged as

2 x + (�4 x) + (�5) + 10 = 2 x � 4 x � 5 + 10.

If we pay attention to the sign in front of each term in a polynomial, the like terms can be
combined without rearranging the terms, as demonstrated in the next example.

To add two polynomials, combine like terms, as illustrated in the next example.

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310 CHAPTER 5 POLYNOMIALS AND EXPONENTS

EXAMPLE 4 Adding polynomials

Add (x 3 – 3x 2 + 7x – 4) + (4 x 3 – 5x + 9) by combining like terms.

Solution
Remove parentheses and identify like terms with their signs as shown.

x 3 � 3x 2 � 7x � 4 � 4 x 3 � 5x � 9

When like terms (of the same color) are added, the resulting sum is

5x 3 � 3x 2 � 2 x � 5.

Now Try Exercise 41

Polynomials can also be added vertically, as demonstrated in the next example.

EXAMPLE 5 Adding polynomials vertically

Simplify (3x 2 – 3x + 5) + ( – x 2 + x – 6).

Solution
Write the polynomials in a vertical format and then add each column of like terms.

3x 2 – 3

x + 5

– x 2 + x – 6
2 x 2 – 2 x – 1 Add like terms in each column.

Regardless of the method used, the same answer should be obtained. However, adding ver-
tically requires that like terms be placed in the same column.

Now Try Exercise 47

Subtraction of Polynomials
To subtract one integer from another, add the first integer and the additive inverse or oppo-
site of the second integer. For example, 3 – 5 is evaluated as follows.

3 – 5 = 3 + ( – 5) Add the opposite.
= – 2 Simplify.

Similarly, to subtract one polynomial from another, add the first polynomial and the
opposite of the second polynomial. To find the opposite of a polynomial, simply negate
each term. Table 5.3 lists some polynomials and their opposites.

READING CHECK

• How do you subtract one
polynomial from another?

CRITICAL THINKING

What is the result when a
polynomial and its opposite
are added?

TABLE 5.3 Opposites of Polynomials

Polynomial Opposite

2 x – 4 – 2

x + 4

– x 2 – 2 x + 9 x 2 + 2 x – 9

6 x 3 – 12 – 6 x 3 + 12

– 3x 4 – 2 x 2 – 8 x + 3 3x 4 + 2 x 2 + 8 x

– 3

3115.2 ADDITION AND SUBTRACTION OF POLYNOMIALS

EXAMPLE 6 Subtracting polynomials

Simplify each expression.
(a) (3x – 4) – (5x + 1)
(b) (5x 2 + 2 x – 3) – (6 x 2 – 7x + 9)
(c) (6 x 3 + x 2) – ( – 3x 3 – 9)

Solution
(a) To subtract (5x + 1) from (3x – 4), we add the opposite of (5x + 1), or ( – 5x – 1).

(3x – 4) – (5x + 1) = (3x – 4) + (�5x – 1)
= (3 � 5)x + (�4 � 1)
= �2 x � 5

(b) The opposite of (6 x 2 – 7x + 9) is ( – 6 x 2 + 7x – 9).

(5x 2 + 2 x – 3) – (6 x 2 – 7x + 9) = (5x 2 + 2 x – 3) + ( – 6 x 2 + 7x – 9)
= (5 – 6)x 2 + (2 + 7)x + ( – 3 – 9)
= – x 2 + 9x – 12

(6 x 3 + x 2) – ( – 3x 3 – 9) = (6 x 3 + x 2) + (3x 3 + 9)
= (6 + 3)x 3 + x 2 + 9
= 9x 3 + x 2 + 9

Now Try Exercises 57, 59, 61

NOTE: Some students prefer to subtract one polynomial from another by noting that a sub-
traction sign in front of parentheses changes the signs of all of the terms within the paren-
theses. For example, part (a) of the previous example could be worked as follows.

(3x – 4) – (5x + 1) = 3x – 4 – 5x – 1
= (3 – 5)x + ( – 4 – 1)
= – 2 x – 5

EXAMPLE 7 Subtracting polynomials vertically

Simplify (5x 2 – 2 x + 7) – ( – 3x 2 + 3).

Solution
To subtract one polynomial from another vertically, simply add the first polynomial and the
opposite of the second polynomial. No x-term occurs in the second polynomial, so insert 0x.

5x 2 – 2 x + 7
3x 2 + 0x – 3 The opposite of – 3x 2 + 3 is 3x 2 – 3 or 3x 2 + 0x – 3.
8 x 2 – 2 x + 4 Add like terms in each column.

Now Try Exercise 69

Evaluating Polynomial Expressions
Frequently, monomials and polynomials represent formulas that may be evaluated. We
illustrate such applications in the next two examples.

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312 CHAPTER 5 POLYNOMIALS AND EXPONENTS

EXAMPLE 8 Writing and evaluating a monomial

Write the monomial that represents the volume of the box having a square bottom, as shown
in Figure 5.5. Find the volume of the box if x = 3 feet and y = 2 feet.

Solution
The volume of a box is found by multiplying the length, width, and height together. Because
the length and width are both x and the height is y, the monomial xxy represents the volume
of the box. This can be written x 2y. To calculate the volume, let x = 3 and y = 2 in the
monomial x 2y.

x 2y = 32 # 2 = 9 # 2 = 18 cubic feet
Now Try Exercise 73

y
x
x

Figure 5.5

EXAMPLE 9 Modeling sales of personal computers

Worldwide sales of personal computers have increased dramatically in recent years, as
illustrated in Figure 5.6. The polynomial

0.7868 x 2 + 16.72 x + 122.58

approximates the number of computers sold in millions, where x = 0 corresponds to 2000,
x = 1 to 2001, and so on. Estimate the number of personal computers sold in 2008 by
using both the graph and the polynomial. (Source: International Data Corporation.)

Solution
From the graph shown in Figure 5.7, it appears that personal computer sales were slightly
more than 300 million, or about 310 million, in 2008.

Figure 5.6

4 6 8 10

100
0
200

300

400

x
y

Year (0 ↔ 2000)

C
om

pu
te

r
S

al
es

(
m

il
li

on
s)

Worldwide Computer Sales

Figure 5.7

4 6 8 10
100
0
200
300
400
x
y
Year (0 ↔ 2000)
C
om
pu
te
r
S
al
es
(
m
il
li
on
s)
Worldwide Computer Sales
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3135.2 ADDITION AND SUBTRACTION OF POLYNOMIALS

5.2 Putting It All Together

Monomial A number, variable, or product of
numbers and variables raised to natu-
ral number powers

The degree is the sum of the exponents.

The coefficient is the number in a
monomial.

4 x 2y Degree: 3; coefficient: 4

– 6 x 2 Degree: 2; coefficient: – 6
– a4 Degree: 4; coefficient: – 1

x Degree: 1; coefficient: 1

– 8 Degree: 0; coefficient: – 8

Polynomial A monomial or the sum of two or
more monomials

4 x 2 + 8 xy2 + 3y2 Trinomial
– 9x 4 + 100 Binomial
– 3x 2y3 Monomial

Like Terms Monomials containing the same vari-
ables raised to the same powers

10x and – 2 x, 4 x 2 and

3x 2

5ab 2 and – ab 2, 5z and 12 z

CONCEPT EXPLANATION EXAMPLES

Addition of Polynomials To add polynomials, combine like
terms.

(x 2 + 3x + 1) + (2 x 2 – 2 x + 7)
= (1 + 2)x 2 + (3 – 2)x + (1 + 7)
= 3x 2 + x + 8

3xy + 5xy = (3 + 5)xy = 8 xy

Opposite of a Polynomial To obtain the opposite of a polyno-
mial, negate each term.

Polynomial Opposite

– 2 x 2 + x – 6 2 x 2 – x + 6
a2 – b 2 – a2 + b 2

– 3x – 18 3x + 18

Subtraction of
Polynomials

To subtract one polynomial from
another, add the first polynomial
and the opposite of the second
polynomial.

(x 2 + 3x) – (2 x 2 – 5x)
= (x 2 + 3x) + ( – 2 x 2 + 5x)
= (1 – 2)x 2 + (3 + 5)x
= – x 2 + 8 x

Evaluating a Polynomial To evaluate a polynomial in x, sub-
stitute a value for x in the expression
and simplify.

To evaluate the polynomial

3x 2 – 2 x + 1 for x = 2,

substitute 2 for x and simplify.

3(2)2 – 2(2) + 1 = 9

The year 2008 corresponds to x = 8 in the given polynomial, so substitute 8 for x and
evaluate the resulting expression.

0.7868 x 2 + 16.72 x + 122.58 = 0.7868(8)2 + 16.72(8) + 122.58
� 307 million

The graph and the polynomial give similiar results.

Now Try Exercise 71

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314 CHAPTER 5 POLYNOMIALS AND EXPONENTS

5.2 Exercises

CONCEPTS AND VOCABULARY

1. A(n) _____ is a number, a variable, or a product of
numbers and variables raised to a natural number
power.

2. A(n) _____ is a monomial or a sum of monomials.

3. The _____ of a monomial is the sum of the exponents
of the variables.

4. The _____ of a polynomial is the degree of the term
with the greatest degree.

5. A polynomial with two terms is called a(n) _____.

6. A polynomial with three terms is called a(n) _____.

7. Two monomials with the same variables raised to the
same powers are terms.

8. To add two polynomials, combine terms.

9. To subtract two polynomials, add the first polynomial
to the _____ of the second polynomial.

10. Polynomials can be added horizontally or _____.

PROPERTIES OF POLYNOMIALS

Exercises 11–18: Identify the degree and coefficient of
the monomial.

11. 3x 2 12. y

13. – ab 14. – 2 xy

15. – 5rt 16. 8 x 2y5

17. 6 18. – 12

Exercises 19–28: Determine whether the expression is a
polynomial. If it is, state how many terms and variables
the polynomial contains. Then state its degree.

19. – x 20. 7z

21. 4 x 2 – 5x + 9 22. x 3 – 9

23. x +
1
x

24.
5

xy + 1

25. 3x -2y -3 26. 52a3b 4

27. – 23a4bc 28. – 7y -1z-3

Exercises 29–36: State whether the given pair of expres-
sions are like terms. If they are like terms, add them.

29. 5x, – 4 x 30. x 2, 8 x 2

31. x 3, – 6 x 3 32. 4 xy, – 9xy

33. 9x, – xy 34. 5x 2y, – 3xy2

35. ab, ba 36. rt 2, – 2t 2r

ADDITION OF POLYNOMIALS

Exercises 37– 46: Add the polynomials.

37. (3x + 5) + ( – 4 x + 4)

38. ( – x + 5) + (2 x – 5)

39. (3x 2 + 4 x + 1) + (x 2 + 4 x)

40. ( – x 2 – x) + (2 x 2 + 3x – 1)

41. ( y3 + 3y2 – 5) + (3y3 + 4y – 4)

42. (4z4 + z2 – 10) + ( – z4 + 4z – 5)

43. ( – xy + 5) + (5xy – 4)

44. (2a2 + b 2) + (3a2 – 5b 2)

45. (a3b 2 + a2b 3) + (a2b 3 – a3b 2)

46. (a2 + ab + b 2) + (a2 – ab + b 2)

Exercises 47–50: Add the polynomials vertically.

47. 4 x 2 – 2 x + 1
5x 2 + 3x – 7

48. 8 x 2 + 3x + 5
– x 2 – 3x – 9

49. – x 2 + x
2 x 2 – 8 x – 1

50. a3 – 3a2b + 3ab 2 – b 3
a3 + 3a2b + 3ab 2 + b 3

SUBTRACTION OF POLYNOMIALS

Exercises 51–56: Write the opposite of the polynomial.

51. 5x 2 52. 17x + 12

53. 3a2 – a + 4

54. – b 3 + 3b

55. – 2t 2 – 3t + 4 56. 7t 2 + t – 10

Exercises 57– 66: Subtract the polynomials.

57. (3x + 1) – ( – x + 3)

58. ( – 2 x + 5) – (x + 7)

59. ( – x 2 + 6 x) – (2 x 2 + x – 2)

3155.2 ADDITION AND SUBTRACTION OF POLYNOMIALS

60. (2y2 + 3y – 2) – ( y2 – y)

61. (z3 – 2z2 – z) – (4z2 + 5z + 1)

62. (3z4 – z) – ( – z4 + 4z2 – 5)

63. (4 xy + x 2y2) – (xy – x 2y2)

64. (a2 + b 2) – ( – a2 + b 2)

65. (ab 2) – (ab 2 + a3b)

66. (x 2 + 3xy + 4y2) – (x 2 – xy + 4y2)

Exercises 67–70: Subtract the polynomials vertically.

67. (x 2 + 2 x – 3) – (2 x 2 + 7x + 1)

68. (5x 2 – 9x – 1) – (x 2 – x + 3)

69. (3x 3 – 2 x) – (5x 3 + 4 x + 2)

70. (a2 + 3ab + 2b 2) – (a2 – 3ab + 2b 2)

APPLICATIONS

71. Exercise and Heart Rate The polynomial given by
1.6t 2 – 28t + 200 calculates the heart rate shown in
Figure 5.1(b) in A Look Into Math for this section,
where t represents the elapsed time in minutes since
exercise stopped.
(a) What is the heart rate when the athlete first stops

exercising?
(b) What is the heart rate after 5 minutes?
(c) Describe what happens to the heart rate after

exercise stops.

72. Cellular Phone Subscribers In the early years of cel-
lular phone technology—from 1986 through 1991—
the number of subscribers in millions could be mod-
eled by the polynomial 0.163x 2 – 0.146 x + 0.205,
where x = 1 corresponds to 1986, x = 2 to 1987,
and so on. The graph illustrates this growth.

1987
0

1989 1991

1
2
3
4
5
6
7
x
y

Year

C
el

l
P

ho
ne

S
ub

sc
ri

be
rs

(m
il

li
on

(a) Use the graph to estimate the number of cellular
phone subscribers in 1990.

(b) Use the polynomial to estimate the number of cel-
lular phone subscribers in 1990.

73. Areas of Squares Write a monomial that equals the
sum of the areas of the squares. Then calculate this
sum for z = 10 inches.

74. Areas of Rectangles Find a monomial that equals the
sum of the areas of the three rectangles. Find this sum
for a = 5 yards and b = 3 yards.

a
b
a
b
a
b

75. Area of a Figure Find a polynomial that equals the
area of the figure. Calculate its area for x = 6 feet.

2x
x
x
x

76. Area of a Rectangle Write a polynomial that gives
the area of the rectangle. Calculate its area for x = 3
feet.

7
x

77. Areas of Circles Write a polynomial that gives the
sum of the areas of two circles, one with radius x and
the other with radius y. Find this sum for x = 2 feet
and y = 3 feet. Leave your answer in terms of p.

