College Algebra pot

EXPONENTS AND RADICALS
x
m
x
n
ϭ x
mϩn
1xy2
n
ϭ x
n
y
n
SPECIAL PRODUCTS
1x ϩ y21x Ϫ y2ϭ x
2
Ϫ y
2
1x ϩ y2
2
ϭ x
2
ϩ 2xy ϩ y
2
1x Ϫ y2
2
ϭ x
2
Ϫ 2xy ϩ y
2

1x ϩ y2
3
ϭ x
3
ϩ 3x
2
y ϩ 3xy
2
ϩ y
3
1x Ϫ y2
3
ϭ x
3
Ϫ 3x
2
y ϩ 3xy
2
Ϫ y
3
FACTORING FORMULAS
x
2
Ϫ y
2
ϭ 1x ϩ y21x Ϫ y2
x
2
ϩ 2xy ϩ y
2

ϭ 1x ϩ y2
2
x
2
Ϫ 2xy ϩ y
2
ϭ 1x Ϫ y2
2
x
3
ϩ y
3
ϭ 1x ϩ y21x
2
Ϫ xy ϩ y
2
2
x
3
Ϫ y
3
ϭ 1x Ϫ y21x
2
ϩ xy ϩ y
2
2
QUADRATIC FORMULA
If ax
2
ϩ bx ϩ c ϭ 0, then

INEQUALITIES AND ABSOLUTE VALUE
If a Ͻ b and b Ͻ c, then a Ͻ c.
If a Ͻ b, then a ϩ c Ͻ b ϩ c.
If a Ͻ b and c Ͼ 0, then ca Ͻ cb.
If a Ͻ b and c Ͻ 0, then ca Ͼ cb.
If a Ͼ 0, then
means x ϭ a or x ϭϪa.
means Ϫa Ͻ x Ͻ a.
means x Ͼ a or x ϽϪa.0x 0Ͼ a
0x 0Ͻ a
0x 0ϭ a
x ϭ
Ϫb Ϯ
2b
2
Ϫ 4ac
2a
2
m
1
n
x ϭ 2
n
1
m
x ϭ 1
n
x
m
B

n
x
y
ϭ
1
n
x
1
n
y
1
n
xy ϭ 1
n
x 1
n
y
x
m/n
ϭ 1
n
x
m
ϭ A1
n
xB
m
x
1/ n
ϭ 1

n
x
a
x
y
b
n
ϭ
x
n
y
n
x
Ϫn
ϭ
1
x
n
1x
m
2
n
ϭ x
m

n
x
m
x
n

ϭ x
mϪn
DISTANCE AND MIDPOINT FORMULAS
Distance between P
1
1x
1
, y
1
2and P
2
1x
2
, y
2
2:
Midpoint of P
1
P
2
:
LINES
Slope of line through
P
1
1x
1
, y
1
2and P

2
1x
2
, y
2
2
Point-slope equation of line y Ϫ y
1
ϭ m1x Ϫ x
1
2
through P
1
1x
1
, y
1
2with slope m
Slope-intercept equation of y ϭ mx ϩ b
line with slope m and y-intercept b
Two-intercept equation of line
with x-intercept a and y-intercept b
The lines y ϭ m
1
x ϩ b
1
and y ϭ m
2
x ϩ b
2

are
Parallel if the slopes are the same m
1
ϭ m
2
Perpendicular if the slopes are m
1
ϭϪ1/m
2
negative reciprocals
LOGARITHMS
y ϭ log
a
x means a
y
ϭ x
log
a
a
x
ϭ xa
log
a
x
ϭ x
log
a
1 ϭ 0log
a
a ϭ 1

Common and natural logarithms
log x ϭ log
10
x ln x ϭ log
e
x
Laws of logarithms
log
a
xy ϭ log
a
x ϩ log
a
y
log
a
x
b
ϭ b log
a
x
Change of base formula
log
b
x ϭ
log

a
x
log

a
b
log
a
a
x
y
bϭ log
a
x Ϫ log
a
y
x
a
ϩ
y
b
ϭ 1
m ϭ
y
2
Ϫ y
1
x
2
Ϫ x
1
a
x
1

ϩ x
2
2
,
y
1
ϩ y
2
2
b
d ϭ 21x
2
Ϫ x
1
2
2
ϩ 1y
2
Ϫ y
1
2
2
GRAPHS OF FUNCTIONS
Linear functions: f1x2ϭ mx ϩ b
Power functions: f1x2ϭ x
n
Root functions:
Reciprocal functions: f1x2ϭ 1/x
n
Ï=

1
≈
x
y
Ï=
1
x
x
y
Ï=
£
œ
∑
x
x
y
Ï=œ
∑
x
x
y
f1x2ϭ 1
n
x
Ï=x∞
x
y
Ï=x¢
x
y

Ï=x£
x
y
Ï=≈
x
y
Ï=mx+b
b
x
y
Ï=b
b
x
y
Exponential functions: fÓxÔ ϭ a
x
Logarithmic functions: fÓxÔ ϭ log
a
x
Absolute value function Greatest integer function
SHIFTING OF FUNCTIONS
Vertical shifting
Horizontal shifting
c>0
c
y=f(x+c)
y=Ï
y=f(x-c)
c
y

x
c
c>0
y=Ï
y=Ï+c
c
y
x
y=Ï-c
Ï=“x‘
1
1
x
y
Ï=|x|
x
y
1
y
x
0
Ï=log
a
x
Ï=log
a
x
1
y
x

