Algebra trigonometry 12e earl swokowski

ALGEBRA
QUADRATIC FORMULA

SPECIAL PRODUCT FORMULAS

SPECIAL FACTORING FORMULAS

If a 0, the roots of
ax 2 ϩ bx ϩ c ϭ 0 are

͑x ϩ y͒͑x Ϫ y͒ ϭ x 2 Ϫ y 2

x 2 Ϫ y 2 ϭ ͑x ϩ y͒͑x Ϫ y͒

͑x ϩ y͒2 ϭ x 2 ϩ 2xy ϩ y 2

x 2 ϩ 2xy ϩ y 2 ϭ ͑x ϩ y͒2

͑x Ϫ y͒2 ϭ x 2 Ϫ 2xy ϩ y 2

x 2 Ϫ 2xy ϩ y 2 ϭ ͑x Ϫ y͒2

͑x ϩ y͒3 ϭ x 3 ϩ 3x 2y ϩ 3xy 2 ϩ y 3

x 3 Ϫ y 3 ϭ ͑x Ϫ y͒͑x 2 ϩ xy ϩ y 2͒

͑x Ϫ y͒3 ϭ x 3 Ϫ 3x 2y ϩ 3xy 2 Ϫ y 3

x 3 ϩ y 3 ϭ ͑x ϩ y͒͑x 2 Ϫ xy ϩ y 2͒

BINOMIAL THEOREM

INEQUALITIES

xϭ

Ϫb Ϯ 2b Ϫ 4ac
2a
2

EXPONENTS AND RADICALS

aman ϭ amϩn

n
a1/n ϭ 2
a

͑am͒n ϭ amn

am/n ϭ 2 am

͑ab͒ ϭ a b
n

ͩͪ
a
b

n

n n

ϭ

n

a
bn

am
ϭ amϪn
an
1
aϪn ϭ n
a

ϭ ͑ 2a ͒
n

m/n

a

n

n

n

m

n

n
n!
ϭ
k
k!͑n Ϫ k͒!

where

n
a
2a
ϭ n
b
2b

͙2 a ϭ
n

n nϪ1
n nϪ2 2
x yϩ
x y ϩ
1
2

n nϪk k

иии ϩ
x y ϩ и и и ϩ y n,
k

m

2 ab ϭ 2 a 2 b

ͱ

ͩͪ ͩͪ
ͩͪ
ͩͪ

͑x ϩ y͒n ϭ x n ϩ

n

If a Ͼ b and b Ͼ c, then a Ͼ c
If a Ͼ b, then a ϩ c Ͼ b ϩ c
If a Ͼ b and c Ͼ 0, then ac Ͼ bc
If a Ͼ b and c Ͻ 0, then ac Ͻ bc

mn

2a

ABSOLUTE VALUE ͑d Ͼ 0͒

SEQUENCES

EXPONENTIALS AND LOGARITHMS

͉x͉ Ͻ d if and only if
Ϫd Ͻ x Ͻ d

nth term of an arithmetic sequence with first
term a1 and common difference d

y ϭ loga x

͉x͉ Ͼ d if and only if either
x Ͼ d or x Ͻ Ϫd
MEANS

an ϭ a1 ϩ ͑n Ϫ 1͒d
Sum Sn of the first n terms of an arithmetic
sequence
n
Sn ϭ ͑a1 ϩ an͒
2

Arithmetic mean A of n numbers
Aϭ

a1 ϩ a2 ϩ и и и ϩ an
n

Geometric mean G of n numbers
G ϭ ͑a1a2 и и и an͒ , ak Ͼ 0

1/n

or

Sn ϭ

n
͓2a1 ϩ ͑n Ϫ 1͒d͔
2

nth term of a geometric sequence with first
term a1 and common ratio r
an ϭ a1r nϪ1
Sum Sn of the first n terms of a geometric
sequence
Sn ϭ