78. Squares and Circles Write a polynomial that gives

the sum of the areas of a square having sides of length
x and a circle having diameter x. Approximate this
sum to the nearest hundredth of a square foot for
x = 6 feet.

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316 CHAPTER 5 POLYNOMIALS AND EXPONENTS

79. World Population The table lists actual and projected
world population P in billions for selected years t.

(a) Find the slope of each line segment connecting
consecutive data points in the table. Can these
data be modeled with a line? Explain.

(b) Does the polynomial 0.077t – 148 give good
estimates for the world population in year t?
Explain how you decided.

Source: U.S. Census Bureau.

t 1974 1987 1999 2012

P 4 5 6 7

Source: U.S. Postal Service.

t 1963 1975 1987 2002 2007 2011

P 5¢ 13¢ 25¢ 37. 41¢ 44¢

(a) Does the polynomial 0.835t – 1635 model the
data in the table exactly?

(b) Does it give approximations that are within 1.5¢
of the actual values?

WRITING ABOUT MATHEMATICS

81. Explain what the terms monomial, binomial, trino-
mial, and polynomial mean. Give an example of each.

82. Explain how to determine the degree of a polynomial
having one variable. Give an example.

83. Explain how to obtain the opposite of a polynomial.
Give an example.

84. Explain how to subtract two polynomials. Give an
example.

80. Price of a Stamp The table lists the price P of a first-
class postage stamp for selected years t.

Checking Basic ConceptsSECTIONS5.1 and 5.2
1. Evaluate each expression.
(a) – 52 (b) 32 – 23

2. Simplify each expression.
(a) 103 # 105 (b) (3x 2) ( – 4 x 5)
(c) (a3b)2 (d) a x

z3
b4

3. Simplify each expression.
(a) (4y3)0 (b) (x 3)2(3x 4)2
(c) 2a2(5a3 – 7)

4. State the number of terms and variables in the
polynomial 5x 3y – 2 x 2y + 5. What is its degree?

5. A box has a rectangular bottom twice as long as it
is wide.

(a) If the bottom has width w and the box has
height h, write a monomial that gives the vol-
ume of the box.

(b) Find the volume of the box for w = 12
inches and h = 10 inches.

6. Simplify each expression.
(a) (2a2 + 3a – 1) + (a2 – 3a + 7)
(b) (4z 3 + 5z) – (2z 3 – 2z + 8)
(c) (x 2 + 2 xy + y2) – (x 2 – 2 xy + y2)

5.3 Multiplication of Polynomials
Multiplying Monomials ● Review of the Distributive Properties ●
Multiplying Monomials and Polynomials ● Multiplying Polynomials

A LOOK INTO MATH N The study of polynomials dates back to Babylonian civilization in about 1800–1600 B.C.
Many eighteenth-century mathematicians devoted their entire careers to the study of poly-
nomials. Today, polynomials still play an important role in mathematics, often being used
to approximate unknown quantities. In this section we discuss the basics of multiplying
polynomials. (Source: Historical Topics for the Mathematics Classroom, Thirty-first Yearbook, NCTM.)

3175.3

MULTIPLICATION OF POLYNOMIALS

Multiplying Monomials
A monomial is a number, a variable, or a product of numbers and variables raised to natural
number powers. To multiply monomials, we often use the product rule for exponents.

EXAMPLE 1 Multiplying monomials

Multiply.
(a) – 5x 2 # 4 x 3 (b) (7xy4)(x 3y2)
Solution
(a) – 5x 2 # 4 x 3 = ( – 5)(4)x 2x 3 Commutative property
= – 20x 2 + 3 The product rule
= – 20x 5 Simplify.

(b) (7xy4)(x 3y2) = 7xx 3y4y2 Commutative property
= 7x 1 + 3y4 + 2 The product rule
= 7x 4y6 Simplify.

Now Try Exercises 9, 13

READING CHECK

• Which rule for exponents is commonly used to multiply monomials?

Review of the

Distributive Properties

Distributive properties are used frequently for multiplying monomials and polynomials. For
all real numbers a, b, and c,

a(b + c) = ab + ac and a(b – c) = ab – ac.

The first distributive property above can be visualized geometrically. For example,

3(x + 2) = 3x + 6

is illustrated in Figure 5.8. The dimensions of the large rectangle are 3 by x + 2, and its
area is 3(x + 2). The areas of the two small rectangles, 3x and 6, equal the area of the large
rectangle. Therefore 3(x + 2) = 3x + 6.

In the next example we use the distributive properties to multiply expressions.

EXAMPLE 2 Using distributive properties

Multiply.
(a) 2(3x + 4) (b) (3x 2 + 4)5 (c) – x(3x – 6)

Solution

(a) 2(3x + 4) = 2 # 3x + 2 # 4 = 6 x + 8
(b) (3x 2 + 4)5 = 3x 2 # 5 + 4 # 5 = 15x 2 + 20
(c) – x(3x – 6) = – x # 3x + x # 6 = – 3x 2 +

6 x

Now Try Exercises 15, 19, 21

Figure 5.8 Area: 3x + 6

x + 2

3 3x 6

x 2

Distributive Property
3(x � 2) � 3x � 6

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318 CHAPTER 5 POLYNOMIALS AND EXPONENTS

Multiplying Monomials and Polynomials
A monomial consists of one term, whereas a polynomial consists of one or more terms
separated by + or – signs. To multiply a monomial by a polynomial, we apply the distribu-
tive properties and the product rule.

READING CHECK

• What properties are com-
monly used to multiply a
monomial and a polynomial?

EXAMPLE 3 Multiplying monomials and polynomials

Multiply.
(a) 9x(2 x 2 – 3) (b) (5x – 8)x 2
(c) – 7(2 x 2 – 4 x + 6) (d) 4 x 3(x 4 + 9x 2 – 8)

Solution

(a) 9x(2 x 2 – 3) = 9x # 2 x 2 – 9x # 3 Distributive property
= 18 x 3 – 27x The product rule

(b) (5x – 8)x 2 = 5x # x 2 – 8 # x 2 Distributive property
= 5x 3 – 8 x 2 The product rule

(d) 4 x 3(x 4 + 9x 2 – 8) = 4 x 3 # x 4 + 4 x 3 # 9x 2 – 4 x 3 # 8 Distributive property
= 4 x 7 + 36 x 5 – 32 x 3 The product rule

Now Try Exercises 23, 25, 27, 29

We can also multiply monomials and polynomials that contain more than one variable.

EXAMPLE 4 Multiplying monomials and polynomials

Multiply.
(a) 2 xy(7x 2y3 – 1) (b) – ab(a2 – b 2)

Solution

(a) 2 xy(7x 2y3 – 1) = 2 xy # 7x 2y3 – 2 xy # 1 Distributive property
= 14 xx 2yy3 – 2 xy Commutative property
= 14 x 3y4 – 2 xy The product rule

(b) – ab(a2 – b 2) = – ab # a2 + ab # b 2 Distributive property
= – aa2b + abb 2 Commutative property
= – a3b + ab 3 The product rule

Now Try Exercises 31, 35

Multiplying Polynomials
Monomials, binomials, and trinomials are examples of polynomials. Recall that a monomial
has one term, a binomial has two terms, and a trinomial has three terms. In the next example
we multiply two binomials, using both geometric and symbolic techniques.

3195.3 MULTIPLICATION OF POLYNOMIALS

EXAMPLE 5 Multiplying binomials

Multiply (x + 4)(x + 2)
(a) geometrically and (b) symbolically.

Solution
(a) To multiply (x + 4)(x + 2) geometrically, draw a rectangle x + 4 long and x + 2

wide, as shown in Figure 5.9(a). The area of this rectangle equals length times width, or
(x + 4)(x + 2). The large rectangle can be divided into four smaller rectangles, which
have areas of x 2, 4 x, 2 x, and 8, as shown in Figure 5.9(b).

Thus

(x + 4)(x + 2) = x 2 � 4 x � 2 x + 8
= x 2 � 6 x + 8.

(b) To multiply (x + 4)(x + 2) symbolically, apply the distributive property two times.

(x � 4) (x + 2) = (x � 4) (x) + (x � 4) (2)
= x # x + 4 # x + x # 2 + 4 # 2
= x 2 � 4 x � 2 x + 8
= x 2 � 6 x + 8

r
Now Try Exercise 39

The distributive properties used in part (b) of the previous example show that if we
want to multiply (x + 4) by (x + 2), we should multiply every term in x + 4 by every
term in x + 2.

(x + 4)(x + 2) = x 2 + 2 x + 4 x + 8
= x 2 + 6 x + 8

NOTE: This process of multiplying binomials is sometimes called FOIL. This acro-
nym may be used to remind us to multiply the first terms (F ), outside terms (O), inside
terms (I ), and last terms (L). The FOIL process is a shortcut for the process used in
Example 5(b).

Multiply the First terms to obtain x 2. (x + 4)(x + 2)

Multiply the Outside terms to obtain 2 x. (x + 4)(x + 2)

Multiply the Inside terms to obtain 4 x. (x + 4)(x + 2)

Multiply the Last terms to obtain 8. (x + 4)(x + 2)

MULTIPLYING POLYNOMIALS

The product of two polynomials may be found by multiplying every term in the first
polynomial by every term in the second polynomial and then combining like terms.

READING CHECK

• What kind of polynomials can be multiplied using the FOIL method?

Figure 5.9

x + 2
x + 4

(a) Area = (x + 4)(x + 2)

2 2x

x2
8

x 4

4xx

(b) Area = x2 + 4x + 2x + 8

The following statement summarizes how to multiply two polynomials in general.

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320 CHAPTER 5 POLYNOMIALS AND EXPONENTS

EXAMPLE 6 Multiplying binomials

Multiply. Draw arrows to show how each term is found.
(a) (3x + 2)(x + 1) (b) (1 – x)(1 + 2 x) (c) (4 x – 3)(x 2 – 2 x)

Solution

(a) (3x + 2)(x + 1) = 3x # x + 3x # 1 + 2 # x + 2 # 1
= 3x 2 � 3x � 2 x + 2
= 3x 2 � 5x + 2

(b) (1 – x)(1 + 2 x) = 1 # 1 + 1 # 2 x – x # 1 – x # 2 x
= 1 � 2 x � x – 2 x 2

= 1 � x – 2 x 2

Now Try Exercises 51, 53, 59

The FOIL process may be helpful for remembering how to multiply two binomials, but it
cannot be used for every product of polynomials. In the next example, the general process
for multiplying polynomials is used to find products of binomials and trinomials.

EXAMPLE 7 Multiplying polynomials

Multiply.
(a) (2 x + 3)(x 2 + x – 1) (b) (a – b)(a2 + ab + b 2)
(c) (x 4 + 2 x 2 – 5)(x 2 + 1)

Solution
(a) Multiply every term in (2 x + 3) by every term in (x 2 + x – 1).

(2 x + 3)(x 2 + x – 1) = 2 x # x 2 + 2 x # x – 2 x # 1 + 3 # x 2 + 3 # x – 3 # 1
= 2 x 3 � 2 x 2 � 2 x � 3x 2 � 3x – 3
= 2 x 3 � 5x 2 � x – 3

(b) (a – b)(a2 + ab + b 2) = a # a2 + a # ab + a # b 2 – b # a2 – b # ab – b # b 2
= a3 � a2b � ab 2 � a2b � ab 2 – b 3

= a3 – b 3

(c) (x 4 + 2 x 2 – 5)(x 2 + 1) = x 4 # x 2 + x 4 # 1 + 2 x 2 # x 2 + 2 x 2 # 1 – 5 # x 2 – 5 # 1
= x 6 � x 4 � 2 x 4 � 2 x 2 � 5x 2 – 5
= x 6 � 3x 4 � 3x 2 – 5

Now Try Exercises 63, 67, 69

STUDY TIP

Even if you know exactly how to do a math problem correctly, a simple computational error will
often cause you to get an incorrect answer. Be sure to take your time on simple calculations.

3215.3 MULTIPLICATION OF POLYNOMIALS

Polynomials can be multiplied vertically in a manner similar to multiplication of real
numbers. For example, multiplication of 123 times 12 is performed as follows.

1 2 3

* 1 2
2 4 6

1 2 3

1 4 7 6

A similar method can be used to multiply polynomials vertically.

EXAMPLE 8 Multiplying polynomials vertically

Multiply 2 x 2 – 4 x + 1 and x + 3 vertically.

Solution
Write the polynomials vertically. Then multiply every term in the first polynomial by each
term in the second polynomial. Arrange the results so that like terms are in the same column.

2 x 2 – 4 x + 1
x + 3

6 x 2 – 12 x + 3
2 x 3 – 4 x 2 + x
2 x 3 + 2 x 2 – 11x + 3

Multiply top polynomial by 3.
Multiply top polynomial by x.

Add each column.

Now Try Exercise 71
MAKING CONNECTIONS

Vertical and Horizontal Formats

Whether you decide to add, subtract, or multiply polynomials vertically or horizontally,
remember that the same answer is obtained either way.

EXAMPLE 9 Finding the volume of a box

A box has a width 3 inches less than its height and a length 4 inches more than its height.
(a) If h represents the height of the box, write a polynomial that represents the volume of

the box.
(b) Use this polynomial to calculate the volume of the box if h = 10 inches.

Solution
(a) If h is the height, then h – 3 is the width and h + 4 is the length, as illustrated in

Figure 5.10. Its volume equals the product of these three expressions.

h(h – 3)(h + 4) = (h 2 – 3h)(h + 4)

= h 2 # h + h 2 # 4 – 3h # h – 3h # 4
= h 3 + 4h 2 – 3h 2 – 12

= h 3 + h 2 – 12h

(b) If h = 10, then the volume is

103 + 102 – 12(10) = 1000 + 100 – 120 = 980 cubic inches.

Now Try Exercise 79

Figure 5.10

h – 3

h + 4

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322 CHAPTER 5 POLYNOMIALS AND EXPONENTS

5.3 Putting It All Together

Distributive Properties For all real numbers a, b, and c,

a(b + c) = ab + ac and
a(b – c) = ab – ac.

5(x + 3) = 5x + 15,
3(x – 6) = 3x – 18, and
– 2 x(3 – 5x 3) = – 6 x + 10x 4

Multiplying Polynomials The product of two polynomials may
be found by multiplying every term in
the first polynomial by every term in
the second polynomial and then com-
bining like terms.

3x(5x 2 + 2 x – 7)
= 3x # 5x 2 + 3x # 2 x – 3x # 7
= 15x 3 + 6 x 2 – 21x

(x + 2)(7x – 3)
= x # 7x – x # 3 + 2 # 7x – 2 # 3
= 7x 2 – 3x + 14 x – 6
= 7x 2 + 11x – 6

CONCEPT EXPLANATION EXAMPLES

5.3 Exercises

CONCEPTS AND VOCABULARY

1. The equation x 2 # x 3 = x 5 illustrates what rule of
exponents?

2. The equation 3(x – 2) = 3x – 6 illustrates what
property?

3. The product of two polynomials may be found by
multiplying every in the first polynomial by
every in the second polynomial and then com-
bining like terms.