0
0<a<1a>1
y
x
0
1
Ï=a˛
y
x
0
1
Ï=a˛
a>1 0<a<1
FIFTH EDITION
College Algebra
JAMES STEWART received his MS
from Stanford University and his PhD
from the University of Toronto.He did
research at the University of London
and was influenced by the famous
mathematician, George Polya, at
Stanford University. Stewart is
currently a Professor of Mathematics
at McMaster University, and his
research field is harmonic analysis.
James Stewart is the author of a best-
selling calculus textbook series
published by Brooks/Cole, Cengage
Learning, including Calculus,Calculus:
Early Transcendentals, and Calculus:

Concepts and Contexts, a series of
precalculus texts, as well as a series of
high-school mathematics textbooks.
LOTHAR REDLIN grew up on
Vancouver Island, received a Bachelor
of Science degree from the University
of Victoria, and a PhD from McMaster
University in 1978. He subsequently
did research and taught at the
University of Washington, the
University of Waterloo, and California
State University,Long Beach.
He is currently Professor of
Mathematics at The Pennsylvania
State University,Abington Campus.
His research field is topology.
SALEEM WATSON received his
Bachelor of Science degree from
Andrews University in Michigan. He
did graduate studies at Dalhousie
University and McMaster University,
where he received his PhD in 1978.
He subsequently did research at the
Mathematics Institute of the Uni-
versity of Warsaw in Poland.He also
taught at The Pennsylvania State
University.He is currently Professor of
Mathematics at California State
University,Long Beach. His research
field is functional analysis.

The authors have also published Precalculus: Mathematics for Calculus, Algebra and Trigonometry,and Trigonometry.
ABOUT THE COVER
The building portrayed on the cover is 30 St. Mary Axe in London,
England.More commonly known as “the Gherkin,”it was designed
by the renowned architect Sir Norman Foster and completed in
2004. Although the building gives an overall curved appearance,
its exterior actually contains only one curved piece of glass—the
lens-shaped cap at the very top.In fact, the striking shape of this
building hides a complex mathematical structure. Mathematical
curves have been used in architecture throughout history, for
structural reasons as well as for their intrinsic beauty.In Focus on
Modeling: Conics in Architecture (pages 595–598) we see how
parabolas,ellipses, and hyperbolas are used in architecture.
ABOUT THE AUTHORS
College Algebra
James Stewart
McMaster University
Lothar Redlin
The Pennsylvania State University, Abington Campus
Saleem Watson
California State University, Long Beach
FIFTH EDITION
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
College Algebra, Fifth Edition
James Stewart, Lothar Redlin, Saleem Watson
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© 2009, 2004 Brooks/Cole, Cengage Learning
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Printed in Canada
1 2 3 4 5 6 7 12 11 10 09 08
v
CONTENTS
PREFACE ix
TO THE STUDENT xiv
CALCULATORS AND CALCULATIONS xv
P Prerequisites 1

P.1 Modeling the Real World with Algebra 2
P.2 Real Numbers and Their Properties 7
P.3 The Real Number Line and Order 13
P.4 Integer Exponents 19
P.5 Rational Exponents and Radicals 26
P.6 Algebraic Expressions 32
Ǡ
DISCOVERY PROJECT Visualizing a Formula 38
P.7 Factoring 39
P.8 Rational Expressions 45
CHAPTER P Review 54
CHAPTER P Test 58
Ǡ
FOCUS ON PROBLEM SOLVING General Principles 59
1 Equations and Inequalities 65
1.1 Basic Equations 66
1.2 Modeling with Equations 74
Ǡ
DISCOVERY PROJECT Equations Through the Ages 86
1.3 Quadratic Equations 87
1.4 Complex Numbers 97
1.5 Other Types of Equations 103
1.6 Inequalities 112
1.7 Absolute Value Equations and Inequalities 121
CHAPTER 1 Review 125
CHAPTER 1 Test 129
Ǡ
FOCUS ON MODELING Making the Best Decisions 130
2 Coordinates and Graphs 137
2.1 The Coordinate Plane 138

Ǡ
DISCOVERY PROJECT Visualizing Data 145
2.2 Graphs of Equations in Two Variables 147
2.3 Graphing Calculators: Solving Equations and
Inequalities Graphically 156
2.4 Lines 166
2.5 Making Models Using Variation 179
CHAPTER 2 Review 185
CHAPTER 2 Test 189
CUMULATIVE REVIEW TEST: Chapters 1 and 2 190
Ǡ
FOCUS ON MODELING Fitting Lines to Data 192
3 Functions 203
3.1 What is a Function? 204
3.2 Graphs of Functions 214
Ǡ
DISCOVERY PROJECT Relations and Functions 225
3.3 Getting Information from the Graph of a Function 227
3.4 Average Rate of Change of a Function 236
3.5 Transformations of Functions 243
3.6 Combining Functions 254
Ǡ
DISCOVERY PROJECT Iteration and Chaos 263
3.7 One-to-One Functions and Their Inverses 265
CHAPTER 3 Review 273
CHAPTER 3 Test 278
Ǡ
FOCUS ON MODELING Modeling with Functions 280
4 Polynomial and Rational Functions 291
4.1 Quadratic Functions and Models 292