a1͑1 Ϫ r n͒
1Ϫr

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means ay ϭ x

loga xy ϭ loga x ϩ loga y
loga

x
ϭ loga x Ϫ loga y
y

loga x r ϭ r loga x
alog x ϭ x
a

loga ax ϭ x
loga 1 ϭ 0
loga a ϭ 1
log x ϭ log10 x
ln x ϭ loge x
logb u ϭ

loga u
loga b


FORMULAS FROM GEOMETRY
area A

perimeter P circumference C

volume V curved surface area S

RIGHT TRIANGLE

TRIANGLE

c

c

a

altitude h

EQUILATERAL TRIANGLE

s

a

h

radius r

s
h

b
s

b

Pythagorean Theorem: c2 ϭ a2 ϩ b2
RECTANGLE

A ϭ 12 bh

Pϭaϩbϩc

hϭ

23

2

Aϭ

s

23

4

s2

TRAPEZOID

PARALLELOGRAM

a
w

h

l

A ϭ lw

h

b

P ϭ 2l ϩ 2w

b

A ϭ bh

CIRCLE

Aϭ

CIRCULAR SECTOR

u

r

1
2 ͑a

CIRCULAR RING

s

r
R

r

A ϭ ␲r 2

1

A ϭ 2 r 2␪

C ϭ 2 ␲r

s ϭ r␪

SPHERE

RECTANGULAR BOX

ϩ b͒h

h

A ϭ ␲ ͑R 2 Ϫ r 2͒
RIGHT CIRCULAR CYLINDER

h

r
w
l
r

V ϭ lwh

S ϭ 2͑hl ϩ lw ϩ hw͒

RIGHT CIRCULAR CONE

V ϭ 43 ␲r 3

S ϭ 4␲ r 2

FRUSTUM OF A CONE

V ϭ ␲ r 2h

S ϭ 2␲ rh

PRISM

r
h
h

r

Vϭ

1
2
3 ␲r h

h

R

S ϭ ␲ r 2r ϩ h
2

2

Vϭ

1
2
3 ␲h͑r

ϩ rR ϩ R 2͒

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V ϭ Bh with B the area of the base


ANALYTIC GEOMETRY
DISTANCE FORMULA

EQUATION OF A CIRCLE

d͑P1, P2͒ ϭ 2͑x2 Ϫ x1͒2 ϩ ͑y2 Ϫ y1͒2

͑x Ϫ h͒2 ϩ ͑ y Ϫ k͒2 ϭ r 2

y

y
r
(h, k)

P2(x2, y2)

P1(x1, y1)

x
x

SLOPE m OF A LINE
y

mϭ

l
(x1, y1)

GRAPH OF A QUADRATIC FUNCTION

y ϭ ax 2, a Ͼ 0

y2 Ϫ y1
x2 Ϫ x1

y ϭ ax 2 ϩ bx ϩ c, a Ͼ 0
y

y
c

(x2, y2)
x
x

b
2a

CONSTANTS

POINT-SLOPE FORM OF A LINE

␲ Ϸ 3.14159

y Ϫ y1 ϭ m͑x Ϫ x1͒

y

Ϫ

e Ϸ 2.71828

l
(x1, y1)

CONVERSIONS
x

1 centimeter Ϸ 0.3937 inch
1 meter Ϸ 3.2808 feet

SLOPE-INTERCEPT FORM OF A LINE

1 kilometer Ϸ 0.6214 mile

y ϭ mx ϩ b

y

1 gram Ϸ 0.0353 ounce

l

1 kilogram Ϸ 2.2046 pounds

(0, b)

1 liter Ϸ 0.2642 gallon
x

1 milliliter Ϸ 0.0381 fluid ounce
1 joule Ϸ 0.7376 foot-pound

INTERCEPT FORM OF A LINE

y
x

ϩ ϭ1
a
b

y

l

1 newton Ϸ 0.2248 pound
͑a

0, b

0͒

1 lumen Ϸ 0.0015 watt
1 acre ϭ 43,560 square feet

(0, b)
(a, 0)
x

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x


CLASSIC TWELFTH EDITION

ALGEBRA AND

TRIGONOMETRY
WITH ANALYTIC GEOMETRY

E A R L W. S W O K O W S K I
JEFFERY A. COLE
Anoka Ramsey Community College

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

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Algebra and Trigonometry with
Analytic Geometry,
Classic Twelfth Edition
Earl W. Swokowski, Jeffery A. Cole
Mathematics Editor: Gary Whalen
Assistant Editor: Cynthia Ashton
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© 2010, 2006 Brooks/Cole, Cengage Learning
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To the memory of Earl W. Swokowski

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CONTENTS
Preface
CHAPTER

1 Fundamental Concepts of Algebra
1.1
1.2
1.3
1.4

CHAPTER

53

Equations
54
Applied Problems
61
Quadratic Equations
73
Complex Numbers
87
Other Types of Equations
94
Inequalities
102
More on Inequalities
111
Chapter 2 Review Exercises
119
Chapter 2 Discussion Exercises
122

3 Functions and Graphs

3.1
3.2
3.3
3.4
3.5
3.6
3.7

1

Real Numbers
2
Exponents and Radicals
16
Algebraic Expressions
27
Fractional Expressions
40
Chapter 1 Review Exercises
49
Chapter 1 Discussion Exercises
51