4. Polynomials can be multiplied horizontally or _____.

MULTIPLICATION OF MONOMIALS

Exercises 5–14: Multiply.

5. x 2 # x 5 6. – a # a5
7. – 3a # 4a 8. 7x # 5x
9. 4 x 3 # 5x 2 10. 6b 6 # 3b 5
11. xy2 # 4 xy 12. 3ab # ab 2
13. ( – 3xy2)(4 x 2y) 14. ( – r 2t 2)( – r 3t)

MULTIPLICATION OF MONOMIALS
AND POLYNOMIALS

Exercises 15–36: Multiply and simplify the expression.

15. 3(x + 4) 16. – 7(4 x – 1)

19. (4 – z)z 20. 3z(1 – 5z)

17. – 5(9x + 1)

18. 10(1 – 6 x)

21. – y(5 + 3y)

22. (2y – 8)2y

23. 3x(5x 2 – 4) 24. – 6 x(2 x 3 + 1)

27. – 8(4t 2 + t + 1)

28. 7(3t 2 – 2t – 5)

31. xy(x + y)

32. ab(2a – 3b)

25. (6 x – 6)x 2

26. (1 – 2 x 2)3x 2

29. n2( – 5n2 + n – 2)

30. 6n3(2 – 4n + n2)

33. x 2(x 2y – xy2)

34. 2y2(xy – 5)

35. – ab(a3 – 2b 3)

36. 5rt(r 2 + 2rt + t 2)

3235.3 MULTIPLICATION OF POLYNOMIALS

MULTIPLICATION OF POLYNOMIALS

Exercises 37–42: (Refer to Example 5.) Multiply the given
expression (a) geometrically and (b) symbolically.

37. x(x + 3)

38. 2 x(x + 5)

39. (x + 2)(x + 2)

40. (x + 1)(x + 3)

41. (x + 3)(x + 6)

42. (x + 5)(x + 2)

43. (x + 3)(x + 5)

44. (x – 4)(x – 7)

45. (x – 8)(x – 9)

46. (x + 10)(x + 10)

51. (10y + 7)( y – 1) 52. ( y + 6)(2y + 7)

57. (x – 1)(x 2 + 1)

58. (x + 2)(x 2 – x)

47. (3z – 2)(2z – 5) 48. (z + 6)(2z – 1)

53. (5 – 3a)(1 – 2a) 54. (4 – a)(5 + 3a)

59. (x 2 + 4)(4 x – 3)

60. (3x 2 – 1)(3x 2 + 1)

49. (8b – 1)(8b + 1) 50. (3t + 2)(3t – 2)

55. (1 – 3x)(1 + 3x) 56. (10 – x)(5 – 2 x)

61. (2n + 1)(n2 + 3) 62. (2 – n2)(1 + n2)

Exercises 43–70: Multiply and simplify the expression.

63. (m + 1)(m2 + 3m + 1)

64. (m – 2)(m2 – m + 5)

65. (3x – 2)(2 x 2 – x + 4)

66. (5x + 4)(x 2 – 3x + 2)

67. (x + 1)(x 2 – x + 1)

68. (x – 2)(x 2 + 4 x + 4)

69. (4b 2 + 3b + 7)(b 2 + 3)

70. ( – 3a2 – 2a + 1)(3a2 – 3)

Exercises 71–76: Multiply the polynomials vertically.

71. (x + 2)(x 2 – 3x + 1)

72. (2y – 3)(3y2 – 2y – 2)

73. (a – 2)(a2 + 2a + 4)

74. (b – 3)(b 2 + 3b + 9)

75. (3x 2 – x + 1)(2 x 2 + 1)

76. (2 x 2 – 3x – 5)(2 x 2 + 3)

77. Thinking Generally If a polynomial with m terms
and a polynomial with n terms are multiplied, how
many terms are there in the product before like terms
are combined?

78. Thinking Generally When a polynomial with m
terms is multiplied by a second polynomial, the product
contains k terms before like terms are combined. How
many terms does the second polynomial contain?

APPLICATIONS

79. Volume of a Box (Refer to Example 9.) A box has a
width 4 inches less than its height and a length 2 inches
more than its height.

(a) If h is the height of the box, write a polynomial

that represents the volume of the box.
(b) Use this polynomial to calculate the volume for

h = 25 inches.

80. Surface Area of a Box Use the drawing of the box to
write a polynomial that represents each of the following.

b + 1

2b + 1

(a) The area of its bottom
(b) The area of its front
(c) The area of its right side
(d) The total area of its six sides

81. Perimeter of a Pen A rectangular pen for a pet has a
perimeter of 100 feet. If one side of the pen has length
x, then its area is given by x(50 – x).
(a) Multiply this expression.
(b) Evaluate the expression obtained in part (a) for

x = 25.

82. Rectangular Garden A rectangular garden has a
perimeter of 500 feet.
(a) If one side of the garden has length x, then write

a polynomial expression that gives its area. Mul-
tiply this expression completely.

(b) Evaluate the expression for x = 50 and interpret
your answer.

83. Surface Area of a Cube Write a polynomial that rep-
resents the total area of the six sides of the cube hav-
ing edges with length x + 1.

x + 1

x + 1
x + 1

84. Surface Area of a Sphere The surface area of a
sphere with radius r is 4pr 2. Write a polynomial that
gives the surface area of a sphere with radius x + 2.
Leave your answer in terms of p.

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324 CHAPTER 5 POLYNOMIALS AND EXPONENTS

85. Toy Rocket A toy rocket is shot straight up into the
air. Its height h in feet above the ground after t sec-
onds is represented by the expression t(64 – 16t).
(a) Multiply this expression.
(b) Evaluate the expression obtained in part (a) and

the given expression for t = 2.
(c) Are your answers in part (b) the same? Should

they be the same?

86. Toy Rocket on the Moon (Refer to the preceding
exercise.) If the same toy rocket were flown on the
moon, then its height h in feet after t seconds would
be t164 – 52 t2.
(a) Multiply this expression.
(b) Evaluate the expression obtained in part (a) and

the given expression for t = 2. Did the rocket go
higher on the moon?

WRITING ABOUT MATHEMATICS

87. Explain how the acronym FOIL relates to multiplying
two binomials, such as x + 3 and 2 x + 1.

88. Does the FOIL method work for multiplying a bino-
mial and a trinomial? Explain.

89. Explain in words how to multiply any two polynomi-
als. Give an example.

90. Give two properties of real numbers that are used
for multiplying 3x(5x 2 – 3x + 2). Explain your
answer.

Working with Real DataGroup Activity

Biology Some types of worms have a remarkable
ability to live without moisture. The following table
from one study shows the number of worms W surviv-
ing after x days without moisture.

(a) Use the equation W = – 0.0014 x 2 – 0.076 x + 50
to find W for each x-value in the table.

Directions: Form a group of 2 to 4 people. Select someone to record the group’s responses for
this activity. All members of the group should work cooperatively to answer the questions. If
your instructor asks for your results, each member of the group should be prepared to respond.

(b) Discuss how well this equation approximates the
data.

Source: D. Brown and P. Rothery, Models in Biology.

x (days) 0 20 40 80 120 160

W (worms) 50 48 45 36 20 3

5.4 Special Products
Product of a Sum and Difference ● Squaring Binomials ● Cubing Binomials

A LOOK INTO MATH N Polynomials are often used to approximate real-world phenomena in applications. Poly-
nomials have played an important role in the development of everyday products such as
tablet computers, cell phones, and automobiles. Even digital images in computers and
interest calculations at a bank make use of polynomials. In this section we discuss how to
multiply some special types of binomials.

Product of a Sum and Difference

Products of the form (a + b)(a – b) occur frequently in mathematics. Other examples include

(x + y)(x – y) and (2r + 3t)(2r – 3t).

3255.4 SPECIAL PRODUCTS

These products can always be multiplied by using the techniques discussed in Section 5.3.
However, there is a faster way to multiply these special products.

(a + b)(a – b) = a # a � a # b � b # a – b # b
= a2 � ab � ba – b 2

= a2 – b 2

In words, the product of a sum of two numbers and their difference equals the difference of
their squares. We generalize this method as follows.

STUDY TIP

In mathematics, there are often several correct ways to perform a particular process. If your
instructor does not require you to use a specified method, choose the one that works best
for you.

READING CHECK

• Explain in words how you
can find the product of the
sum of two numbers and
their difference.

PRODUCT OF A SUM AND DIFFERENCE

For any real numbers a and b,

(a + b)(a – b) = a2 – b 2.

EXAMPLE 1 Finding products of sums and differences

Multiply.
(a) (x + y)(x – y) (b) (z – 2)(z + 2)
(c) (2r + 3t)(2r – 3t) (d) (5m2 – 4n2)(5m2 + 4n2)

Solution
(a) If we let a = x and b = y, then we can apply the rule

(a + b)(a – b) = a2 – b 2.

Thus

(x + y)(x – y) = (x)2 – ( y)2

= x 2 – y2.

(b) Because the expressions (z + 2)(z – 2) and (z – 2)(z + 2) are equal by the commu-
tative property, we can apply the formula for the product of a sum and difference.

(z – 2)(z + 2) = (z)2 – (2)2

= z2 – 4

(2r + 3t)(2r – 3t) = (2r)2 – (3t)2

= 4r 2 – 9t 2

(d) (5m2 – 4n2)(5m2 + 4n2) = (5m2)2 – (4n2)2 = 25m4 – 16n4

Now Try Exercises 7, 13, 17

The next example demonstrates how the product of a sum and difference can be used to
find some products of numbers mentally.

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326 CHAPTER 5 POLYNOMIALS AND EXPONENTS

EXAMPLE 2 Finding a product

Use the product of a sum and difference to find 22 # 18.
Solution
Because 22 = 20 + 2 and 18 = 20 – 2, rewrite and evaluate 22 # 18 as follows.
22 # 18 = (20 + 2)(20 – 2) Rewrite 22 as 20 + 2 and 18 as 20 – 2.
= 202 – 22 Product of a sum and difference
= 400 – 4 Evaluate exponents.
= 396 Subtract.

Now Try Exercise 21

Squaring Binomials

Because each side of the square shown in Figure 5.11 has length (a + b), its area equals

(a + b)(a + b),

which can be written as (a + b)2. We can multiply this expression as follows.

(a + b)2 = (a + b)(a + b)
= a2 � ab � ba + b 2

= a2 � 2ab + b 2

This result is illustrated geometrically in Figure 5.12, where the area of the large square
is (a + b)2. This area can also be found by adding the areas of the four small rectangles.

a2 � ab � ba + b 2 = a2 � 2ab + b 2

The geometric and symbolic results are the same. Note that to obtain the middle term, 2ab,
we can multiply the two terms, a and b, in the binomial and double the result.

A similar product that is also the square of a binomial can be calculated as

(a – b)2 = (a – b)(a – b)
= a2 � ab � ba + b 2

= a2 � 2ab + b 2.

These results are summarized as follows.

READING CHECK

• Explain in words how you
can square a binomial.

SQUARING A BINOMIAL

For any real numbers a and b,

(a + b)2 = a2 + 2ab + b 2 and
(a – b)2 = a2 – 2ab + b 2.

That is, the square of a binomial equals the square of the first term, plus (or minus)
twice the product of the two terms, plus the square of the last term.

NOTE: (a + b)2 3 a2 + b 2. Do not forget the middle term when squaring a binomial.

EXAMPLE 3 Squaring a binomial

Multiply.
(a) (x + 3)2 (b) (2 x – 5)2 (c) (1 – 5y)2 (d) (7a2 + 3b)2

Figure 5.11

a + b
a + b

Area � (a � b)2

Figure 5.12

a
b
a2

a b

(a � b)2 � a2 � 2ab � b 2

3275.4 SPECIAL PRODUCTS

Solution
(a) Let a = x and b = 3 in the formula (a + b)2 = a2 + 2ab + b 2.

(x + 3)2 = (x)2 + 2(x)(3) + (3)2

= x 2 + 6 x + 9

(b) Apply the formula (a – b)2 = a2 – 2ab + b 2 with a = 2 x and b = 5.

(2 x – 5)2 = (2 x)2 – 2(2 x)(5) + (5)2

= 4 x 2 – 20x + 25

(d) (7a2 + 3b)2 = (7a2)2 + 2(7a2)(3b) + (3b)2 = 49a4 + 42a2b + 9b 2

Now Try Exercises 27, 29, 35, 39

MAKING CONNECTIONS

Multiplying Binomials and Special Products

If you forget these special products, you can still multiply polynomials by using earlier
techniques. For example, the binomial in Example 3(b) can be multiplied as

(2 x – 5)2 = (2 x – 5)(2 x – 5)

= 2 x # 2 x – 2 x # 5 – 5 # 2 x + 5 # 5
= 4 x 2 – 10x – 10x + 25
= 4 x 2 – 20x + 25.

N REAL-WORLD CONNECTION NASA first developed digital pictures because they were easy
to transmit through space and because they provided clear images. A digital image from
outer space is shown in Figure 5.13.

Today, digital cameras are readily available, and the Internet uses digital images exclu-
sively. The next example shows how polynomials relate to digital pictures.

EXAMPLE 4 Calculating the size of a digital picture

A digital picture comprises tiny square units called pixels. Shading individual pixels creates
a picture. A simplified version of a digital picture of the letter T is shown in Figure 5.14. This
picture includes an image of the letter T that measures 3 pixels by 3 pixels and a 1-pixel
border.
(a) If a square digital image measures x pixels by x pixels, then a picture that includes the

image and a 1-pixel border will measure x + 2 pixels by x + 2 pixels. Find a polyno-
mial that gives the total number of pixels in the picture, including the border.

(b) Let x = 3 and evaluate the polynomial. Does it agree with Figure 5.14?

Solution
(a) The total number of pixels equals (x + 2) times (x + 2), or (x + 2)2.

(x + 2)2 = x 2 + 4 x + 4

(b) For x = 3, the polynomial evaluates to 32 + 4 # 3 + 4 = 25, the total number of pixels.
This result agrees with Figure 5.14, which has a total of 5 # 5 = 25 pixels with a 3 pixel
by 3 pixel image of the letter T inside.

Now Try Exercise 79

Figure 5.13 Digital Picture

Figure 5.14

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328 CHAPTER 5 POLYNOMIALS AND EXPONENTS

Cubing Binomials
To calculate the volume of the cube shown in Figure 5.15, we find the product of its length,
width, and height. Because all sides have the same measure, its volume is (x + 2)3. That is,
the volume equals the cube of x + 2.

To multiply the expression (x + 2)3, we proceed as follows.