4.2 Polynomial Functions and Their Graphs 300
4.3 Dividing Polynomials 315
4.4 Real Zeros of Polynomials 321
Ǡ
DISCOVERY PROJECT Zeroing in on a Zero 333
4.5 Complex Zeros and the Fundamental Theorem of Algebra 335
4.6 Rational Functions 343
CHAPTER 4 Review 359
CHAPTER 4 Test 363
Ǡ
FOCUS ON MODELING Fitting Polynomial Curves
to Data 364
5 Exponential and Logarithmic Functions 369
5.1 Exponential Functions 370
Ǡ
DISCOVERY PROJECT Exponential Explosion 383
5.2 Logarithmic Functions 384
5.3 Laws of Logarithms 394
5.4 Exponential and Logarithmic Equations 400
5.5 Modeling with Exponential and Logarithmic Functions 411
CHAPTER 5 Review 423
CHAPTER 5 Test 428
CUMULATIVE REVIEW TEST: Chapters 3,4,and 5 429
vi Contents
Ǡ
FOCUS ON MODELING Fitting Exponential and
Power Curves to Data 431
6 Systems of Equations and Inequalities 441
6.1 Systems of Equations 442
6.2 Systems of Linear Equations in Two Variables 450

6.3 Systems of Linear Equations in Several Variables 457
Ǡ
DISCOVERY PROJECT Best Fit Versus Exact Fit 467
6.4 Partial Fractions 469
6.5 Systems of Inequalities 474
CHAPTER 6 Review 481
CHAPTER 6 Test 485
Ǡ
FOCUS ON MODELING Linear Programming 486
7 Matrices and Determinants 493
7.1 Matrices and Systems of Linear Equations 494
7.2 The Algebra of Matrices 507
Ǡ
DISCOVERY PROJECT Will the Species Survive? 519
7.3 Inverses of Matrices and Matrix Equations 520
7.4 Determinants and Cramer’s Rule 530
CHAPTER 7 Review 541
CHAPTER 7 Test 546
Ǡ
FOCUS ON MODELING Computer Graphics 547
8 Conic Sections 551
8.1 Parabolas 552
Ǡ
DISCOVERY PROJECT Rolling Down a Ramp 562
8.2 Ellipses 563
8.3 Hyperbolas 572
8.4 Shifted Conics 581
CHAPTER 8 Review 589
CHAPTER 8 Test 592
CUMULATIVE REVIEW TEST: Chapters 6,7,and 8 593

Ǡ
FOCUS ON MODELING Conics in Architecture 595
9 Sequences and Series 599
9.1 Sequences and Summation Notation 600
9.2 Arithmetic Sequences 610
9.3 Geometric Sequences 616
Ǡ
DISCOVERY PROJECT Finding Patterns 625
9.4 Mathematics of Finance 626
9.5 Mathematical Induction 632
9.6 The Binomial Theorem 638
Contents vii
CHAPTER 9 Review 647
CHAPTER 9 Test 651
Ǡ
FOCUS ON MODELING Modeling with Recursive
Sequences 652
10 Counting and Probability 657
10.1 Counting Principles 658
10.2 Permutations and Combinations 662
10.3 Probability 672
Ǡ
DISCOVERY PROJECT Small Samples, Big Results 684
10.4 Binomial Probability 685
10.5 Expected Value 690
CHAPTER 10 Review 692
CHAPTER 10 Test 696
CUMULATIVE REVIEW TEST: Chapters 9 and 10 697
Ǡ
FOCUS ON MODELING The Monte Carlo Method 699

ANSWERS A1
INDEX I1
PHOTO CREDITS P1
viii Contents
ix
For many students a College Algebra course represents the first opportunity to discover the
beauty and practical power of mathematics. Thus instructors are faced with the challenge
of teaching the concepts and skills of algebra while at the same time imparting a sense of
its utility in the real world. In this edition, as in the previous four editions, our aim is to
provide instructors and students with tools they can use to meet this challenge.
The emphasis is on understanding concepts. Certainly all instructors are committed to
encouraging conceptual understanding. For many this is implemented through the rule of
four: “Topics should be presented geometrically, numerically, algebraically, and verbally.”
Technology facilitates the learning of geometrical and numerical concepts, extended proj-
ects and group learning help students explore their understanding of algebraic concepts,
writing exercises emphasize the verbal or descriptive point of view, and modeling can clar-
ify a concept by connecting it to real life. Underlying all these approaches is an emphasis
on algebra as a problem-solving endeavor. In this book we have used all these methods of
presenting college algebra as enhancements to a central core of fundamental skills. These
methods are tools to be used by instructors and students in navigating their own course of
action toward the goal of conceptual understanding.
In writing this fifth edition one of our main goals was to encourage students to be ac-
tive learners. So, for instance, each example in the text is now linked to an exercise that
will reinforce the student’s understanding of the example. New concept exercises at the be-
ginning of each exercise set encourage students to work with the basic concepts of the sec-
tion and to use algebra vocabulary appropriately. We have also reorganized and rewritten
some chapters (as described below) with the goal of further focusing the exposition on the
main concepts. In all these changes and numerous others (small and large) we have re-
tained the main features that have contributed to the success of this book. In particular, our
premise continues to be that conceptual understanding, technical skill, and real-world ap-