2 Equations and Inequalities
2.1
2.2
2.3
2.4
2.5
2.6

2.7

CHAPTER

viii

123

Rectangular Coordinate Systems
124
Graphs of Equations
130
Lines
140
Definition of Function
155
Graphs of Functions
171
Quadratic Functions
185
Operations on Functions
197
Chapter 3 Review Exercises
205
Chapter 3 Discussion Exercises
211

iv

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Contents

CHAPTER

4 Polynomial and Rational Functions
4.1
4.2
4.3
4.4
4.5
4.6

CHAPTER

CHAPTER

213

Polynomial Functions of Degree Greater Than 2
214
Properties of Division
222
Zeros of Polynomials
229
Complex and Rational Zeros of Polynomials
241
Rational Functions
248

Variation
265
Chapter 4 Review Exercises
272
Chapter 4 Discussion Exercises
275

5 Inverse, Exponential, and Logarithmic Functions
5.1
5.2
5.3
5.4
5.5
5.6

277

Inverse Functions
278
Exponential Functions
287
The Natural Exponential Function
299
Logarithmic Functions
308
Properties of Logarithms
323
Exponential and Logarithmic Equations
330
Chapter 5 Review Exercises

342
Chapter 5 Discussion Exercises
345

6 The Trigonometric Functions
6.1
6.2
6.3
6.4
6.5
6.6
6.7

v

347

Angles
348
Trigonometric Functions of Angles
358
Trigonometric Functions of Real Numbers
375
Values of the Trigonometric Functions
393
Trigonometric Graphs
400
Additional Trigonometric Graphs
412
Applied Problems

420
Chapter 6 Review Exercises
433
Chapter 6 Discussion Exercises
439

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vi

CONTENTS

CHAPTER

7 Analytic Trigonometry
7.1
7.2
7.3
7.4
7.5
7.6

CHAPTER

Verifying Trigonometric Identities
442
Trigonometric Equations
447
The Addition and Subtraction Formulas

457
Multiple-Angle Formulas
467
Product-to-Sum and Sum-to-Product Formulas
The Inverse Trigonometric Functions
482
Chapter 7 Review Exercises
496
Chapter 7 Discussion Exercises
499

8 Applications of Trigonometry
8.1
8.2
8.3
8.4
8.5
8.6

CHAPTER

441

501

The Law of Sines
502
The Law of Cosines
512
Vectors

522
The Dot Product
536
Trigonometric Form for Complex Numbers
De Moivre’s Theorem and nth Roots of
Complex Numbers
552
Chapter 8 Review Exercises
557
Chapter 8 Discussion Exercises
560

9 Systems of Equations and Inequalities
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10

546

563

Systems of Equations
564

Systems of Linear Equations in Two Variables
Systems of Inequalities
582
Linear Programming
590
Systems of Linear Equations in More Than Two
Variables
598
The Algebra of Matrices
614
The Inverse of a Matrix
623
Determinants
628
Properties of Determinants
634
Partial Fractions
642
Chapter 9 Review Exercises
648
Chapter 9 Discussion Exercises
651

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477

573

Contents

CHAPTER

10 Sequences, Series, and Probability

653

10.1
10.2
10.3
10.4
10.5
10.6
10.7

Infinite Sequences and Summation Notation
Arithmetic Sequences
664
Geometric Sequences
671
Mathematical Induction
680
The Binomial Theorem
686
Permutations
695
Distinguishable Permutations and
Combinations
702

10.8 Probability
709
Chapter 10 Review Exercises
723
Chapter 10 Discussion Exercises
725
CHAPTER

11 Topics from Analytic Geometry
11.1
11.2
11.3
11.4
11.5
11.6

I
II
III

727

Parabolas
728
Ellipses
737
Hyperbolas
750
Plane Curves and Parametric Equations
Polar Coordinates

772
Polar Equations of Conics
786
Chapter 11 Review Exercises
792
Chapter 11 Discussion Exercises
794

Appendixes

654

762

797

Common Graphs and Their Equations
798
A Summary of Graph Transformations
800
Graphs of Trigonometric Functions and
Their Inverses
802
Values of the Trigonometric Functions of
Special Angles on a Unit Circle
804

IV

Answers to Selected Exercises

Index of Applications
Index

A79

A84

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A1

vii


PREFACE

The classic edition of Algebra and Trigonometry with Analytic Geometry is a
special version of the twelfth edition of the same title. It has been written for
professors seeking to teach a traditional course which requires only a scientific
calculator. Both editions improve upon the eleventh edition in several ways.
This edition includes over 120 new or revised examples and exercises,
many of these resulting from suggestions of users and reviewers of the
eleventh edition. All have been incorporated without sacrificing the mathematical soundness that has been paramount to the success of this text.
Below is a brief overview of the chapters, followed by a short description
of the College Algebra course that I teach at Anoka Ramsey Community College, and then a list of the general features of the text.

Overview
Chapter 1

This chapter contains a summary of some basic algebra topics. Students

should be familiar with much of this material, but also challenged by some of
the exercises that prepare them for calculus.