(x + 2)3 = (x + 2)(x � 2)2 a3 = a # a2

= (x + 2)(x 2 � 4 x � 4) Square the binomial.

= x # x 2 + x # 4 x + x # 4 + 2 # x 2 + 2 # 4 x + 2 # 4 Multiply the polynomials.
= x 3 + 4 x 2 + 4 x + 2 x 2 + 8 x + 8 Simplify terms.
= x 3 + 6 x 2 + 12 x + 8 Combine like terms.

Figure 5.15
Volume = (x + 2)3

x + 2

x + 2
x + 2

EXAMPLE 5 Cubing a binomial

Multiply (2z – 3)3.

Solution

(2z – 3)3 = (2z – 3)(2z � 3)2 a3 = a # a2

= (2z – 3)(4z2 � 12z � 9) Square the binomial.

= 8z3 – 24z2 + 18z – 12z2 + 36z – 27 Multiply the polynomials.
= 8z3 – 36z2 + 54z – 27 Combine like terms.

NOTE: (2z – 3)3 3 (2z)3 – (3)3 = 8z3 – 27.

Now Try Exercise 47
CRITICAL THINKING

Suppose that a student is convinced that the expressions

(x + y)3 and x3 + y 3

are equal. How could you convince the student otherwise?

EXAMPLE 6 Calculating interest

If a savings account pays x percent annual interest, where x is expressed as a decimal, then
after 3 years a sum of money will grow by a factor of (1 + x)3.
(a) Multiply this expression.
(b) Evaluate the expression for x = 0.10 (or 10%), and interpret the result.

Solution
(a) (1 + x)3 = (1 + x)(1 + x)2 a3 = a # a2
= (1 + x)(1 + 2 x + x 2) Square the binomial.
= 1 + 2 x + x 2 + x + 2 x 2 + x 3 Multiply the polynomials.
= 1 + 3x + 3x 2 + x 3 Combine like terms.

3295.4 SPECIAL PRODUCTS

(b) Let x = 0.1 in the expression 1 + 3x + 3x 2 + x 3.

1 + 3(0.1) + 3(0.1)2 + (0.1)3 = 1.331

The sum of money will increase by a factor of 1.331. For example, $1000 deposited in
this account will grow to $1331 in 3 years.

Now Try Exercise 75

5.4 Putting It All Together

Product of a Sum and
Difference

For any real numbers x and y,

(x + y)(x – y) = x 2 – y2.
(x + 6)(x – 6) = x 2 – 36,
(2 x – 3)(2 x + 3) = 4 x 2 – 9, and
(x 2 + y2)(x 2 – y2) = x 4 – y4

Squaring a Binomial For all real numbers x and y,

(x + y)2 = x 2 + 2 xy + y2 and
(x – y)2 = x 2 – 2 xy + y2.

(x + 4)2 = x 2 + 8 x + 16,
(5x – 2)2 = 25x 2 – 20x + 4, and
(x 2 + y2)2 = x 4 + 2 x 2y2 + y4

Cubing a Binomial Multiply the binomial by its square. (x + 3)3

= (x + 3)(x + 3)2

= (x + 3)(x 2 + 6 x + 9)
= x 3 + 6 x 2 + 9x + 3x 2 + 18 x + 27
= x 3 + 9x 2 + 27x + 27

CONCEPT EXPLANATION EXAMPLES

5.4 Exercises

CONCEPTS AND VOCABULARY

1. (a + b)(a – b) = _____

2. (a + b)2 = _____

3. (a – b)2 = _____

4. (a + b)3 = (a + b) # _____
5. (True or False?) The two expressions (x + y)2 and

x 2 + y2 are equal for all real numbers x and y.

6. (True or False?) The two expressions (r – t)2 and
r 2 – t 2 are equal for all real numbers r and t.

PRODUCT OF A SUM AND DIFFERENCE

Exercises 7–20: Multiply.

7. (x – 3)(x + 3)

8. (x + 6)(x – 6)

9. (4 x – 1)(4 x + 1) 10. (10x + 3)(10x – 3)

11. (1 + 2a)(1 – 2a) 12. (4 – 9b)(4 + 9b)

13. (2 x + 3y)(2 x – 3y) 14. (5r – 6t)(5r + 6t)

15. (ab – 5)(ab + 5)

16. (2 xy + 7)(2 xy – 7)

17. (a2 – b 2)(a2 + b 2)

18. (3x 2 + y2)(3x 2 – y2)

19. (x 3 – y3)(x 3 + y3)

20. (2a4 + b 4)(2a4 – b 4)

Exercises 21–26: (Refer to Example 2.) Use the product
of a sum and a difference to evaluate the expression.

21. 101 # 99 22. 52 # 48
23. 23 # 17 24. 29 # 31
25. 90 # 110 26. 38 # 42

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330 CHAPTER 5 POLYNOMIALS AND EXPONENTS

SQUARING BINOMIALS

Exercises 27– 40: Multiply.

27. (a – 2)2

28. (x – 7)2

31. (3b + 5)2 32. (7t + 10)2

35. (1 – b)2 36. (1 – 4a)2

29. (2 x + 3)2

30. (7x – 2)2

33. 134 a – 422

37. (5 + y3)2 38. (9 – 5x 2)2

34. 115 a + 122

39. (a2 + b)2

40. (x 3 – y3)2

CUBING BINOMIALS

Exercises 41– 48: Multiply.

41. (a + 1)3

42. (b + 4)3

43. (x – 2)3

44. ( y – 7)3

45. (2 x + 1)3

46. (4z + 3)3

47. (6u – 1)3 48. (5v + 3)3

MULTIPLICATION OF POLYNOMIALS

Exercises 49–66: Multiply, using any appropriate method.

(b) Find the sum of the areas of the smaller rectan-
gles inside the large square.

49. 4(5x + 9) 50. (2 x + 1)(3x – 5)

51. (x – 5)(x + 7)

52. (x + 10)(x + 10)

53. (3x – 5)2 54. (x – 3)(x + 9)

55. (5x + 3)(5x + 4) 56. – x 3(x 2 – x + 1)

57. (4b – 5)(4b + 5) 58. (x + 5)3

59. – 5x(4 x 2 – 7x + 2)

60. (4 x 2 – 5)(4 x 2 + 5)

61. (4 – a)3 62. 2 x(x – 3)3

63. x(x + 3)2

64. (x – 1)2(x + 1)

65. (x + 2)(x – 2)(x + 1)(x – 1)

66. (x – y)(x + y)(x 2 + y2)

67. Thinking Generally Multiply (an + b n)(an – b n).

68. Thinking Generally Multiply (an + b n)2.

APPLICATIONS

Exercises 69–72: Do each part and verify that your
answers are the same.

(a) Find the area of the large square by multiplying
its length and width.

69.
2

x
x 2

70.
4

a
a 4

71.

2x
3

2x 3

72.
x

3y x

Exercises 73 and 74: Find a polynomial that represents
the following.

(a) The outside surface area given by the six sides of
the cube

(b) The volume of the cube

73.

x + 5

x + 5
x + 5

74.

2x + 1

2x + 1
2x + 1

75. Compound Interest (Refer to Example 6.) If a sum
of money is deposited in a savings account that is
paying x percent annual interest (expressed as a deci-
mal), then this sum of money increases by a factor of
(1 + x)2 after 2 years.
(a) Multiply this expression.
(b) Evaluate the polynomial expression found in

part (a) for an annual interest rate of 10%, or
x = 0.10, and interpret the answer.

76. Compound Interest If a sum of money is deposited
in a savings account that is paying x percent annual
interest, then this sum of money increases by a factor
of 11 + 1100×23 after 3 years.
(a) Multiply this expression.

(b) Evaluate the polynomial expression in part (a)
for an annual interest rate of 8%, or x = 8, and
interpret the answer.

77. Probability If there is an x percent chance of rain
on each of two consecutive days, then the expres-
sion (1 – x)2 gives the percent chance that neither
day will have rain. Assume that all percentages are
expressed as decimals.

3315.4 SPECIAL PRODUCTS

(a) Multiply this expression.
(b) Evaluate the polynomial expression in part (a)

for a 50% chance of rain, or x = 0.50, and inter-
pret the answer.

78. Probability If there is an x percent chance of rolling
a 6 with one die, then the expression (1 – x)3 gives
the percent chance of not rolling a 6 with three dice.
Assume that all percentages are expressed as deci-
mals or fractions.
(a) Multiply this expression.
(b) Evaluate the polynomial expression found in part

(a) for a 16.6% chance of rolling a 6, or x = 16 ,
and interpret the answer.

79. Swimming Pool A square swimming pool has an
8-foot-wide sidewalk around it.
(a) If the sides of the pool have length z, as shown in

the accompanying figure, find a polynomial that
gives the area of the sidewalk.

(b) Evaluate the polynomial in part (a) for z = 60
and interpret the answer.

8
z
z

80. Digital Picture (Refer to Example 4.) Suppose that
a digital picture, including its border, is x + 2 pixels
by x + 2 pixels and that the actual image inside the
border is x – 2 pixels by x – 2 pixels, as shown in
the following figure.

x + 2

x – 2

(a) Find a polynomial that gives the number of pixels
in the border.

(b) Evaluate the polynomial in part (a) for x = 5.
(c) Sketch a digital picture of the letter H with

x = 5. Does the picture agree with the answer in
part (b)?

(d) Digital pictures typically have large values for x.
If a picture has x = 500, find the total number of
pixels in its border.

WRITING ABOUT MATHEMATICS

81. Explain why (a + b)2 does not equal a2 + b 2 in
general for real numbers a and b.

82. Explain how to find the cube of a binomial.

Checking Basic ConceptsSECTIONS5.3 and 5.4
1. Multiply each expression.
(a) ( – 3xy4)(5x 2y) (b) – x(6 – 4 x)
(c) 3ab(a2 – 2ab + b 2)

2. Multiply each expression.
(a) (x + 3)(4 x – 3)
(b) (x 2 – 1)(2 x 2 + 2)
(c) (x + y)(x 2 – xy + y2)

3. Multiply each expression.
(a) (5x + 2)(5x – 2) (b) (x + 3)2

4. Complete each part and verify that your answers
are the same.

(a) Find the area of the large square by squaring
the length of one of its sides.

(b) Find the sum of the areas of the smaller rect-

angles inside the large square.

5
m

m 5

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332 CHAPTER 5 POLYNOMIALS AND EXPONENTS

NEW VOCABULARY

n Scientific notation

5.5 Integer Exponents and the Quotient Rule
Negative Integers as Exponents ● The Quotient Rule ● Other Rules for Exponents ●
Scientific Notation

A LOOK INTO MATH N In 2009, astronomers discovered a large planet that orbits a distant star. The planet, named
WASP-17b, is about 5,880,000,000,000,000 miles from Earth. Also in 2009, the H1N1
virus was identified in a worldwide influenza pandemic. A typical flu virus measures about
0.00000468 inch across. In this section, we will discuss how integer exponents can be used
to write such numbers in scientific notation. (Source: Scientific American.)

READING CHECK

• How is a negative integer
power on a base related
to the corresponding posi-
tive integer power on that
base?

TABLE 5.4 Powers of 2

Power of 2 Value
21 2

20 1

2-1 12

=
1

2-2 14 =
1
22

Decrease exponent by 1

Decrease exponent by 1
Decrease exponent by 1

Divide by 2

Divide by 2
Divide by 2

Table 5.4 shows that 2-1 = 1
21

and 2-2 = 1
22

. In other words, if the exponent on 2 is nega-
tive, then the expression is equal to the reciprocal of the corresponding expression with a
positive exponent on 2. This discussion suggests the following definition for negative inte-
ger exponents.

NEGATIVE INTEGER EXPONENTS

Let a be a nonzero real number and n be a positive integer. Then

-n =
1

an
.

That is, a -n is the reciprocal of an.

STUDY TIP

Mathematics often builds on concepts that have already been studied. Try to get in the regular
habit of reviewing topics from earlier parts of the text.

Negative Integers as Exponents
So far we have defined exponents that are whole numbers. For example,

50 = 1 and 23 = 2 # 2 # 2 = 8.
What if an exponent is a negative integer? To answer this question, consider Table 5.4,
which shows values for decreasing powers of 2. Note that each time the exponent on 2
decreases by 1, the resulting value is divided by 2.

3335.5 INTEGER EXPONENTS AND

THE QUOTIENT RULE

EXAMPLE 1 Evaluating negative exponents

Simplify each expression.
(a) 2-3 (b) 7-1 (c) x -2 (d) (x + y)-8

Solution

(a) Because a�n =
1

an
, 2�3 =

23
=

2 # 2 # 2 =
1

8
.

(b) 7�1 =
1

1
=

1
7

x 2

(d) (x + y)�8 =
1

(x + y)8

Now Try Exercises 7, 19, 25(b)

TECHNOLOGY NOTE

Negative Exponents
Calculators can be used to evaluate negative exponents. The figure shows how a graphing cal-
culator evaluates the expressions in parts (a) and (b) of Example 1.

2^(�3)�Frac
1/8

7^(�1)�Frac
1/7

CALCULATOR HELP
To use the fraction feature (Frac), see
Appendix A (pages AP-1 and AP-2).

The rules for exponents discussed in this chapter so far also apply to expressions hav-
ing negative exponents. For example, we can apply the product rule, am # an = am + n, as
follows.

Add

2-3 # 22 = 2-3 + 2 = 2-1 = 1
2

We can check this result by evaluating the expression without using the product rule.

2-3 # 22 = 1
23

# 22 = 1
8

# 4 = 4
8
=

1
2

EXAMPLE 2 Using the product rule with negative exponents

Evaluate each expression.
(a) 52 # 5-4 (b) 3-2 # 3-1

Solution
T

Add

(a) 52 # 5�4 = 52 + (�4) = 5�2 = 1
52

=
1
25

(b) 3-2 # 3-1 = 3-2 + (-1) = 3-3 = 1
33

=
1

27
Now Try Exercise 9

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334 CHAPTER 5 POLYNOMIALS AND EXPONENTS

EXAMPLE 3 Using the rules of exponents

Simplify the expression. Write the answer using positive exponents.
(a) x 2 # x -5 (b) ( y3)-4 (c) (rt)-5 (d) (ab)-3(a -2b)3

Solution
(a) Using the product rule, am # an = am�n, gives

x 2 # x�5 = x 2 + (�5) = x�3 = 1
x 3

(b) Using the power rule, (am)n = amn, gives

( y3)�4 = y3(�4) = y�12 =
1

y12
.

(rt)�5 = r�5t�5 =
1

r 5
# 1

t 5
=

r 5t 5

This expression could also be simplified as follows.