plications all go hand in hand, each reinforcing the others.
NEW for the Fifth Edition
■
New chapter openers emphasize how algebra topics in the chapter are used in the
real world.
■
New study aids include Learning Objectives at the beginning of each section and
expanded Review sections at the end of each chapter. The review includes a sum-
mary of the main Properties and Formulas of the chapter and a Concept Summary
keyed to specific review exercises.
■
A new Practice What You’ve Learned feature at the end of each example directs
students to a related exercise, allowing them to immediately reinforce the concept
in the example.
■
Approximately 15% of the exercises are new. New Concept exercises at the begin-
ning of each exercise set are designed to encourage students to work with the ba-
sic concepts of the section and to use mathematical vocabulary appropriately.
PREFACE
The art of teaching
is the art of assisting discovery.
MARK VAN DOREN
■
New Cumulative Review Tests appear after Chapters 2, 5, 8, and 10 and help stu-
dents gauge their progress and gain experience in taking tests that cover a broad
range of concepts and skills.
■
Chapter P, Prerequisites, has been revised to provide a more complete review of
the prerequisite basic algebra needed for this course. The properties of real num-
bers and the real number line now appear in two separate sections (Sections P.2

and P.3).
■
Chapter 3, Functions, has been rewritten to focus more sharply on the concept of
function itself. It now includes a new section entitled “Getting Information from
the Graph of a Function.” (The material on quadratic functions now appears in the
chapter on polynomial functions.)
■
Chapter 4, Polynomial and Rational Functions, now begins with a section en-
titled “Quadratic Functions and Models.” (This section previously appeared in the
chapter on functions.)
■
In Chapter 6, Systems of Equations and Inequalities, the order of the sections
“Partial Fractions” and “Systems of Inequalities” has been switched; the material
on systems of inequalities now immediately precedes the section on linear
programming.
Special Features
Exercise Sets Themost important way to foster conceptual understanding is through the
problems that the instructor assigns. To that end we have provided a wide selection of exer-
cises. Each exercise set is carefully graded, progressing from basic conceptual exercises and
skill-development problems to more challenging problems requiring synthesis of previ-
ously learned material with new concepts. To help students use the exercise sets effectively,
each example in the text is keyed to a specific exercise via the PracticeWhatYou’ve Learned
feature; this encourages students to “learn by doing” as they read through the text.
Real-World Applications We have included substantial applications of algebra that
we believe will capture the interest of students. These are integrated throughout the text in
the chapter openers, examples, exercises, Discovery Projects, and Focus on Modeling sec-
tions. In the exercise sets, applied problems are grouped together under the label Applica-
tions. (See, for example, pages 31, 120, 178, and 234.)
Discovery,Writing, and Group Learning Each exercise set ends with a block of ex-
ercises labeled Discovery • Discussion • Writing. These exercises are designed to encour-

age the students to experiment, preferably in groups, with the concepts developed in the
section, and then to write out what they have learned, rather than simply look for “the an-
swer.” (See, for example, pages 26, 121, 166, and 224.)
Graphing Calculators and Computers Calculator and computer technology ex-
tends in a powerful way our ability to calculate and to visualize mathematics. We have in-
tegrated the use of the graphing calculator throughout the text—to graph and analyze func-
tions, families of functions, and sequences; to calculate and graph regression curves; to
perform matrix algebra; to graph linear inequalities; and other such powerful uses. We also
exploit the programming capabilities of the graphing calculator to provide simple pro-
grams that model real-life situations (see, for instance, pages 264, 549, and 701). The
graphing calculator sections, subsections, examples, and exercises, all marked with the
special symbol , are optional and may be omitted without loss of continuity.
Focus on Modeling In addition to many applied problems where students are given a
model to analyze, we have included several sections and subsections in which students are
required to construct models of real-life situations. In addition, we have concluded each
chapter with a section entitled Focus on Modeling, where we present ways in which
x Preface
algebra is used to model real-life situations. For example, the Focus on Modeling after
Chapter 2 introduces the basic idea of modeling a real-life situation by fitting lines to data
(linear regression). Other Focus sections discuss modeling with polynomial, power, and
exponential functions, as well as applications of algebra to architecture, computer graph-
ics, optimization, and others. Chapter P concludes with a section entitled Focus on Prob-
lem Solving.
Projects One way to engage students and make them active learners is to have them
work (perhaps in groups) on extended projects that give a feeling of substantial accom-
plishment when completed. Each chapter contains one or more Discovery Projects (listed
in the Contents); these provide a challenging but accessible set of activities that enable stu-
dents to explore in greater depth an interesting aspect of the topic they have just studied.
Mathematical Vignettes Throughout the book we provide short biographies of inter-
esting mathematicians as well as applications of mathematics to the real world. The biog-