Chapter 2

Equations and inequalities are solved algebraically in this chapter. Students
will extend their knowledge of these topics; for example, they have worked
with the quadratic formula, but will be asked to relate it to factoring and work
with coefficients that are not real numbers (see Examples 10 and 11 in Section 2.3).

Chapter 3

Two-dimensional graphs and functions are introduced in this chapter. See the
updated Example 10 in Section 3.5 for a topical application (taxes) that relates
tables, formulas, and graphs.

Chapter 4

This chapter begins with a discussion of polynomial functions and some polynomial theory. A thorough treatment of rational functions is given in Section
4.5. This is followed by a section on variation, which includes graphs of simple polynomial and rational functions.

viii

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Preface

ix

Chapter 5

Inverse functions are the first topic of discussion (see new Example 4 in Section 5.1 for a relationship to rational functions), followed by several sections
that deal with exponential and logarithmic functions. Modeling an exponential
function is given additional attention in this chapter (see Example 8 in Section
5.2) as well as in Chapter 9.

Chapter 6

Angles are the first topic in this chapter. Next, the trigonometric functions are
introduced using a right triangle approach and then defined in terms of a unit
circle. Basic trigonometric identities appear throughout the chapter. The chapter concludes with sections on trigonometric graphs and applied problems.

Chapter 7

This chapter consists mostly of trigonometric identities, formulas, and equations. The last section contains definitions, properties, and applications of the
inverse trigonometric functions.

Chapter 8

The law of sines and the law of cosines are used to solve oblique triangles.
Vectors are then introduced and used in applications. The last two sections relate the trigonometric functions and complex numbers.

Chapter 9

Systems of inequalities and linear programming immediately follow solving
systems by substitution and elimination. Next, matrices are introduced and
used to solve systems. This chapter concludes with a discussion of determinants and partial fractions.

Chapter 10

This chapter begins with a discussion of sequences. Mathematical induction
and the binomial theorem are next, followed by counting topics (see Example
3 in Section 10.7 for an example involving both combinations and permutations). The last section is about probability and includes topics such as odds
and expected value.

Chapter 11

Sections on the parabola, ellipse, and hyperbola begin this chapter. Two different ways of representing functions are given in the next sections on parametric equations and polar coordinates.

My Course
At Anoka Ramsey Community College in Coon Rapids, Minnesota, College
Algebra I is a one-semester 3-credit course. For students intending to take Calculus, this course is followed by a one-semester 4-credit course, College Algebra II and Trigonometry. This course also serves as a terminal math course
for many students.
The sections covered in College Algebra I are
3.1–3.7, 4.1, 4.5 (part), 4.6, 5.1–5.6, 9.1–9.4, 10.1–10.3, and 10.5–10.8.

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x

PREFACE

Chapters 1 and 2 are used as review material in some classes, and the remaining sections are taught in the following course. A graphing calculator is required in some sections and optional in others.

Features
Illustrations Brief demonstrations of the use of definitions, laws, and theorems are provided in the form of illustrations.
Charts Charts give students easy access to summaries of properties, laws,
graphs, relationships, and definitions. These charts often contain simple illustrations of the concepts that are being introduced.

Examples Titled for easy reference, all examples provide detailed solutions of

problems similar to those that appear in exercise sets. Many examples include
graphs, charts, or tables to help the student understand procedures and solutions.
Step-by-Step Explanations In order to help students follow them more easily,
many of the solutions in examples contain step-by-step explanations.
Discussion Exercises Each chapter ends with several exercises that are suit-

able for small-group discussions. These exercises range from easy to difficult
and from theoretical to application-oriented.
Checks The solutions to some examples are explicitly checked, to remind

students to verify that their solutions satisfy the conditions of the problems.
Applications To arouse student interest and to help students relate the exer-

cises to current real-life situations, applied exercises have been titled. One
look at the Index of Applications in the back of the book reveals the wide array
of topics. Many professors have indicated that the applications constitute one
of the strongest features of the text.
Exercises Exercise sets begin with routine drill problems and gradually

progress to more difficult problems. An ample number of exercises contain
graphs and tabular data; others require the student to find a mathematical
model for the given data. Many of the new exercises require the student to understand the conceptual relationship of an equation and its graph.
Applied problems generally appear near the end of an exercise set, to
allow students to gain confidence in working with the new ideas that have been
presented before they attempt problems that require greater analysis and synthesis of these ideas. Review exercises at the end of each chapter may be used
to prepare for examinations.