(rt)-5 =
1

(rt)5
=

1
r 5t 5

(d) (ab)-3(a -2b)3 = a -3b -3a -6b 3

= a -3 + (-6)b -3 + 3

= a -9b 0

=
1

a9
# 1

=
1
a9

Now Try Exercises 21, 27, 29(a)

The Quotient Rule
Consider the division problem

32
=

3 # 3 # 3 # 3
3 # 3 =

3
3

# 3
3

# 3 # 3 = 1 # 1 # 32 = 32.
Because there are two more 3s in the numerator than in the denominator, the result is

T
Subtract

34 – 2 = 32.

That is, to divide exponential expressions having the same base, subtract the exponent of
the denominator from the exponent of the numerator and keep the same base. This rule is
called the quotient rule, which we express in symbols as follows.

THE QUOTIENT RULE

For any nonzero number a and integers m and n,

an
= am – n.

3355.5 INTEGER EXPONENTS AND THE QUOTIENT RULE

EXAMPLE 4 Using the quotient rule

Simplify each expression. Write the answer using positive exponents.

(a)

45
(b)

6a7

3a4
(c)

xy7

x 2y5
Solution

T
Subtract

(a)
43

45
= 43 – 5 = 4-2 =

42
=

1
16

(b)
6a7

3a4
=

3
# a7

a4
= 2a7 – 4 = 2

x 2y5
=

x 1

x 2
# y7

y5
= x 1 – 2y7 – 5 = x -1y2 =

y2
x

Now Try Exercises 13(b), 31(b), 33(a)

MAKING CONNECTIONS

The Quotient Rule and Simplifying Quotients

Some quotients can be simplified mentally. Because

x 5

x 3
=

x # x # x # x # x
x # x # x ,

the quotient x
5

x 3
has five factors of x in the numerator and three factors of x in the denominator.

There are two more factors of x in the numerator than in the denominator, 5 – 3 = 2, so
this expression simplifies to x 2. Similarly,

x 3

x 5
=

x # x # x
x # x # x # x # x

has two more factors of x in the denominator than in the numerator. This quotient x
3

x 5
sim-

plifies to 1
x 2

. Use this technique to simplify the expressions

z4
,

a8
, and

x 6y2

x 3y7
.

Other Rules for Exponents
Other rules can be used to simplify expressions with negative exponents.

QUOTIENTS AND

NEGATIVE EXPONENTS

The following three rules hold for any nonzero real numbers a and b and positive
integers m and n.

1.
1

a -n

= an 2.

a -n

b -m
=

b m

an
3. aa

b
b-n = ab

a
bn

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336 CHAPTER 5 POLYNOMIALS AND EXPONENTS

We demonstrate the validity of these rules as follows.

1.
1

a -n
=

1
1
an

= 1 # a
n

1
= an

2.
a -n

b -m
=
1
an
1
b m
=
1

an
# b m

1
=
b m
an

3. aa
b
b-n = a -n

b -n
=

1
an
1
b n
=
1

an
# b n

1
=
b n

an
= ab

a
bn

EXAMPLE 5 Working with quotients and negative exponents

Simplify each expression. Write the answer using positive exponents.

(a)
1

2-5
(b)

3-3

4-2
(c)

5x -4y2

10x 2y -4
(d) a 2

z2
b-4

Solution
(a)
1

2�5
= 25 = 2 # 2 # 2 # 2 # 2 = 32 (b) 3

�3

4�2
=

42
33
=
16
27

10x 2y�4
=

y2y4

2 x 2x 4
=

2 x 6
(d) a 2

z2
b�4 = az2

2
b4 =

24
=

z8
16

Now Try Exercises 15(b), 17, 37, 47

The rules for natural number exponents that are summarized in Section 5.1 on page 303
also hold for integer exponents. Additional rules for integer exponents are summarized as
follows.

RULES FOR INTEGER EXPONENTS

The following rules hold for nonzero real numbers a and b, and positive integers m and n.

Description Rule Example

Negative Exponents (1)

a -n =
1

an
9-2 =

92
=

The Quotient Rule
am

an
= am – n

2-2
= 23 – (-2) = 25

Negative Exponents (2)
1

a -n
= an

7-5
= 75

Negative Exponents (3)
a -n

b -m
=
b m

4-3

2-5
=

25
43

Negative Exponents (4) aa
b
b – n = ab

a
bn a1

5
b – 2 = a5

1
b2 = 25

3375.5 INTEGER EXPONENTS AND THE QUOTIENT RULE

Scientific Notation
Powers of 10 are important because they are used in science to express numbers that are
either very small or very large in absolute value. Table 5.5 lists some powers of 10. Note that
if the power of 10 decreases by 1, the result decreases by a factor of 110, or equivalently, the
decimal point is moved one place to the left. Table 5.6 shows the names of some important
powers of 10.

READING CHECK

• What kinds of numbers are
often expressed in scientific
notation?

TABLE 5.5 Powers of 10

Power of 10 Value

103 1000

102 100

101 10

100 1

10-1 110 = 0.1

10-2 1100 = 0.01

10-3 11000 = 0.001

TABLE 5.6 Important Powers of 10

Power of 10 Name

1012 Trillion

109 Billion

106 Million

103 Thousand

10-1 Tenth

10-2 Hundredth

10-3 Thousandth

10-6 Millionth

Recall that numbers written in decimal notation are sometimes said to be in standard
form. Decimal numbers that are either very large or very small in absolute value can be
expressed in scientific notation.

TECHNOLOGY NOTE

Powers of 10
Calculators make use of sci-
entific notation, as illustrated
in the accompanying figure.
The letter E denotes a power
of 10. That is,

2.5E13 = 2.5 : 1013 and
5E�6 = 5 : 10�6.

25000000000000
2.5E13

.000005
5E�6

Note: The calculator has
been set in scientific mode.

REAL-WORLD CONNECTION As mentioned in A Look Into Math for this section, the distance
to the planet WASP-17b is about 5,880,000,000,000,000 miles. This distance can be written
in scientific notation as 5.88 * 1015 because

5,880,000,000,000,000 = 5.88 * 1015.
8
15 decimal places

The 1015 indicates that the decimal point in 5.88 should be moved 15 places

to the right.

A typical virus is about 0.00000468 inch in diameter, which can be written in scientific

notation as 4.68 * 10-6 because

0.00000468 = 4.68 * 10-6.
(1)1*
6 decimal places

The 10�6 indicates that the decimal point in 4.68 should be moved 6 places to the left.
The following definition provides a more complete explanation of scientific notation.

SCIENTIFIC NOTATION

A real number a is in scientific notation when a is written in the form b * 10n,
where 1 … 0b 0 6 10 and n is an integer.

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338 CHAPTER 5 POLYNOMIALS AND EXPONENTS

EXAMPLE 6 Converting scientific notation to standard form

Write each number in standard form.
(a) 5.23 * 104 (b) 8.1 * 10-3 (c) 6 * 10-2

Solution
(a) The positive exponent 4 indicates that the decimal point in 5.23 is to be moved 4 places

to the right.

5.23 * 104 = 5. 2 3 0 0. = 52,300
1 2 3 4

(b) The negative exponent – 3 indicates that the decimal point in 8.1 is to be moved
3 places to the left.

8.1 * 10�3 = 0. 0 0 8. 1 = 0.0081
1 2 3

Now Try Exercises 61, 63

The following steps can be used for writing a positive number a in scientific notation.

NOTE: The scientific notation for a negative number a is the opposite of the scientific nota-
tion of 0a 0 . For example, 450 = 4.5 * 102 and – 450 = – 4.5 * 102.

WRITING A POSITIVE NUMBER IN SCIENTIFIC NOTATION

For a positive, rational number a expressed as a decimal, if 1 … a 6 10, then
a = a * 100. Otherwise, use the following process to write a in scientific notation.

1. Move the decimal point in a until it becomes a number b such that 1 … b 6 10.
2. Let the positive integer n be the number of places the decimal point was moved.
3. Write a in scientific notation as follows.
• If a Ú 10, then a = b * 10n.
• If a 6 1, then a = b * 10-n.

EXAMPLE 7 Writing a number in scientific notation

Write each number in scientific notation.
(a) 308,000,000 (U.S. population in 2010)
(b) 0.001 (Approximate time in seconds for sound to travel one foot)

Solution
(a) Move the assumed decimal point in 308,000,000 eight places to obtain 3.08.

3. 0 8 0 0 0 0 0 0.

1 2 3 4 5 6 7 8

Since 308,000,000 Ú 10, the scientific notation is 3.08 * 108.

3395.5 INTEGER EXPONENTS AND THE QUOTIENT RULE

Numbers in scientific notation can be multiplied by applying properties of real numbers
and properties of exponents.

(6 * 104) # (3 * 103) = (6 # 3) * (104 # 103) Properties of real numbers
= 18 * 107 Product rule
= 1.8 * 108 Scientific notation

Division can also be performed with scientific notation.

6 * 104

3 * 103
=

6
3

*
104

103
Property of fractions

= 2 * 101 Quotient rule

These results are supported in Figure 5.16, where the calculator is in scientific mode.

In the next example we show how to use scientific notation in an application.

CALCULATOR HELP
To display numbers in scientific
notation, see Appendix A (page
AP-2).

Figure 5.16

(6�10^4)(3�10^3)

1.8E8
(6�10^4)/(3�10^3
)

2E1

EXAMPLE 8 Analyzing the cost of Internet advertising

In 2009, a total of $2.38 * 1010 was spent on Internet advertising in the United States. At
that time the population of the United States was 3.05 * 108. Determine how much was
spent per person on Internet advertising. (Source: New York Times.)

Solution
To determine the amount spent per person, divide $2.38 * 1010 by 3.05 * 108.

2.38 * 1010

3.05 * 108
=

2.38

3.05
* 1010 – 8 � 0.78 * 102 = 78

In 2009, about $78 per person was spent on Internet advertising.

Now Try Exercise 97

CRITICAL THINKING

Estimate the number of
seconds that you have been
alive. Write your answer in
scientific notation.

(b) Move the decimal point in 0.001 three places to obtain 1.

0. 0 0 1.

1 2 3

Since 0.001 6 1, the scientific notation is 1 * 10�3.

Now Try Exercises 75, 79

5.5 Putting It All Together

For the rules for integer exponents in this table, assume that a and b are nonzero real num-
bers and that m and n are integers.

CONCEPT EXPLANATION EXAMPLES

Negative Integer
Exponents a

-n =
1

an
2-4 =

1
24
=
1

16
, a -8 =

a8
, and

(xy)-2 =
1

(xy)2
=

x 2y2

continued on next page

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CHAPTER 5 POLYNOMIALS AND EXPONENTS340

CONCEPT EXPLANATION EXAMPLES

Quotient Rule am

an
= am – n 7

74
= 72 – 4 = 7-2 =

72
=

49
and

x 6

x 3
= x 6 – 3 = x 3

Quotients and Negative
Integer Exponents

1.
1

a -n
= an

2.
a -n
b -m
=
b m
an

3. aa
b
b-n = ab

a
bn
1.
1

5-2
= 52 = 25

x -4

y -2

y2
x 4

3. a2
3
b-3 = a3

2
b3 = 33

23
=
27
8

Scientific Notation Write a as b * 10n, where
1 … 0b 0 6 10 and n is an integer. 23,500 = 2.35 * 104,0.0056 = 5.6 * 10-3, and

1000 = 1 * 103

5.5 Exercises

CONCEPTS AND VOCABULARY

Exercises 1–5: Complete the given rule for integer expo-
nents m and n, where a and b are nonzero real numbers.

1. a -n = 2.
1

a -n
=

3.
am

an
= 4.

a -n
b -m
=

5. aa
b
b-n =

6. To write a positive number a in scientific notation as
b * 10n, the number b must satisfy _____.

NEGATIVE EXPONENTS

Exercises 7–18: Simplify the expression.

7. (a) 4-1 (b) a1
3
b-2

8. (a) 6-2 (b) 2.5-1

9. (a) 23 # 2-2 (b) 10-1 # 10-2

10. (a) 3-4 # 32 (b) 104 # 10-2
11. (a) 3-2 # 3-1 # 3-1 (b) (23)-1
12. (a) 2-3 # 25 # 2-4 (b) (3-2)-2

13. (a) (3243)-1 (b)
45

14. (a) (2-232)-2 (b)
55

15. (a)
19

17
(b)

4-3

16. (a)
– 64

6
(b)

6-2

17. (a)
5-2

5-4
(b) a2

7
b-2

18. (a)
7-3

7-1
(b) a3

4
b-3

continued from previous page

3415.5 INTEGER EXPONENTS AND THE QUOTIENT RULE

Exercises 19–50: Simplify the expression. Write the
answer using positive exponents.

19. (a) x -1 (b) a -4

20. (a) y -2 (b) z-7

21. (a) x -2 # x -1 # x (b) a -5 # a -2 # a -1
22. (a) y -3 # y4 # y -5 (b) b 5 # b -3 # b -6
23. (a) x 2y -3x -5y6 (b) (xy)-3

24. (a) a -2b -6b 3a -1 (b) (ab)-1

25. (a) (2t)-4 (b) (x + 1)-7

26. (a) (8c)-2 (b) (a + b)-9

27. (a) (a -2)-4 (b) (rt 3)-2

28. (a) (4 x 3)-3 (b) (xy -3)-2

29. (a) (ab)2(a2)-3 (b)
x 4

x 2

30. (a) (x 3)-2(xy)4y -5 (b)
y9

31. (a)
a10

a -3
(b)

2z4

32. (a)
b 5

b -2
(b)

12 x 2

24 x 7

33. (a)
– 4 xy5

6 x

3y2

(b)

x -4

x -1

34. (a)
12a6b 2

8ab 3
(b)

y -2

y -7

35. (a)
10b -4

5b -5
(b) aa

b
b3

36. (a)
8a -2

2a -3
(b) a2 x

y
b5

37. (a)
6 x 2y -4

18 x -5y4
(b)

16a -3b -5

4a -8b

38. (a)
m2n4

3m-5n4
(b)

7x -3y -5

x -3y -2

39. (a)
1

y -5
(b)

2t -3

40. (a)
1

z-6
(b)

10b -5

41. (a)
3a4

(2a -2)3
(b)

(2b 5)-3

4b -6

42. (a)
(2 x 4)-2

5x -2
(b)

2y5

(3y -4)-2

43. (a)
1

(xy)-2
(b)

(a2b)-3

44. (a)
1

(ab)-1
(b)

(rt 4)-2

45. (a)
(3m4n)-2

(2mn-2)3
(b)

( – 4 x 4y)2

(xy -5)-3

46. (a)
(x 4y2)2

( – 2 x 2y -2)3
(b)

(m2n-6)-2

(4m2n-4)-3

47. (a) aa
b
b-2 (b) a u

4v
b-1

48. (a) a2 x
y
b-3 (b) a5u

3v
b-2

49. (a) a 3a4b
2ab -2

b-2 (b) a 4m4n
5m-3n2

b2
50. (a) a 2 x 4y2

3x 3y -3
b3 (b) a a -5b 2

2ab -2
b-2

51. Thinking Generally For positive integers m and n

show that
an

am
=

am – n
.