raphies often include a key insight that the mathematician discovered and which is relevant
to algebra. (See, for instance, the vignettes on Viète, page 89; Salt Lake City, page 139; and
radiocarbon dating, page 402.) The vignettes serve to enliven the material and show that
mathematics is an important, vital activity, and that even at this elementary level it is fun-
damental to everyday life. A series of vignettes, entitled Mathematics in the Modern World,
emphasizes the central role of mathematics in current advances in technology and the sci-
ences. (See pages 106, 462, and 554, for example.)
Check Your Answer The Check Your Answer feature is used, wherever possible, to em-
phasize the importance of looking back to check whether an answer is reasonable. (See,
for instance, pages 81 and 105.)
Review Sections and Chapter Tests Each chapter ends with an extensive review sec-
tion, including a Chapter Test designed to help students gauge their progress. Brief an-
swers to odd-numbered exercises in each section (including the review exercises), and to
all questions in the Chapter Tests, are given in the back of the book. The review material
in each chapter begins with a summary of the main Properties and Formulas and a Con-
cept Summary. These two features provide a concise synopsis of the material in the chap-
ter. Cumulative Review Tests follow Chapters 2, 5, 8, and 10.
Ancillaries
College Algebra, Fifth Edition, is supported by a complete set of ancillaries developed un-
der our direction. Each piece has been designed to enhance student understanding and to
facilitate creative instruction. New to this edition is Enhanced WebAssign (EWA), our
Web-based homework system that allows instructors to assign, collect, grade and record
homework assignments online, minimizing workload and streamlining the grading pro-
cess. EWA also gives students the ability to stay organized with assignments and have up-
to-date grade information. For your convenience, the exercises available in EWA are indi-
cated in the instructor’s edition by a blue square.
Acknowledgments
We thank the following reviewers for their thoughtful and constructive comments.
Reviewers of the First Edition Barry W. Brunson, Western Kentucky University;
Gay Ellis, Southwest Missouri State University; Martha Ann Larkin, Southern Utah Uni-

versity; Franklin A. Michello, Middle Tennessee State University; Kathryn Wetzel, Ama-
rillo College.
Reviewers of the Second Edition David Watson, Rutgers University; Floyd Downs,
Arizona State University at Tempe; Muserref Wiggins, University of Akron; Marjorie
Preface xi
Kreienbrink, University of Wisconsin; Richard Dodge, Jackson Community College;
Christine Panoff, University of Michigan at Flint; Arnold Volbach, University of Houston,
University Park; Keith Oberlander, Pasadena City College; Tom Walsh, City College of
San Francisco; and George Wang, Whittier College.
Reviewers of the Third Edition Christine Oxley, Indiana University; Linda B.
Hutchison, Belmont University; David Rollins, University of Central Florida; and Max
Warshauer, Southwest Texas State University.
Reviewers of the Fourth Edition Mohamed Elhamdadi, University of South Florida;
Carl L. Hensley, Indian River Community College; Scott Lewis, Utah Valley State College;
Beth-Allyn Osikiewicz, Kent State University—Tuscarawas Campus; Stanley Stascinsky,
Tarrant County College; Fereja Tahir, Illinois Central College; and Mary Ann Teel, Uni-
versity of North Texas.
Reviewers of the Fifth Edition Faiz Al-Rubaee, University of North Florida;
Christian Barrientos, Clayton State University; Candace Blazek, Anoka-Ramsey Commu-
nity College; Catherine May Bonan-Hamada, Mesa State College; José D. Flores, Univer-
sity of South Dakota; Christy Leigh Jackson, University of Arkansas, Little Rock; George
Johnson, St. Phillips College; Gene Majors, Fullerton College; Theresa McChesney,
Johnson County Community College; O. Michael Melko, Northern State University; Terry
Nyman, University of Wisconsin—Fox Valley; Randy Scott, Santiago Canyon College;
George Rust, West Virginia State University; Alicia Serfaty de Markus, Miami Dade Col-
lege—Kendell Campus; Vassil Yorgov, Fayetteville State University; Naveed Zaman, West
Virginia State University; Xiaohong Zhang, West Virginia State University.
We have benefited greatly from the suggestions and comments of our colleagues who
have used our books in previous editions. We extend special thanks in this regard to Larry
Brownson, Linda Byun, Bruce Chaderjian, David Gau, Daniel Hernandez,YongHee Kim-