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Preface

xi

Guidelines Boxed guidelines enumerate the steps in a procedure or technique
to help students solve problems in a systematic fashion.
Warnings Interspersed throughout the text are warnings to alert students to

common mistakes.
Text Art Forming a total art package that is second to none, figures and graphs
have been computer-generated for accuracy, using the latest technology. Colors
are employed to distinguish between different parts of figures. For example, the
graph of one function may be shown in blue and that of a second function in
red. Labels are the same color as the parts of the figure they identify.
Text Design The text has been designed to ensure that discussions are easy to

follow and important concepts are highlighted. Color is used pedagogically to
clarify complex graphs and to help students visualize applied problems. Previous adopters of the text have confirmed that the text strikes a very appealing
balance in terms of color use.
Endpapers The endpapers in the front and back of the text provide useful
summaries from algebra, geometry, and trigonometry.
Appendixes Appendix I, “Common Graphs and Their Equations,” is a pictorial summary of graphs and equations that students commonly encounter in
precalculus mathematics. Appendix II, “A Summary of Graph Transformations,” is an illustrative synopsis of the basic graph transformations discussed
in the text: shifting, stretching, compressing, and reflecting. Appendix III,
“Graphs of Trigonometric Functions and Their Inverses,” contains graphs,
domains, and ranges of the six trigonometric functions and their inverses.
Appendix IV, “Values of the Trigonometric Functions of Special Angles on a
Unit Circle,” is a full-page reference for the most common angles on a unit

circle—valuable for students who are trying to learn the basic trigonometric
functions values.
Answer Section The answer section at the end of the text provides answers for

most of the odd-numbered exercises, as well as answers for all chapter review
exercises. Considerable thought and effort were devoted to making this section
a learning device for the student instead of merely a place to check answers.
For instance, proofs are given for mathematical induction problems. Numerical answers for many exercises are stated in both an exact and an approximate
form. Graphs, proofs, and hints are included whenever appropriate. Authorprepared solutions and answers ensure a high degree of consistency among the
text, the solutions manuals, and the answers.

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xii

PREFACE

Teaching Tools for the Instructor
Instructor’s Solutions Manual by Jeff Cole (ISBN 0-495-56071-5) This author-prepared manual includes answers to all exercises and detailed solutions to most exercises. The manual has
been thoroughly reviewed for accuracy.
Test Bank (ISBN 0-495-38233-7)

The Test Bank includes multiple tests per chapter as well as
final exams. The tests are made up of a combination of multiple-choice, true/false, and fill-inthe-blank questions.

ExamView (ISBN 0-495-38234-5) Create, deliver, and customize tests and study guides (both
in print and online) in minutes with this easy-to-use assessment and tutorial system, which contains all questions for the Test Bank in electronic format.
Enhanced WebAssign Developed by teachers for teachers, WebAssign® allows instructors to

focus on what really matters—teaching rather than grading. Instructors can create assignments
from a ready-to-use database of algorithmic questions based on end-of-section exercises, or
write and customize their own exercises. With WebAssign®, instructors can create, post, and review assignments; deliver, collect, grade, and record assignments instantly; offer more practice
exercises, quizzes, and homework; assess student performance to keep abreast of individual
progress; and capture the attention of online or distance learning students.

Learning Tools for the Student
Student Solutions Manual by Jeff Cole (ISBN 0-495-56072-3) This author-prepared manual
provides solutions for all of the odd-numbered exercises, as well as strategies for solving additional exercises. Many helpful hints and warnings are also included.
Website The Book Companion Website contains study hints, review material, instructions for

using various graphing calculators, a tutorial quiz for each chapter of the text, and other materials for students and instructors.

Acknowledgments
Many thanks go to the reviewers of this edition:
Brenda Burns-Williams, North Carolina State University
Gregory Cripe, Spokane Falls Community College
George DeRise, Thomas Nelson Community College
Ronald Dotzel, University of Missouri, St. Louis
Hamidullah Farhat, Hampton University
Sherry Gale, University of Cincinnati
Carole Krueger, University of Texas, Arlington