52. Thinking Generally For positive integers m and n

show that
a -n

a -m
= am – n.

SCIENTIFIC NOTATION

Exercises 53–58: (Refer to Table 5.6.) Write the value of
the power of 10 in words.

53. 103 54. 106

55. 109 56. 10-1

57. 10-2 58. 10-6

Exercises 59–70: Write the expression in standard form.

59. 2 * 103 60. 5 * 102

61. 4.5 * 104 62. 7.1 * 106

63. 8 * 10-3 64. 9 * 10-1

65. 4.56 * 10-4 66. 9.4 * 10-2

67. 3.9 * 107 68. 5.27 * 106

69. – 5 * 105 70. – 9.5 * 103

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342 CHAPTER 5 POLYNOMIALS AND EXPONENTS

Exercises 71–82: Write the number in scientific notation.

71. 2000 72. 11,000

73. 567 74. 9300

75. 12,000,000 76. 600,000

77. 0.004 78. 0.0008

79. 0.000895 80. 0.0123

81. – 0.05 82. – 0.934

Exercises 83–90: Evaluate the expression. Write the
answer in standard form.

83. (5 * 103)(3 * 102)

84. (2.1 * 102)(2 * 104)

85. ( – 3 * 10-3)(5 * 102)

86. (4 * 102)(1 * 103)(5 * 10-4)

87.
4 * 105

2 * 102
88.

9 * 102

3 * 106

89.
8 * 10-6

4 * 10-3
90.

6.3 * 102

2 * 10-3

APPLICATIONS

91. Light-year The distance that light travels in 1 year is
called a light-year. Light travels at 1.86 * 105 miles
per second, and there are about 3.15 * 107 seconds
in 1 year.
(a) Estimate the number of miles in 1 light-year.
(b) Except for the sun, Alpha Centauri is the near-

est star, and its distance is 4.27 light-years from
Earth. Estimate its distance in miles. Write your
answer in scientific notation.

92. Milky Way It takes 2 * 108 years for the sun to
make one orbit around the Milky Way galaxy. Write
this number in standard form.

93. Speed of the Sun (Refer to the two previous exer-
cises.) Assume that the sun’s orbit in the Milky Way
galaxy is circular with a diameter of 105 light-years.
Estimate how many miles the sun travels in 1 year.

94. Distance to the Moon The moon is about 240,000
miles from Earth.

(a) Write this number in scientific notation.
(b) If a rocket traveled at 4 * 104 miles per hour,

how long would it take for it to reach the moon?

95. Online Exploration In 1997, the creators of the

Internet search engine BackRub renamed it Google.
This new name is a play on the word googol, which
is a very large number. Look up a googol and write
it in scientific notation.

96. Online Exploration An astronomical unit (AU) is
based on the distance from Earth to the sun. Look
up the distance in kilometers from Earth to the sun.

(a) Write an astronomical unit in standard form to
the nearest million kilometers.

(b) Convert your rounded answer from part (a) to
scientific notation.

97. Gross Domestic Product The gross domestic prod-
uct (GDP) is the total national output of goods and
services valued at market prices within the United
States. The GDP of the United States in 2005 was
$12,460,000,000,000. (Source: Bureau of Economic Analysis.)

(a) Write this number in scientific notation.
(b) In 2005, the U.S. population was 2.98 * 108. On

average, how many dollars of goods and services
were produced by each individual?

98. Average Family Net Worth A family refers to a
group of two or more people related by birth, mar-
riage, or adoption who reside together. In 2000, the
average family net worth was $280,000, and there
were about 7.2 * 107 families. Calculate the total
family net worth in the United States in 2000. (Source:
U.S. Census Bureau.)

WRITING ABOUT MATHEMATICS

99. Explain what a negative exponent is and how it is
different from a positive exponent. Give an example.

100. Explain why scientific notation is helpful for writ-
ing some numbers.

3435.6 DIVISION OF POLYNOMIALS

Working with Real DataGroup Activity

Water in a Lake East Battle Lake in Minnesota covers
an area of about 1950 acres, or 8.5 * 107 square feet,
and its average depth is about 3.2 * 101 feet.
(a) Estimate the cubic feet of water in the lake. (Hint:

volume = area * average depth.)
(b) One cubic foot of water equals about 7.5 gallons.

How many gallons of water are in this lake?

(c) The population of the United States is about
3.1 * 108, and the average American uses about
1.5 * 102 gallons of water per day. Could this
lake supply the American population with water
for 1 day?

Directions: Form a group of 2 to 4 people. Select a person to record the group’s responses for
this activity. All members of the group should work cooperatively to answer the questions. If
your instructor asks for your results, each member of the group should be prepared to respond.

5.6 Division of Polynomials
Division by a Monomial ● Division by a Polynomial

A LOOK INTO MATH N The study of polynomials has occupied the minds of mathematicians for centuries. During
the sixteenth century, Girolamo Cardano and other Italian mathematicians discovered how
to solve higher degree polynomial equations. In this section we demonstrate how to divide
polynomials. Division is often needed to factor polynomials and to solve polynomial equa-
tions. (Source: H. Eves, An Introduction to the History of Mathematics.)

Division by a Monomial
To add two fractions with like denominators, we use the property

d
+

d
=

a + b
d

For example, 17 +
3
7 =

1 + 3
7 =

4
7.

To divide a polynomial by a monomial, we use the same property, only in reverse. That is,

a + b
d

=
a

d
+
b

d
.

Note that each term in the numerator is divided by the monomial in the denominator. The
next example shows how to divide a polynomial by a monomial.

Girolamo Cardano (1501–1576)

EXAMPLE 1 Dividing a polynomial by a monomial

Divide.

(a)
a5 + a3

a2
(b)

5x 4 +

10x

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344 CHAPTER 5 POLYNOMIALS AND EXPONENTS

Solution

(a)

a5 � a3

a2
=

a2
�

a2
= a5 – 2 + a3 – 2 = a3 + a

(b)
5x 4 � 10x

x
=

5x 4

10x
�

10x
10x
=
x 3

2
+ 1

6y
=

3y2

6y
�

6y
�
12
6y
=
y

2
+

3
–

2
y

Now Try Exercises 17, 19, 21

MAKING CONNECTIONS

Division and Simplification

A common mistake made when dividing expressions is to “cancel” incorrectly. Note in
Example 1(b) that

5x 4 + 10x
10x

3 5x 4 +
10x

10x
.

The monomial must be divided into every term in the numerator.

When dividing two natural numbers, we can check our work by multiplying. For exam-
ple, 105 = 2, and we can check this result by finding the product 5 # 2 = 10. Similarly,
to check

a5 � a3

a2
= a3 � a

in Example 1(a) we can multiply a2 and a3 � a.

a2(a3 � a) = a2 # a3 + a2 # a Distributive property
= a5 � a3 ✓ It checks.

EXAMPLE 2 Dividing and checking

Divide the expression 8 x
3 – 4 x 2 + 6 x

2 x 2
and then check the result.

Solution
Be sure to divide 2 x 2 into every term in the numerator.

8 x 3 � 4 x 2 � 6 x

2 x 2
=

8 x 3

2 x 2
�

4 x 2

2 x 2
�
6 x

2 x 2
= 4 x – 2 +

3
x

Check:

2 x 2a4 x – 2 + 3
x
b = 2 x 2 # 4 x – 2 x 2 # 2 + 2 x 2 # 3

= 8 x 3 – 4 x 2 + 6 x ✓

Now Try Exercise 23

3455.6 DIVISION OF POLYNOMIALS

EXAMPLE 3 Finding the length of a rectangle

The rectangle in Figure 5.17 has an area A = x 2 + 2 x and width x. Write an expression for
its length l in terms of x.

STUDY TIP

Do you have enough time
to study your notes and
complete your assign-
ments? One way to man-
age your time is to make
a list of your time com-
mitments and determine
the amount of time that
each activity requires.
Remember to include time
for eating, sleeping, and
relaxing!

Figure 5.17

x A = x2 + 2x

Solution
The area A of a rectangle equals length l times width w, or A = lw. Solving for l gives

l =
A
w

Thus to find the length of the given rectangle, divide the area by the width.

l =
x 2 + 2 x

x
=
x 2

x
+

2 x
x

= x + 2

The length of the rectangle is x + 2. The answer checks because x(x + 2) = x 2 + 2 x.

Now Try Exercise 49

To check this result, we find the product of the quotient and divisor and then add the remain-
der. Because 67 # 4 + 3 = 271, the answer checks. The quotient and remainder can also be
expressed as 6734. Division of polynomials is similar to long division of natural numbers.

Division by a Polynomial
To understand division by a polynomial better, we first need to review some terminology
related to long division of natural numbers. To compute 271 , 4, we complete long divi-
sion as follows.

67 R 3

4�271
24

Quotient h v

Remainder

Divisorh v Dividend

EXAMPLE 4 Dividing polynomials

Divide 6 x
2 + 13x + 3

3x + 2 and check.

Solution
Begin by dividing the first term of 3x + 2 into the first term of 6 x 2 + 13x + 3. That is,
divide 3x into 6 x 2 to obtain 2 x. Then find the product of 2 x and 3x + 2, or 6 x 2 + 4 x,
place it below 6 x 2 + 13x, and subtract. Bring down the 3.

2x
3x + 2�6 x 2 + 13x + 3

6 x 2 +

4 x

9x + 3

6 x 2

3x
= 2 x

2 x(3x + 2) = 6 x 2 + 4 x
Subtract: 13x – 4 x = 9x. Bring down the 3.

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346 CHAPTER 5 POLYNOMIALS AND EXPONENTS

In the next step, divide 3x into the first term of 9x + 3 to obtain 3. Then find the product of
3 and 3x + 2, or 9x + 6, place it below 9x + 3, and subtract.

2 x + 3
3x + 2�6 x 2 + 13x + 3

6 x 2 + 4 x
9x + 3
9x + 6

– 3

9x
3x

= 3

3(3x + 2) = 9x + 6
Subtract: 3 – 6 = – 3.

The quotient is 2 x + 3 with remainder – 3. This result can also be written as

2 x + 3 +
– 3

3x + 2
, Quotient +

Remainder

Divisor

in the same manner that 67 R 3 was written as 6734.
Check polynomial division by adding the remainder to the product of the divisor and

the quotient. That is,

(Divisor)(Quotient) + Remainder = Dividend.

For this example, the equation becomes

(3x � 2)(2 x � 3) + (�3) = 3x # 2 x + 3x # 3 + 2 # 2 x + 2 # 3 � 3
= 6 x 2 + 9x + 4 x + 6 – 3
= 6 x 2 � 13x � 3. ✓ It checks.

Now Try Exercise 27

READING CHECK

• How can you check a polynomial division problem?

EXAMPLE 5 Dividing polynomials having a missing term

Simplify (3x 3 + 2 x – 4) , (x – 2).

Solution
Because the dividend does not have an x 2-term, insert 0x 2 as a “place holder.” Then begin
by dividing x into 3x 3 to obtain 3x 2.

3x 2

x – 2�3x 3 + 0 x 2 + 2 x – 4

3x 3 – 6 x 2

6 x 2 + 2 x

3x 3

x
= 3x 2

3x 2(x – 2) = 3x 3 – 6 x 2

Subtract 0x 2 – ( – 6 x 2) = 6 x 2. Bring down 2x.

In the next step, divide x into 6 x 2.

3x 2 + 6 x
x – 2�3x 3 + 0 x 2 + 2 x – 4

3x 3 – 6 x 2

6 x 2 + 2 x
6 x 2 – 12 x

14 x – 4

6 x 2

x
= 6 x

6 x(x – 2) = 6 x 2 – 12 x
Subtract: 2 x – ( – 12 x) = 14 x. Bring down – 4.

3475.6 DIVISION OF POLYNOMIALS

Now divide x into 14 x.

3x 2 + 6 x + 14
x – 2�3x 3 + 0 x 2 + 2 x – 4

3x 3 – 6 x 2

6 x 2 + 2 x
6 x 2 – 12 x

14 x – 4
14 x – 28

The quotient is 3x 2 + 6 x + 14 with remainder 24. This result can also be written as

3x 2 + 6 x + 14 +
24

x – 2
.

Now Try Exercise 37

EXAMPLE 6 Dividing when the divisor is not linear

Divide x 3 – 3x 2 + 3x + 2 by x 2 + 1.

Solution
Begin by writing x 2 + 1 as x 2 + 0x + 1.

x – 3
x 2 + 0x + 1�x 3 – 3x 2 + 3x + 2

x 3 + 0x 2 + x
– 3x 2 + 2 x + 2
– 3x 2 + 0x – 3

2 x + 5

The quotient is x – 3 with remainder 2 x + 5. This result can also be written as

x – 3 +
2 x + 5
x 2 + 1

.
Now Try Exercise 41

5.6 Putting It All Together

Division by a Monomial Use the property

a + b
d
=
a
d
+
b
d
.

Be sure to divide the denominator into
every term in the numerator.

2 x 3 + 4 x
2 x 2

=
2 x 3

2 x 2
+

4 x

2 x 2
= x +

2
x

and

a2 –

=
a2

4a
–

4a
=

4
–

1
2
CONCEPT EXPLANATION EXAMPLES
continued on next page

14 x
x

= 14

14(x – 2) = 14 x – 28
Subtract: – 4 – ( – 28) = 24.

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley.
Copyright © 2013 by Pearson Education, Inc.

CONCEPT EXPLANATION EXAMPLES

Division by a
Polynomial

Is done similarly to the way long divi-
sion of natural numbers is performed

If either the divisor or the dividend is
missing a term, be sure to insert as a
“place holder” the missing term with
coefficient 0.

Divide x 2 + 3x + 3 by x + 1.

x + 2
x + 1�x 2 + 3x + 3

x 2 + x
2 x + 3
2 x + 2

The quotient is x + 2 with remainder 1,
which can be expressed as

x + 2 +
1

x + 1
.

Checking a Result Dividend = (Divisor)(Quotient) +
Remainder

When x 2 + 3x + 3 is divided by x + 1,
the quotient is x + 2 with remainder 1. Thus

(x + 1)(x + 2) + 1 = x 2 + 3x + 3,

and the answer checks.

continued from previous page

CHAPTER 5 POLYNOMIALS AND EXPONENTS348

CONCEPTS AND VOCABULARY

1. a + bd = 2.
a + b – c

d =

3. When dividing a polynomial by a monomial, the mono-
mial must be divided into every of the polynomial.

4. (True or False?) The expressions 5x
2 + 2 x

2 x and 5x
2 + 1

are

equal.

5. (True or False?) The expressions 5x
2 + 2 x

2 x and
5x 2
2 x are

equal.