Park, Daniel Martinez, David McKay, Robert Mena, Kent Merryfield, Viet Ngo, Marilyn
Oba, Alan Safer, Robert Valentini, and Derming Wang, from California State University,
Long Beach; to Karen Gold, Betsy Huttenlock, Cecilia McVoy, Mike McVoy, Samir
Ouzomgi, and Ralph Rush, of The Pennsylvania State University, Abington College; and
to Fred Safier, of the City College of San Francisco. We have learned much from our stu-
dents; special thanks go to Devaki Shah and Ellen Newman for their helpful suggestions.
Ann Ostberg read the entire manuscript and did a masterful job of checking the cor-
rectness of the examples and answers to exercises. We extend our heartfelt thanks for her
timely and accurate work. We also thank Phyllis Panman-Watson for solving the exercises
and checking the answer manuscript. We thank Andy Bulman-Fleming and Doug Shaw for
their inspired work in producing the solutions manuals and the supplemental study guide.
We especially thank Martha Emry, our production service, for her tireless attention to
quality and detail. Her energy, devotion, experience, and intelligence were essential
components in the creation of this book. We thank Terri Wright for her diligent and beau-
tiful photo research. At Matrix, we thank Jade Myers and his staff for their elegant
graphics. We also thank Precision Graphics for bringing many of our illustrations to life.
At Brooks/Cole, our thanks go to Senior Developmental Editor Jay Campbell, Assistant
Editor Natasha Coats, Editorial Assistant Rebecca Dashiell, Editorial Production Project
Manager Jennifer Risden, Executive Marketing Manager Joseph Rogove, and Marketing
Coordinator Ashley Pickering. They have all done an outstanding job.
Finally, we thank our editor Gary Whalen for carefully and thoughtfully guiding this
book through the writing and production process. His support and editorial insight played
a crucial role in completing this edition.
xii Preface
xiii
Ancillaries For College Algebra, Fifth Edition
INSTRUCTOR RESOURCES
PRINTED
Instructor’s Solutions Manual
ISBN-10: 0-495-56525-3; ISBN-13: 978-0-495-56525-3

Contains solutions to all even-numbered text exercises.
Test Bank
ISBN-10: 0-495-56526-1; ISBN-13: 978-0-495-56526-0
The Test Bank includes six tests per chapter as well as three final exams. The tests are made up of a
combination of multiple-choice, free-response, true/false, and fill-in-the-blank questions.
Instructor’s Guide
ISBN-10: 0-495-56527-X; ISBN-13: 978-0-495-56527-7
This Instructor’s Guide is the ideal course companion. Each section of the main text is discussed
from several viewpoints, including suggested time to allot, points to stress, text discussion topics,
core materials for lecture, workshop/discussion suggestions, group-work exercises in a form suit-
able for handout, and suggested homework problems. An electronic version is available on the
PowerLecture CD-ROM.
TESTING SOFTWARE
ExamView® for Algorithmic Equations (Windows®/Macintosh®)
Create, deliver, and customize tests and study guides (both print and online) in minutes with this
easy-to-use assessment and tutorial software on CD. Includes complete questions from the College
Algebra Test Bank. Included on the PowerLecture™ CD.
PowerLecture™ CD
ISBN-10: 0-495-56529-6; ISBN-13: 978-0-495-56529-1
Contains PowerPoint
®
lecture outlines, a database of all the art in the text, ExamView, and elec-
tronic copies of the Instructor’s Guide and Instructor’s Solution Manual.
VIDEO
Text-Specific DVDs
ISBN-10: 0-495-56522-9; ISBN-13: 978-0-495-56522-2
This set of DVDs is available free upon adoption of the text. Each video offers one chapter of the
text and is broken down into 10- to 20-minute problem-solving lessons that cover each section of
the chapter. These videos can be used as additional explanations for each chapter of the text.
STUDENT RESOURCES

PRINTED
Student Solutions Manual
ISBN-10: 0-495-56524-5; ISBN-13: 978-0-495-56524-6
Contains solutions to all odd-numbered text exercises.
Study Guide
ISBN-10: 0-495-56523-7; ISBN-13: 978-0-495-56523-9
Reinforces student understanding with detailed explanations, worked-out examples, and practice
problems. Lists key ideas to master and builds problem-solving skills. There is a section in the
Study Guide corresponding to each section in the text.
xiv
This textbook was written for you to use as a guide to mastering College Algebra. Here are
some suggestions to help you get the most out of your course.
First of all, you should read the appropriate section of text before you attempt your
homework problems. Reading a mathematics text is quite different from reading a novel,
a newspaper, or even another textbook. You may find that you have to reread a passage sev-
eral times before you understand it. Pay special attention to the examples, and work them
out yourself with pencil and paper as you read. Then do the linked exercise(s) referred to
in the Practice What You’ve Learned at the end of each example. With this kind of prepa-
ration you will be able to do your homework much more quickly and with more under-
standing.
Don’t make the mistake of trying to memorize every single rule or fact you may come
across. Mathematics doesn’t consist simply of memorization. Mathematics is a problem-
solving art, not just a collection of facts. To master the subject you must solve problems—
lots of problems. Do as many of the exercises as you can. Be sure to write your solutions
in a logical, step-by-step fashion. Don’t give up on a problem if you can’t solve it right
away. Try to understand the problem more clearly—reread it thoughtfully and relate it to
what you have learned from your teacher and from the examples in the text. Struggle with
it until you solve it. Once you have done this a few times you will begin to understand what
mathematics is really all about.
Answers to the odd-numbered exercises, as well as all the answers to each chapter test,