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Preface

xiii

Sheila Ledford, Coastal Georgia Community College
Christopher Reisch, Jamestown Community College
Beverly Shryock, University of North Carolina, Chapel Hill
Hanson Umoh, Delaware State University
Beverly Vredevelt, Spokane Falls Community College
Limin Zhang, Columbia Basin Community College
Thanks are also due to reviewers of past editions, who have helped increase the usefulness of the
text for the students over the years:
Jean H. Bevis, Georgia State University
David Boliver, University of Central Oklahoma
Randall Dorman, Cochise College
Sudhir Goel, Valdosta State University
Karen Hinz, Anoka-Ramsey Community College
John W. Horton, Sr., St. Petersburg College
Robert Jajcay, Indiana State University
Conrad D. Krueger, San Antonio College
Susan McLoughlin, Union County College
Lakshmi Nigam, Quinnipiac University
Wesley J. Orser, Clark College
Don E. Soash, Hillsborough Community College
Thomas A. Tredon, Lord Fairfax Community College
Fred Worth, Henderson State University
In addition, I thank Marv Riedesel and Mary Johnson for their precise accuracy checking of new
and revised examples and exercises; and Mike Rosenborg of Canyonville (Oregon) Christian
Academy and Anna Fox, accuracy checkers for the Instructor’s Solutions Manual.
I am thankful for the excellent cooperation of the staff of Brooks/Cole, especially Acquisitions Editor Gary Whalen, for his helpful advice and support throughout the project. Natasha
Coats and Cynthia Ashton managed the excellent ancillary package that accompanies the text.
Special thanks go to Cari Van Tuinen of Purdue University for her guidance with the new review
exercises and to Leslie Lahr for her research and insightful contributions. Sally Lifland, Gail
Magin, Madge Schworer, and Peggy Flanagan, all of Lifland et al., Bookmakers, saw the book

through all the stages of production, took exceptional care in seeing that no inconsistencies occurred, and offered many helpful suggestions. The late George Morris, of Scientific Illustrators,
created the mathematically precise art package and updated all the art through several editions.
This tradition of excellence is carried on by his son Brian.
In addition to all the persons named here, I would like to express my sincere gratitude to the
many students and teachers who have helped shape my views on mathematics education. Please
feel free to write to me about any aspect of this text—I value your opinion.
Jeffery A. Cole

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1
Fundamental
Concepts of Algebra
1.1 Real Numbers
1.2 Exponents and
Radicals
1.3 Algebraic
Expressions
1.4 Fractional
Expressions

The word algebra comes from ilm al-jabr w’al muqabala, the title of a book
written in the ninth century by the Arabian mathematician al-Khworizimi.
The title has been translated as the science of restoration and reduction,

which means transposing and combining similar terms (of an equation).
The Latin transliteration of al-jabr led to the name of the branch of mathematics we now call algebra.
In algebra we use symbols or letters—such as a, b, c, d, x, y—to denote arbitrary numbers. This general nature of algebra is illustrated by the
many formulas used in science and industry. As you proceed through this
text and go on either to more advanced courses in mathematics or to fields
that employ mathematics, you will become more and more aware of the importance and the power of algebraic techniques.

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2

CHAPTER 1 FUNDAMENTAL CONCEPTS OF ALGEBRA

1.1
Real Numbers

Real numbers are used throughout mathematics, and you should be acquainted
with symbols that represent them, such as
1,

73,

Ϫ5,

49
12 ,

22,

3
2
Ϫ85,

0,

0.33333 . . . ,

596.25,

and so on. The positive integers, or natural numbers, are
1,

2,

3,

4,

....

The whole numbers (or nonnegative integers) are the natural numbers combined with the number 0. The integers are often listed as follows:
...,

Ϫ4,

Ϫ3,

Ϫ2,

Ϫ1,

0,

1,

2,

3,

4,

...

Throughout this text lowercase letters a, b, c, x, y, … represent arbitrary
real numbers (also called variables). If a and b denote the same real number,
we write a ϭ b, which is read “a is equal to b” and is called an equality. The
notation a b is read “a is not equal to b.”
If a, b, and c are integers and c ϭ ab, then a and b are factors, or divisors, of c. For example, since
6 ϭ 2 и 3 ϭ ͑Ϫ2͒͑Ϫ3͒ ϭ 1 и 6 ϭ ͑Ϫ1͒͑Ϫ6͒,
we know that 1, Ϫ1, 2, Ϫ2, 3, Ϫ3, 6, and Ϫ6 are factors of 6.
A positive integer p different from 1 is prime if its only positive factors
are 1 and p. The first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. The Fundamental Theorem of Arithmetic states that every positive integer different
from 1 can be expressed as a product of primes in one and only one way (except for order of factors). Some examples are
12 ϭ 2 и 2 и 3,

126 ϭ 2 и 3 и 3 и 7,

540 ϭ 2 и 2 и 3 и 3 и 3 и 5.

A rational number is a real number that can be expressed in the form
a͞b, where a and b are integers and b 0. Note that every integer a is a rational number, since it can be expressed in the form a͞1. Every real number
can be expressed as a decimal, and the decimal representations for rational
numbers are either terminating or nonterminating and repeating. For example,
we can show by using the arithmetic process of division that
5
4

ϭ 1.25

and

177
55

ϭ 3.2181818 . . . ,

where the digits 1 and 8 in the representation of 177
55 repeat indefinitely (sometimes written 3.218).

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1.1 Re a l N u m b e r s

In technical writing, the use of the
symbol Џ for is approximately
equal to is convenient.