6. Because 379 = 4 with remainder 1, it follows that
37 = # + .

7. Because 2 x 3 – x + 5 divided by x + 1 equals
2 x 2 – 2 x + 1 with remainder 4, it follows that
2 x 3 – x + 5 = _____ # _____ + _____.

8. When dividing 2 x 3 + 3x – 1 by x – 1, insert

into the dividend as a “place holder” for the missing
x 2-term.

DIVISION BY A MONOMIAL

Exercises 9–16: Divide and check.

9.
6 x 2

3x
10.

– 5x 2

10x 4

11.
z4 + z3

z
12.

t 3 – t
t

13.
a5 – 6a3

2a3
14.

b 4 – 4b
4b 2

15.
y + 6y2

3y3
16.

8z2 – z
4z2

Exercises 17–26: Divide.

17.
4 x – 7x 4

x 2
18.

1 + 6 x 4

3x 3

19.
6y2 + 3y

3y3
20.

5z2 – 10z3

5z4

5.6 Exercises

21.
9x 4 – 3x + 6

3x
22.

y3 – 4y + 6
y

23.
12y4 – 3y2 + 6y

3y2
24.

2 x 2 – 6 x + 9
12 x

25.
15m4 – 10m3 + 20m2

5m2
26.

n8 – 8n6 + 4n4

2n5

3495.6 DIVISION OF POLYNOMIALS

47. Thinking Generally If the quotient in a polynomial
division problem is an integer, what must be true
about the degrees of the dividend and divisor?

48. Thinking Generally If the quotient in a polynomial
division problem is a polynomial of degree 1, what must
be true about the degrees of the dividend and divisor?

APPLICATIONS

Exercises 49 and 50: Area of a Rectangle The area of a
rectangle and its width are given. Find an expression for
the length l.

49. 50.

51. Volume of a Box The volume V of a box is 2 x 3 + 4 x 2,
and the area of its bottom is 2 x 2. Find the height of the
box in terms of x. Make a possible sketch of the box,
and label the length of each side.

52. Area of a Triangle A triangle has height h and area
A = 2h 2 – 4h. Find its base b in terms of h. Make
a possible sketch of the triangle, and label the height
and base. (Hint: A = 12 bh.)

WRITING ABOUT MATHEMATICS

53. Suppose that one polynomial is divided into another
polynomial and the remainder is 0. What does the
product of the divisor and quotient equal? Explain.

54. A student simplifies the expression 4 x
3 – 1
4 x 2

to x – 1.
Explain the student’s error.

2x A = 8×2

l
l

x – 1 A = x2 – 1

27.
2 x 2 – 3x + 1

x – 2
28.

4 x 2 – x + 3
x + 2

Exercises 27–34: Divide and check.

29.
x 2 + 2 x + 1

x + 1
30.

4 x 2 – 4 x + 1
2 x – 1

31.
x 3 – x 2 + x – 2

x – 1
32.

2 x 3 + 3x 2 + 3x – 1
2 x + 1

33.
x 3 + x 2 – 7x + 2

x – 2

34.
x 3 + x 2 – 2 x + 12

x + 3

Exercises 35– 46: Divide.

35.
4 x 3 – 3x 2 + 7x + 3

4 x + 1

36.
10x 3 – x 2 – 17x – 7

5x + 2

37.
x 3 – x + 2

x – 2

38.
6 x 3 + 8 x 2 + 4

3x + 4

39. (3x 3 + 2) , (x – 1)

40. ( – 3x 3 + 8 x 2 + x) , (3x + 4)

41. (x 3 + 3x 2 + 1) , (x 2 + 1)

42. (x 4 – x 3 + x 2 – x + 1) , (x 2 – 1)

43.
x 3 + 1

x 2 – x + 1
44.

4 x 3 + 3x + 2
2 x 2 – x + 1

45.
x 3 + 8
x + 2

46.
x 4 – 16
x – 2

Checking Basic ConceptsSECTIONS5.5 and 5.6

1. Simplify each expression. Write the result with
positive exponents.

(a) 9-2 (b)
3

x -3

6 x 4
(c) (4ab -4)-2

2. Simplify each expression. Write the result with
positive exponents.

(a)
1

z-5
(b)

x -3

y -6
(c) a 3

x 2
b-3

3. Write each number in scientific notation.
(a) 45,000 (b) 0.000234 (c) 0.01

4. Write each expression in standard form.
(a) 4.71 * 104 (b) 6 * 10-3

5. Simplify 25a
4 – 15a 3
5a 3

6. Divide 3x 2 – x – 4 by x – 1. State the quotient
and remainder.

7. Divide x 4 + 2 x 3 – 2 x 2 – 5x – 2 by x 2 – 3.
State the quotient and remainder.

8. Distance to the Sun The distance to the sun is

approximately 93 million miles.
(a) Write this distance in scientific notation.
(b) Light travels at 1.86 * 105 miles per second.

How long does it take for the sun’s light to
reach Earth?

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley.
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350 CHAPTER 5 POLYNOMIALS AND EXPONENTS

CHAPTER 5 Summary
SECTION 5.1 . RULES FOR EXPONENTS

Bases and Exponents The expression b n has base b and exponent n and equals the expression
b # b # b # g # b, when n is a natural number.u

n times

Example: 23 has base 2 and exponent 3 and equals 2 # 2 # 2 = 8.
Evaluating Expressions When evaluating expressions, evaluate exponents before performing
addition, subtraction, multiplication, division, or negation. In general, operations within paren-
theses should be evaluated before using the order of operations.

1. Evaluate exponents.
2. Perform negation.
3. Do multiplication and division from left to right.
4. Do addition and subtraction from left to right.

Example: – 32 + 3 # 4 = – 9 + 3 # 4 = – 9 + 12 = 3
Zero Exponents For any nonzero number b, b 0 = 1. Note that 00 is undefined.

Examples: 50 = 1 and ax
y
b0 = 1, where x and y are nonzero.

Product Rule For any real number a and natural numbers m and n,

am # an = am + n.
Examples: 34 # 32 = 36 and x 3x 2x 4 = x 9
Power Rules For any real numbers a and b and natural numbers m and n,

(am)n = amn, (ab)n = anb n, and aa
b
bn = an

b n
, b � 0.

Examples: (x 2)3 = x 6, (3x)4 = 34x 4 = 81x 4, and a2
y
b3 = 23

y3
=

8
y3

SECTION 5.2 . ADDITION AND SUBTRACTION OF POLYNOMIALS

Terms Related to Polynomials
Monomial A number, variable, or product of numbers and variables

raised to natural number powers

Degree of a Monomial Sum of the exponents of the variables

Coefficient of a Monomial The number in a monomial

Example: The monomial – 3x 2y3 has degree 5 and coefficient – 3.

Polynomial A monomial or the sum of two or more monomials

Term of a Polynomial Each monomial is a term of the polynomial.

Binomial A polynomial with two terms

Trinomial A polynomial with three terms

Degree of a Polynomial The degree of the term with highest degree

351CHAPTER 5 SUMMARY

Opposite of a Polynomial The opposite is found by negating each term.

Example: 2 x 3 – 4 x + 5 is a trinomial with degree 3. Its
opposite is – 2 x 3 + 4 x – 5.

Like Terms Two monomials with the same variables raised to the same powers

Examples: 3xy2 and – xy2 are like terms.

5x 3 and 3x 3 are like terms.

5x 2 and 5x are unlike terms.

Addition of Polynomials Combine like terms, using the distributive property.

Example: (2 x 2 – 4 x) + ( – x 2 – x) = (2 – 1)x 2 + ( – 4 – 1)x
= x 2 – 5x

Subtraction of Polynomials Add the first polynomial to the opposite of the second polynomial.

Example: (4 x 4 – 5x) – (7x 4 + 6 x) = (4 x 4 – 5x) + ( – 7x 4 – 6 x)
= (4 – 7)x 4 + ( – 5 – 6)x
= – 3x 4 – 11x

SECTION 5.3 . MULTIPLICATION OF POLYNOMIALS

Multiplication of Monomials Use the commutative property and the product rule.

Examples: – 2 x 3 # 3x 2 = – 2 # 3 # x 3 # x 2 = – 6 x 5
(2 xy2)(3x 2y3) = 2 # 3 # x # x 2 # y2 # y3 = 6 x 3y5
c Assumed exponent of 1

Distributive Properties

a(b + c) = ab + ac and a(b – c) = ab – ac

Examples: 4 x(3x + 6) = 4 x # 3x + 4 x # 6 = 12 x 2 + 24 x
ab(a2 – b 2) = ab # a2 – ab # b 2 = a3b – ab 3
Multiplication of Monomials and Polynomials Apply the distributive properties. Be sure to
multiply every term in the polynomial by the monomial.

Example: – 2 x 2(4 x 2 – 5x – 3) = – 8 x 4 + 10x 3 + 6 x 2

Multiplication of Polynomials The product of two polynomials may be found by multiplying
every term in the first polynomial by every term in the second polynomial. Be sure to combine
like terms.

Examples: (x + 3)(2 x – 5) = 2 x 2 – 5x + 6 x – 15

= 2 x 2 + x – 15

(2 x + 1)(x 2 – 5x + 2) = 2 x 3 – 10x 2 + 4 x + x 2 – 5x + 2
= 2 x 3 – 9x 2 – x + 2

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley.
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352 CHAPTER 5 POLYNOMIALS AND EXPONENTS

SECTION 5.4 . SPECIAL PRODUCTS

Product of a Sum and Difference

(a + b)(a – b) = a2 – b 2

Examples: (x + 4)(x – 4) = x 2 – 16
(2r – 3t)(2r + 3t) = (2r)2 – (3t)2 = 4r 2 – 9t 2

Squaring Binomials

(a + b)2 = a2 + 2ab + b 2 and (a – b)2 = a2 – 2ab + b 2

Examples: (2 x + 1)2 = (2 x)2 + 2(2 x)1 + 12 = 4 x 2 + 4 x + 1
(z2 – 2)2 = (z2)2 – 2z2(2) + 22 = z4 – 4z2 + 4

Cubing Binomials To multiply (a + b)3, write it as (a + b)(a + b)2.

Example: (x + 4)3 = (x + 4)(x + 4)2

= (x + 4)(x 2 + 8 x + 16) Square the binomial.
= x 3 + 8 x 2 + 16 x + 4 x 2 + 32 x + 64 Distributive property
= x 3 + 12 x 2 + 48 x + 64 Combine like terms.

SECTION 5.5 . INTEGER EXPONENTS AND THE QUOTIENT RULE

Negative Integers as Exponents For any nonzero real number a and positive integer n,

a -n =
1
an
.

Examples: 5-2 =
1

52
and x -4 =

1
x 4

The Quotient Rule For any nonzero real number a and integers m and n,

am
an
= am – n.

Examples:
64

62
= 64 – 2 = 62 = 36 and

xy3

x 4y2
= x 1 – 4y3 – 2 = x -3y1 =

y
x 3

Other Rules For any nonzero real numbers a and b and positive integers m and n,

a -n
= an,

a -n
b -m
=
b m

an
, and aa

b
b-n = ab

a
bn.

Examples:
1

4-3
= 43,

x -3

y -2
=

x 3
, and a4

5
b-2 = a5

4
b2

Scientific Notation A real number a written as b * 10n, where 1 … 0b 0 6 10 and n is an integer
Examples: 2.34 * 103 = 2340 Move the decimal point 3 places to the right.
2.34 * 10-3 = 0.00234 Move the decimal point 3 places to the left.

SECTION 5.6 . DIVISION OF POLYNOMIALS

Division of a Polynomial by a Monomial Divide the monomial into every term of the polynomial.

Example:
5x 3 – 10x 2 +

15x

5x
=

5x 3

5x
–

10x 2

5x
+

15x

5x
= x 2 – 2 x + 3

353CHAPTER 5 REVIEW EXERCISES

Division of a Polynomial by a Polynomial Division of polynomials is performed similarly to
long division of natural numbers.

Example: Divide 2 x 3 + 4 x 2 – 3x + 1 by x + 1.

2 x 2 + 2 x – 5
x + 1�2 x 3 + 4 x 2 – 3x + 1

2 x 3 + 2 x 2

2 x 2 – 3x
2 x 2 + 2 x

– 5x + 1
– 5x – 5

The quotient is 2 x 2 + 2 x – 5 with remainder 6, which can be written as

2 x 2 + 2 x – 5 +
6

x + 1
.

CHAPTER 5 Review Exercises
SECTION 5.1

Exercises 1– 6: Evaluate the expression.

1. 53 2. – 34

3. 4( – 2)0 4. 3 + 32 – 30

5.
– 52

5
6. a – 5

5
b2

Exercises 7–24: Simplify the expression.

7. 62 # 63 8. 105 # 107
9. z4 # z5 10. y2 # y # y3
11. 5x 2 # 6 x 7 12. (ab 3)(a3b)
13. (25)2 14. (m4)5

15. (ab)3 16. (x 2y3)4

17. (xy)3(x 2y4)2 18. (a2b 9)0

19. (r – t)4(r – t)5 20. (a + b)2(a + b)4

21. a 3
x – yb2 22. ax + y2 b3

23. 2 x 2(3x – 5) 24. 3x(4 x + x 3)

SECTION 5.2

Exercises 25 and 26: Identify the degree and coefficient
of the monomial.

25. 6 x 7 26. – x 2y3

Exercises 27–30: Determine whether the expression is a
polynomial. If it is, state how many terms and variables
the polynomial contains. Then state its degree.

27. 8y 28. 8 x 3 – 3x 2 + x – 5

29. a2 + 2ab + b 2 30.
1
xy

31. Add the polynomials vertically.

3x 2 + 4 x + 8
2 x 2 – 5x – 5

32. Write the opposite of 6 x 2 – 3x – 7.

Exercises 33– 40: Simplify.

33. (4 x – 3) + ( – x + 7)

34. (3x 2 – 1) – (5x 2 + 12)

35. (x 2 + 5x + 6) – (3x 2 – 4 x + 1)

36. (x 2 + 3x – 5) + (2 x 2 – 5x – 1)

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354 CHAPTER 5 POLYNOMIALS AND EXPONENTS

37. (a3 + 4a2) + (a3 – 5a2 + 7a)

38. (4 x 3 – 2 x + 6) – (4x 3 – 6)

39. (xy + y2) + (4y2 – 4 xy)

40. (7x 2 + 2 xy + y2) – (7x 2 – 2 xy + y2)

SECTION 5.3

Exercises 41–54: Multiply and simplify.