appear at the back of the book. If your answer differs from the one given, don’t immedi-
ately assume that you are wrong. There may be a calculation that connects the two answers
and makes both correct. For example, if you get but the answer given is
, your answer is correct, because you can multiply both numerator and denomi-
nator of your answer by to change it to the given answer.
The symbol is used to warn against committing an error. We have placed this sym-
bol in the margin to point out situations where we have found that many of our students
make the same mistake.
12
ϩ 1
1 ϩ 12
1/A12 Ϫ 1B
TO THE STUDENT
xv
Calculators are essential in most mathematics and science subjects. They free us from per-
forming routine tasks, so we can focus more clearly on the concepts we are studying.
Calculators are powerful tools but their results need to be interpreted with care. In what
follows, we describe the features that a calculator suitable for a College Algebra course
should have, and we give guidelines for interpreting the results of its calculations.
Scientific and Graphing Calculators
For this course you will need a scientific calculator—one that has, as a minimum, the usual
arithmetic operations (ϩ, Ϫ, ϫ, Ϭ) as well as exponential and logarithmic functions (e
x
,
10
x
, ln x,logx). In addition, a memory and at least some degree of programmability will
be useful.
Your instructor may recommend or require that you purchase a graphing calculator.
This book has optional subsections and exercises that require the use of a graphing calcu-

lator or a computer with graphing software. These special subsections and exercises are in-
dicated by the symbol . Besides graphing functions, graphing calculators can also be used
to find functions that model real-life data, solve equations, perform matrix calculations
(which are studied in Chapter 7), and help you perform other mathematical operations.All
these uses are discussed in this book.
It is important to realize that, because of limited resolution, a graphing calculator gives
only an approximation to the graph of a function. It plots only a finite number of points and
then connects them to form a representation of the graph. In Section 2.3, we give guide-
lines for using a graphing calculator and interpreting the graphs that it produces.
Calculations and Significant Figures
Most of the applied examples and exercises in this book involve approximate values. For
example, one exercise states that the moon has a radius of 1074 miles. This does not mean
that the moon’s radius is exactly 1074 miles but simply that this is the radius rounded to
the nearest mile.
One simple method for specifying the accuracy of a number is to state how many sig-
nificant digits it has. The significant digits in a number are the ones from the first nonzero
digit to the last nonzero digit (reading from left to right). Thus, 1074 has four significant
digits, 1070 has three, 1100 has two, and 1000 has one significant digit. This rule may
sometimes lead to ambiguities. For example, if a distance is 200 km to the nearest kilo-
meter, then the number 200 really has three significant digits, not just one. This ambiguity
is avoided if we use scientific notation—that is, if we express the number as a multiple of
a power of 10:
2.00 ϫ10
2
When working with approximate values, students often make the mistake of giving a final
answer with more significant digits than the original data. This is incorrect because you
CALCULATORS
AND CALCULATIONS
cannot “create” precision by using a calculator. The final result can be no more accurate
than the measurements given in the problem. For example, suppose we are told that the two

shorter sides of a right triangle are measured to be 1.25 and 2.33 inches long. By the Py-
thagorean Theorem, we find, using a calculator, that the hypotenuse has length
But since the given lengths were expressed to three significant digits, the answer cannot be
any more accurate. We can therefore say only that the hypotenuse is 2.64 in. long, round-
ing to the nearest hundredth.
In general, the final answer should be expressed with the same accuracy as the least-
accurate measurement given in the statement of the problem. The following rules make this
principle more precise.
As an example, suppose that a rectangular table top is measured to be 122.64 in. by
37.3 in. We express its area and perimeter as follows:
Area ϭ length ϫ width ϭ 122.64 ϫ 37.3 Ϸ 4570 in
2
Three significant digits
Perimeter ϭ 2(length ϩ width) ϭ 2(122.64 ϩ 37.3) Ϸ 319.9 in. Tenths digit
Note that in the formula for the perimeter, the value 2 is an exact value, not an approximate
measurement. It therefore does not affect the accuracy of the final result. In general, if a
problem involves only exact values, we may express the final answer with as many signif-
icant digits as we wish.
Note also that to make the final result as accurate as possible, you should wait until the
last step to round off your answer. If necessary, use the memory feature of your calculator
to retain the results of intermediate calculations.
21.25
2
ϩ 2.33
2
Ϸ 2.644125564 in.
xvi Calculators and Calculations
RULES FOR WORKING WITH APPROXIMATE DATA
1. When multiplying or dividing, round off the final result so that it has as many
significant digits as the given value with the fewest number of significant digits.