3

Real numbers that are not rational are irrational numbers. Decimal representations for irrational numbers are always nonterminating and nonrepeating. One common irrational number, denoted by ␲, is the ratio of the
circumference of a circle to its diameter. We sometimes use the notation
␲ Ϸ 3.1416 to indicate that ␲ is approximately equal to 3.1416.
There is no rational number b such that b2 ϭ 2, where b2 denotes b и b.
However, there is an irrational number, denoted by 22 (the square root of 2),
2
such that ͑ 22 ͒ ϭ 2.
The system of real numbers consists of all rational and irrational numbers. Relationships among the types of numbers used in algebra are illustrated
in the diagram in Figure 1, where a line connecting two rectangles means that
the numbers named in the higher rectangle include those in the lower rectangle. The complex numbers, discussed in Section 2.4, contain all real numbers.

Figure 1 Types of numbers used in algebra

Complex numbers

Real numbers

Rational numbers

Irrational numbers

Integers

Negative integers

0

Positive integers

The real numbers are closed relative to the operation of addition (denoted by ϩ); that is, to every pair a, b of real numbers there corresponds exactly one real number a ϩ b called the sum of a and b. The real numbers are
also closed relative to multiplication (denoted by и); that is, to every pair a,
b of real numbers there corresponds exactly one real number a и b (also denoted by ab) called the product of a and b.
Important properties of addition and multiplication of real numbers are
listed in the following chart.

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4

CHAPTER 1 FUNDAMENTAL CONCEPTS OF ALGEBRA

Properties of Real Numbers

Terminology

General case

(1) Addition is commutative.

aϩbϭbϩa

(2) Addition is associative.

a ϩ ͑b ϩ c͒ ϭ ͑a ϩ b͒ ϩ c

(3) 0 is the additive identity.

aϩ0ϭa

(4) Ϫa is the additive inverse,
or negative, of a.
(5) Multiplication is commutative.

a ϩ ͑Ϫa͒ ϭ 0

(6) Multiplication is associative.

a͑bc͒ ϭ ͑ab͒c

(7) 1 is the multiplicative identity.

aи1ϭa

1
is the
a
multiplicative inverse, or
reciprocal, of a.
(9) Multiplication is distributive
over addition.

(8) If a

0,

ab ϭ ba

a

ͩͪ
1
a

Meaning
Order is immaterial when adding two
numbers.
Grouping is immaterial when adding three
numbers.
Adding 0 to any number yields the same
number.
Adding a number and its negative yields 0.
Order is immaterial when multiplying two
numbers.
Grouping is immaterial when multiplying
three numbers.
Multiplying any number by 1 yields the same
number.

ϭ1

Multiplying a nonzero number by its
reciprocal yields 1.

a͑b ϩ c͒ ϭ ab ϩ ac and
͑a ϩ b͒c ϭ ac ϩ bc

Multiplying a number and a sum of two
numbers is equivalent to multiplying each of

the two numbers by the number and then
adding the products.

Since a ϩ ͑b ϩ c͒ and ͑a ϩ b͒ ϩ c are always equal, we may use
a ϩ b ϩ c to denote this real number. We use abc for either a͑bc͒ or ͑ab͒c.
Similarly, if four or more real numbers a, b, c, d are added or multiplied, we
may write a ϩ b ϩ c ϩ d for their sum and abcd for their product, regardless
of how the numbers are grouped or interchanged.
The distributive properties are useful for finding products of many types
of expressions involving sums. The next example provides one illustration.
EXAMPLE 1

Using distributive properties

If p, q, r, and s denote real numbers, show that
͑ p ϩ q͒͑r ϩ s͒ ϭ pr ϩ ps ϩ qr ϩ qs.
We use both of the
preceding chart:
͑ p ϩ q͒͑r ϩ s͒
ϭ p͑r ϩ s͒ ϩ q͑r ϩ s͒
ϭ ͑ pr ϩ ps͒ ϩ ͑qr ϩ qs͒
ϭ pr ϩ ps ϩ qr ϩ qs

SOLUTION

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distributive properties listed in (9) of the

second distributive property, with c ϭ r ϩ s

first distributive property
remove parentheses

L


1.1 Re a l N u m b e r s

5

The following are basic properties of equality.

Properties of Equality

If a ϭ b and c is any real number, then
(1) a ϩ c ϭ b ϩ c
(2) ac ϭ bc

Properties 1 and 2 state that the same number may be added to both sides
of an equality, and both sides of an equality may be multiplied by the same
number. We will use these properties extensively throughout the text to help
find solutions of equations.
The next result can be proved.

(1) a и 0 ϭ 0 for every real number a.
(2) If ab ϭ 0, then either a ϭ 0 or b ϭ 0.

Products Involving Zero

When we use the word or as we do in (2), we mean that at least one of the factors a and b is 0. We will refer to (2) as the zero factor theorem in future work.