41. – x 2 # x 3 42. – (r 2t 3)(rt)
43. – 3(2t – 5) 44. 2y(1 – 6y)

45. 6 x 3(3x 2 + 5x)

46. – x(x 2 – 2 x + 9)

47. – ab(a2 – 2ab + b 2)

48. (a – 2)(a + 5)

49. (8 x – 3)(x + 2) 50. (2 x – 1)(1 – x)

51. ( y2 + 1)(2y + 1) 52. ( y2 – 1)(2y2 + 1)

53. (z + 1)(z2 – z + 1)

54. (4z – 3)(z2 – 3z + 1)

Exercises 55 and 56: Multiply the expression
(a) geometrically and
(b) symbolically.

55. z(z + 1) 56. 2 x(x + 2)

SECTION 5.4

Exercises 57–72: Multiply.

57. (z + 2)(z – 2) 58. (5z – 9)(5z + 9)

59. (1 – 3y)(1 + 3y) 60. (5x + 4y)(5x – 4y)

61. (rt + 1)(rt – 1)

62. (2m2 – n2)(2m2 + n2)

63. (x + 1)2 64. (4 x + 3)2

65. ( y – 3)2 66. (2y – 5)2

67. (4 + a)2 68. (4 – a)2

69. (x 2 + y2)2

70. (xy – 2)2

71. (z + 5)3

72. (2z – 1)3

Exercises 73 and 74: Use the product of a sum and a dif-
ference to evaluate the expression.

73. 59 # 61 74. 22 # 18
SECTION 5.5

Exercises 75–82: Simplify the expression.

75. 9-1 76. 3-2

77. 43 # 4-2 78. 10-6 # 103

79.
1

6-2
80.

81. (3-1 22)-2 82. (2-4 53)0 1

Exercises 83–98: Simplify the expression. Write the answer
using positive exponents.

83. z-2 84. y -4

85. a -4 # a2 86. x 2 # x -5 # x
87. (2t)-2 88. (ab 2)-3

89. (xy)-2(x -2y)-1 90.
x 6

x 2

91.
4 x

2 x 4
92.

20x 5y3

30xy6

93. aa
b
b5 94. 4

t -4

95.
(3m3n)-2

(2m2n-3)3
96. a x -4y2

3xy -3
b-2

97. ax
y
b-2 98. a3u

2v
b-1

Exercises 99–102: Write the expression in standard form.

99. 6 * 102 100. 5.24 * 104

101. 3.7 * 10-3 102. 6.234 * 10-2

Exercises 103–106: Write the number in scientific notation.

103. 10,000 104. 56,100,000

105. 0.000054 106. 0.001

Exercises 107 and 108: Evaluate the expression. Write
the result in standard form.

107. (4 * 102)(6 * 104) 108.
8 * 103

4 * 104

SECTION 5.6

Exercises 109–116: Divide and check.

109.
5x 2 + 3x

3x
110.

6b 4 – 4b 2 + 2
2b 2

111.
3x 2 – x + 2

x – 1
112.

9x 2 – 6 x – 2
3x + 2

355CHAPTER 5 REVIEW EXERCISES

114.
2 x 3 – x 2 – 1

2 x – 1
115.

x 3 – x 2 – x + 1
x 2 + 1

–

116.
x 4 + 3x 3 + 8 x 2 + 7x + 5

x 2 + x + 1

113.
4 x 3 – 11x 2 – 7x – 1

4 x + 1

APPLICATIONS

117. Heart Rate An athlete starts running and continues
for 10 seconds. The polynomial 12 t

2 + 60 calculates
the heart rate of the athlete in beats per minute t sec-
onds after beginning the run, where t … 10.
(a) What is the athlete’s heart rate when the athlete

first starts to run?
(b) What is the athlete’s heart rate after 10 seconds?
(c) What happens to the athlete’s heart rate while

the athlete is running?

118. Areas of Rectangles Find a monomial equal to the
sum of the areas of the rectangles. Calculate this
sum for x = 3 feet and y = 4 feet.

2x
y
2x
y
2x
y

119. Area of a Rectangle Write a polynomial that gives
the area of the rectangle. Calculate its area for
z = 6 inches.

2
2
5
3

120. Area of a Square Find the area of the square whose
sides have length x 2y.

x2y

121. Compound Interest If P dollars are deposited
in an account that pays 6% annual interest, then
the amount of money after 3 years is given by
P(1 + 0.06)3. Find this amount when P = $700.

122. Volume of a Sphere The expression for the volume
of a sphere with radius r is 43 pr

3. Find a polynomial
that gives the volume of a sphere with radius x + 2.
Leave your answer in terms of p.

x + 2

123. Height Reached by a Baseball A baseball is hit
straight up. Its height h in feet above the ground
after t seconds is given by t(96 – 16t).
(a) Multiply this expression.
(b) Evaluate both the expression in part (a) and the

given expression for t = 2. Interpret the result.

124. Rectangular Building A rectangular building has a
perimeter of 1200 feet.
(a) If one side of the building has length L, write

a polynomial expression that gives its area. (Be
sure to multiply your expression.)

(b) Evaluate the expression in part (a) for L = 50
and interpret the answer.

125. Geometry Complete each part and verify that your
answers are equal.
(a) Find the area of the large square by multiplying

its length and width.
(b) Find the sum of the areas of the smaller rect-

angles inside the large square.
5
x 5
x

126. Digital Picture A digital picture, including its border,
is x + 4 pixels by x + 4 pixels, and the actual picture
inside the border is x – 4 pixels by x – 4 pixels.

x + 4

x – 4

x + 4
x – 4

(a) Find a polynomial that gives the number of pix-
els in the border.

(b) Let x = 100 and evaluate the polynomial.

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356 CHAPTER 5 POLYNOMIALS AND EXPONENTS

127. Federal Debt In 1990, the federal debt held by the
public was $2.19 trillion, and the population of the
United States was 249 million. Use scientific nota-
tion to approximate the national debt per person.
(Source: U.S. Department of the Treasury.)

128. Alcohol Consumption In 2007, about 239 million
people in the United States were age 14 or older.
They consumed, on average, 2.31 gallons of alcohol
per person. Use scientific notation to estimate the
total number of gallons of alcohol consumed by this
age group. (Source: Department of Health and Human Services.)

1. Simplify each expression.
(a) – 50 (b) – 92

2. Evaluate each expression by hand.
(a) – 42 + 10 (b) 8-2

2-3
(d) – 3x 0

3. State how many terms and variables the polynomial
5x 2 – 3xy – 7y3 contains. Then state its degree.

4. Write the opposite of – x 3 + 4 x – 8.

Exercises 5–8: Simplify.

5. ( – 3x + 4) + (7x + 2)

6. ( y3 – 2y + 6) – (4y3 + 5)

7. (5x 2 – x + 3) – (4 x 2 – 2 x + 10)

8. (a3 + 5ab) + (3a3 – 3ab)

Exercises 9–16: Write the given expression with positive
exponents.

9. 6y4 # 4y7 10. (a2b 3)2(ab 2)
11. x 7 # x -3 12. (a -1b 2)-3

13. ab(a2 – b 2) 14. a 3a2
2b -3

b-2
15.

12 xy4

6 x 2y
16. a 2

a + b
b4

Exercises 17–22: Multiply and simplify.

24. Write 6.1 * 10-3 in standard form.

25. Write 5410 in scientific notation.

Exercises 26 and 27: Divide.

CHAPTER 5 Test Step-by-step test solutions are found on the Chapter Test Prep Videos available via the Video Resources on DVD, in , and on (search “RockswoldBeginAlg” and click on “Channels”).

17. 3x 2(4 x 3 – 6 x + 1)

19. (7y2 – 3)(7y2 + 3) 20. (3x – 2)2

18. (z – 3)(2z + 4)

21. (m + 3)3

22. ( y + 2)( y2 – 2y + 3)

23. Evaluate 78 # 82 using the product of a sum and a
difference.

26.
9x 3 – 6 x 2 + 3x

3x 2
27.

x 3 + x 2 – x + 1
x + 2

28. Concert Tickets Tickets for a concert are sold for

$20 each.
(a) Write a polynomial that gives the revenue from

selling t tickets.
(b) Putting on the concert costs management $2000

to hire the band plus $2 for each ticket sold. What
is the total cost of the concert if t tickets are sold?

(c) Subtract the polynomial that you found in part (b)
from the polynomial that you found in part (a).
What does this polynomial represent?

29. Areas of Rectangles Find a polynomial representing

the sum of the areas of two identical rectangles that
have width 2 x and length 3x. Calculate this sum for
x = 10 feet.

3x
2x
3x
2x

30. Volume of a Box Write a polynomial that repre-
sents the volume of the box. Be sure to multiply your
answer completely.

x + 6

x + 3 3x

31. Height Reached by a Golf Ball When a golf ball is
hit into the air, its height in feet above the ground after
t seconds is given by t(88 – 16t).
(a) Multiply this expression.
(b) Evaluate the expression in part (a) for t = 3.

Interpret the result.

357CHAPTER 5 EXTENDED AND DISCOVERY EXERCISES

Exercises 1– 6: Arithmetic and Scientific Notation The
product (4 * 103) * (2 * 102) can be evaluated as

(4 * 2) * (103 * 102) = 8 * 105,

and the quotient (4 * 103) , (2 * 102) can be evalu-
ated as

4 * 103

2 * 102
=

2
*

103

102
= 2 * 101.

How would you evaluate (4 * 103) + (2 * 102)? How
would you evaluate (4 * 103) – (2 * 102)? Make a con-
jecture as to how numbers in scientific notation should be
added and subtracted. Try your method on these prob-
lems and then check your answers with a calculator set in
scientific mode. Does your method work?

1. (4 * 103) + (3 * 103)

2. (5 * 10-2) – (2 * 10-2)

3. (1.2 * 104) – (3 * 103)

4. (2 * 102) + (6 * 101)

5. (2 * 10-1) + (4 * 10-2)

6. (2 * 10-3) – (5 * 10-2)

Exercises 7 and 8: Constructing a Box A box is con-
structed from a rectangular piece of metal by cutting
squares from the corners and folding up the sides. The
square, cutout corners are x inches by x inches.

x
x x

x x
x

7. Suppose that the dimensions of the metal piece are
20 inches by 30 inches.
(a) Write a polynomial that gives the volume of the

box.
(b) Find the volume of the box for x = 4 inches.

8. Suppose that the metal piece is square with sides of
length 25 inches.
(a) Write a polynomial expression that gives the out-

side surface area of the box. (Assume that the
box does not have a top.)

(b) Find this area for x = 3 inches.

Exercises 9–12: Calculators and Polynomials A graph-
ing calculator can be used to help determine whether two
polynomial expressions in one variable are equal. For
example, suppose that a student believes that (x + 2)2
and x 2 + 4 are equal. Then the first two calculator tables
shown demonstrate that the two expressions are not equal
except for x = 0.

Y1�(X�2)2

X Y1
�3 1
�2 0
�1 1
0 4
1 9
2 16
3 25

Y1�X2�4

X Y1
�3 13
�2 8
�1 5
0 4
1 5
2 8
3 13

The next two calculator tables support the fact that
(x + 1)2 and x 2 + 2 x + 1 are equal for all x.

Y1�(X�1)2

X Y1
�3 4
�2 1
�1 0
0 1
1 4
2 9
3 16

Y1�X2�2X�1

X Y1
�3 4
�2 1
�1 0
0 1
1 4
2 9
3 16

Use a graphing calculator to determine whether the
first expression is equal to the second expression. If the
expressions are not equal, multiply the first expression
and simplify it.

9. 3x(4 – 5x), 12 x – 5x

10. (x – 1)2, x 2 – 1

11. (x – 1)(x 2 + x + 1), x 3 – 1

12. (x – 2)3, x 3 – 8

CHAPTER 5 Extended and Discovery Exercises

IS
B
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1-
25
6-
49
08
2-
2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley.
Copyright © 2013 by Pearson Education, Inc.

358 CHAPTER 5 POLYNOMIALS AND EXPONENTS

CHAPTERS 1–5 Cumulative Review Exercises

Exercises 1 and 2: Evaluate each expression by hand.

1. (a) 18 – 2 # 5 (b) 42 , 7 + 2
2. (a) 21 – ( – 8) (b) – 73 , 1 – 149 2
Exercises 3 and 4: Solve the equation. Note that these
equations may have no solutions, one solution, or infi-
nitely many solutions.

3. (a) (x – 3) + x = 4 + x
(b) 2(5x – 4) = 1 + 10x

4. (a) 2 + 6 x = 2(3x + 1)
(b) 11x – 9 = – 31

5. Find the average speed of a car that travels 306 miles
in 4 hours 30 minutes.

6. Write each value as a fraction in lowest terms.
(a) 42% (b) 0.076

7. Graph the equation 4 x – 5y = 20.

8. Sketch a line with slope – 23 that passes through the
point (1, 1).

9. Write the slope–intercept form for the line shown.

–2–3 –1 2 3

–2

–3

–1

1
3
x
y

10. Find the x-and y-intercepts for the graph of the equa-
tion 2y = 3x – 6.

Exercises 11 and 12: Write the slope–intercept form of a
line that satisfies the given information.

11. Parallel to 3x – 6y = 7, passing through (2, – 3)

12. Passing through ( – 2, – 5) and (1, 4)

Exercises 13–16: Solve the system of equations. Note that
these systems may have no solutions, one solution, or
infinitely many solutions.

Exercises 17 and 18: Shade the solution set for the system
of inequalities.

13. 4 x + 3y = – 6
8 x + 6y = 12

14. x – 3y = 5
3x + y = 5

15. x + 4y = – 8
– 3x – 12y = 24

16. x – 5y = 30
2 x + y = – 6

17. x + y 6 3
y Ú x + 2

18. x – 2y 7 4
3x + y 6 6

19. Simplify the expression.
(a) 3x 2 # 5x 3 (b) (x 3y)2(x 4y5)

20. Simplify.
(a) (5x 2 – 3x + 4) – (3x 2 – 2 x + 1)
(b) (7a3 – 4a2 – 5) + (5a3 + 4a2 + a)

21. Multiply and simplify.
(a) (2 x + 3)(x – 7) (b) (y + 3)(y2 – 3y – 1)

22. Simplify the expression. Write the answer using posi-
tive exponents.
(a) x -5 # x 3 # x (b) a 2

x 3
b-3

6 x -2y
(d) (xy -2)3(x -2y)-2

23. Write 24,000,000,000 in scientific notation.

24. Write 4.71 * 10-7 in standard form.

25. Divide.

(a)
8 x 3 – 2 x

2 x
(b)

2 x 2 + x – 14
x + 3

26. Price Decrease If the price of a computer is reduced
from $1200 to $900, find the percent change.

27. Mixing an Acid Solution How many milliliters of a
3% acid solution should be added to 400 milliliters of
a 6% acid solution to dilute it to a 5% acid solution?

28. Surface Area of a Box Use the drawing of the box to
write a polynomial that represents the area of each of
the following.

x + 5

x + 2
2x

(a) The bottom
(b) The front
(c) The right side
(d) All six sides

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