2. When adding or subtracting, round off the final result so that it has its last
significant digit in the decimal place in which the least-accurate given value
has its last significant digit.
3. When taking powers or roots, round off the final result so that it has the same
number of significant digits as the given value.
xvii
cm centimeter mg milligram
dB decibel MHz megahertz
F farad mi mile
ft foot min minute
g gram mL milliliter
gal gallon mm millimeter
h hour N Newton
H henry qt quart
Hz Hertz oz ounce
in. inch s second
J Joule ⍀ ohm
kcal kilocalorie V volt
kg kilogram W watt
km kilometer yd yard
kPa kilopascal yr year
L liter ºC degree Celsius
lb pound ºF degree Fahrenheit
lm lumen K Kelvin
M mole of solute ⇒ implies
per liter of solution ⇔ is equivalent to
m meter
ABBREVIATIONS
xviii
Word Algebra 11

No Smallest or Largest Number in an
Open Interval 16
Diophantus 49
George Polya 59
Einstein’s Letter 62
Bhaskara 63
Euclid 69
François Viète 89
Leonhard Euler 100
Coordinates as Addresses 139
Pierre de Fermat 159
Alan Turing 160
Donald Knuth 220
René Descartes 245
Sonya Kovalevsky 249
Pythagoras 284
Galileo Galilei 293
Evariste Galois 323
Carl Friedrich Gauss 338
Gerolamo Cardano 340
The Gateway Arch 373
John Napier 388
Radiocarbon Dating 402
Standing Room Only 413
Half-Lives of Radioactive
Elements 415
Radioactive Waste 416
pH for Some Common Substances 418
Largest Earthquakes 419
Intensity Levels of Sounds 420

Rhind Papyrus 470
Linear Programming 487
Julia Robinson 508
Olga Taussky-Todd 514
Arthur Cayley 522
David Hilbert 531
Emmy Noether 534
Archimedes 557
Eccentricities of the Orbits
of the Planets 568
Paths of Comets 576
Johannes Kepler 585
Large Prime Numbers 602
Eratosthenes 603
Fibonacci 604
Golden Ratio 607
Srinivasa Ramanujan 618
Blaise Pascal 636
Pascal’s Triangle 639
Sir Isaac Newton 644
Persi Diaconis 659
Ronald Graham 666
Probability Theory 672
The “Contestant’s Dilemma” 702
MATHEMATICS
IN THE MODERN WORLD
Mathematics in the Modern World 23
Error-Correcting Codes 106
Changing Words, Sound, and Pictures
into Numbers 175

Computers 246
Splines 302
Automotive Design 306
Unbreakable Codes 351
Law Enforcement 387
Weather Prediction 447
Global Positioning System (GPS) 462
Mathematical Ecology 527
Looking Inside Your Head 554
Fair Division of Assets 612
Fractals 621
Mathematical Economics 628
Fair Voting Methods 674
MATHEMATICAL VIGNETTES
FIFTH EDITION
College Algebra
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1
CHAPTER P
Prerequisites
P.1 Modeling the Real
World with Algebra
P.2 Real Numbers and
Their Properties
P.3 The Real Number
Line and Order
P.4 Integer Exponents
P.5 Rational Exponents
and Radicals
P.6 Algebraic Expressions

P.7 Factoring
P.8 Rational Expressions
Smart car? This exciting all-electric concept car, called smart fortwo
EV, was designed by Daimler Motors, which plans to introduce it in some
markets in 2008. Will driving this car help to keep the air we breathe
cleaner? What are the cost and environmental impact of producing the
electricity where this car is plugged in for recharging? Will driving this car
save money? (See Exercise 23, Section P.1.) All these questions involve
numbers, and to answer them, we need to know the basic properties of
numbers. Algebra is about these properties. The fundamental idea in
algebra is to use letters to stand for numbers; this helps us to find patterns
in numbers and to answer questions like the ones we asked here. In this
chapter we review some of the basic concepts of algebra.
11
© Daimler AG
2CHAPTER P
|
Prerequisites
Modeling the Real World with Algebra
LEARNING OBJECTIVES
After completing this section, you will be able to:
■ Use an algebra model
■ Make an algebra model
P.1
In algebra we use letters to stand for numbers. This allows us to describe patterns that we
see in the real world.
For example, if we let N stand for the number of hours you work and W stand for your
hourly wage, then the formula
P ϭ NW
gives your pay P. The formula P ϭ NW is a description or model for pay. We can also call

this formula an algebra model. We summarize the situation as follows:
Real World Algebra Model
You work for an hourly wage. You would like to
P ϭ NW
know your pay for any number of hours worked.
The model P ϭ NW gives the pattern for finding the pay for any worker, with any hourly
wage, working any number of hours. That’s the power of algebra: By using letters to stand
for numbers, we can write a single formula that describes many different situations.
We can now use the model P ϭ NW to answer questions such as “I make $10 an hour,
and I worked 35 hours; how much do I get paid?” or “I make $8 an hour; how many hours
do I need to work to get paid $1000?”
In general, a model is a mathematical representation (such as a formula) of a real-world
situation. Modeling is the process of making mathematical models. Once a model has
been made, it can be used to answer questions about the thing being modeled.
The examples we study in this section are simple, but the methods are far reaching. This
will become more apparent as we explore the applications of algebra in the Focus on Mod-
eling sections that follow each chapter starting with Chapter 1.
■ Using Algebra Models
We begin our study of modeling by using models that are given to us. In the next subsec-
tion we learn how to make our own models.
EXAMPLE 1
|
Using a Model for Pay
Use the model P ϭ NW to answer the following question: Aaron makes $10 an hour and
worked 35 hours last week. How much did he get paid?
Ǡ
REAL WORLD
Making a model
Usin
g

the model
MODEL