Some properties of negatives are listed in the following chart.

Properties of Negatives

Property
(1)
(2)
(3)
(4)

Ϫ͑Ϫa͒ ϭ a
͑Ϫa͒b ϭ Ϫ͑ab͒ ϭ a͑Ϫb͒
͑Ϫa͒͑Ϫb͒ ϭ ab
͑Ϫ1͒a ϭ Ϫa

The reciprocal

Illustration
Ϫ͑Ϫ3͒ ϭ 3

͑Ϫ2͒3 ϭ Ϫ͑2 и 3͒ ϭ 2͑Ϫ3͒
͑Ϫ2͒͑Ϫ3͒ ϭ 2 и 3
͑Ϫ1͒3 ϭ Ϫ3

1
of a nonzero real number a is often denoted by aϪ1, as
a

in the next chart.

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6

CHAPTER 1 FUNDAMENTAL CONCEPTS OF ALGEBRA

Notation for Reciprocals

Definition
If a

Illustrations

0, then aϪ1 ϭ

1
.
a

2Ϫ1 ϭ

ͩͪ

Ϫ1

3
4

Note that if a

0, then

ͩͪ

a и aϪ1 ϭ a

1
a

1
2
ϭ

1
4
ϭ
3͞4
3

ϭ 1.

The operations of subtraction ͑Ϫ͒ and division ͑Ϭ͒ are defined as follows.

Subtraction and Division

Definition
a Ϫ b ϭ a ϩ ͑Ϫb͒

ͩͪ

1
b
Ϫ1
ϭ aи b ;b

aϬbϭaи

0

Meaning

Illustration

To subtract one
number from
another, add the
negative.

3 Ϫ 7 ϭ 3 ϩ ͑Ϫ7͒

To divide one
number by a
nonzero number,
multiply by the
reciprocal.

3Ϭ7ϭ3и

ͩͪ

1
7

ϭ 3 и 7Ϫ1

a
for a Ϭ b and refer to a͞b as the quotient of a
b
and b or the fraction a over b. The numbers a and b are the numerator and
denominator, respectively, of a͞b. Since 0 has no multiplicative inverse, a͞b
is not defined if b ϭ 0; that is, division by zero is not defined. It is for this reason that the real numbers are not closed relative to division. Note that
We use either a͞b or

1Ϭbϭ

1
ϭ bϪ1 if
b

b

0.

The following properties of quotients are true, provided all denominators
are nonzero real numbers.

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7

1.1 Re a l N u m b e r s

Properties of Quotients

(1)
(2)
(3)
(4)
(5)
(6)
(7)

Property

Illustration

a
c
ϭ
if ad ϭ bc
b
d
ad
a
ϭ
bd
b
a
Ϫa

a
ϭ
ϭϪ
Ϫb
b
b
a
c
aϩc
ϩ
ϭ
b
b
b
a
c
ad ϩ bc
ϩ
ϭ
b
d
bd
a c
ac
и
ϭ
b d
bd

2

6
ϭ
because 2 и 15 ϭ 5 и 6
5
15
2и3
2
ϭ
5и3
5
2
Ϫ2
2
ϭ
ϭϪ
Ϫ5
5
5
2
9
2ϩ9
11
ϩ
ϭ
ϭ
5
5
5
5
2

4
2и3ϩ5и4
26
ϩ
ϭ
ϭ
5
3
5и3
15
2и7
14
2 7
и
ϭ
ϭ
5 3
5и3
15

a
c
a d
ad
Ϭ
ϭ
и
ϭ
b
d

b c
bc

2
7
2 3
6
Ϭ ϭ и ϭ
5
3
5 7
35

Real numbers may be represented by points on a line l such that to each
real number a there corresponds exactly one point on l and to each point P on
l there corresponds one real number. This is called a one-to-one correspondence. We first choose an arbitrary point O, called the origin, and associate
with it the real number 0. Points associated with the integers are then determined by laying off successive line segments of equal length on either side of
O, as illustrated in Figure 2. The point corresponding to a rational number,
such as 23
5 , is obtained by subdividing these line segments. Points associated
with certain irrational numbers, such as 22, can be found by construction (see
Exercise 45).
Figure 2

O
Ϫ3

Ϫ2

Ϫ1

0

Ϫq
Ϫ1.5
Negative real
numbers

1
͙2

2
2.33

3
p

4

5

B

A

b

a

l

H
Positive real
numbers

The number a that is associated with a point A on l is the coordinate of
A. We refer to these coordinates as a coordinate system and call l a coordinate line or a real line. A direction can be assigned to l by taking the positive
direction to the right and the negative direction to the left. The positive direction is noted by placing an arrowhead on l, as shown in Figure 2.

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