College Algebra

992 Pages·2012·105.65 MB·English

by  Robert F. Blitzer

Robert Blitzer
Miami Dade College
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Library of Congress Cataloging-in-Publication Data
Blilzer, Robert.
College algebra/ Robert Blilzer. - 6thed.
p.cnL
lndudes index.
ISBN 978-().321-78228-t
I. Algebra- Te>1books. l.litle.
QAIS2.3.B642 2014
S12.9-dc23
2012023115
Copyright© 2014.2010, 20CfJ Pearson Education. Inc.
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I 23 4 56 7 8910-00W-1615 14 13 12
PEARSON ISBN 10:0-321-78228-3
wuw.pearsonhighered.com ISBN 13:978-0-321-78228-1
DEDICATION
For Jerld (1985-2012)
And for those who have loved their pets
and have been loved by them
CONTENTS
Preface viii Mid-Chapter Check Point 164
Acknowledgements xi 1.6 Other Types of Equations 167
Dynamic Resources xii 1.7 Unear Inequalities and Absolute Value Inequalities 182
To the Student xiv Summary, Review, and Test 200
About the Author xv Review Exercises 202
Applications Index xvi Chapter 1 Test 206
P Prerequisites: Fundamental
Concepts of Alg_e~b""r'-a"'--1' ______
P.1 Algebraic Expressions, Mathematical Models, and Real
Numbers 2
P.2 Exponents and Scientific Notation 20
P.3 Radicals and Rational Exponents 35
P.4 Polynomials 51
Mid-Chapter Check Point 63
P.s Factoring Polynomials 64
P.6 Rational Expressions 76
Summary, Review, and Test 89
Review Exercises 89
Chapter P Test 91
1 Equations and
Inequalities 93
1.1 Graphs and Graphing Utilities 94
1.2 Unear Equations and Rational Equations 105
1.3 Models and Applications 122
1.4 Complex Numbers 137
1.5 Quadratic Equations 144
iv
Contents v
3 Polynomial and
Rational Functions 329
3.1 Quadratic Functions 330
3.2 POlynOmial Funct1011s and Their Graphs 34 7
3.3 Dov1ding PolynomoalS; Remainder and Factor
Theorems 364
3.4 Zeros of Polynomial Functions 376
Mid-Chapter Check Point 390
3.5 Rational FunctionS and Their Graphs 391
3.6 POlynOmial and Ratoonallnequalitles 410
3.7 Modeling Using vanation 423
Summary, Review, and Test 433
Review Exercises 435
Chapter 3 Test 439
Cumulative Review Exercises (Chapters t -3) 440
4 Exponential and
Logarithmic Functions 441
4.1 Exponential FunctionS 442
2 Functions
4.2 LOgarithmic Functoons 455
=-~a=n=d Graphs 209
4.3 Properties of Logarithms 469
2.1 Basics of Functions and Theor Graphs 210 Mid-Chapter Check Point 479
2.2 More on Functions and Theor Graphs 229 4.4 Exponential and Loganthmic Equabons 4 79
2.3 Unear FunctionS and Slope 244 4.5 Exponential Growth and Decay; Modelong Data 494
2.4 More on SlOpe 259 SUmmary, Review, and Test 506
Mid-Chapter Check Point 269 Review Exercises 510
2. .s Transformations of FunctiOns 270 Chapter 4 Test 513
2.6 Comblnat1011s of Functions: Composite Functions 286 Cumulettve Review Exercises (Chapters 1-4) 514
2.7 tnverse Funcoons 300
2.8 Dostance and Midpoint Formulas; Circles 312
Summary, Review, and Test 321
Review Exercises 323
Chapter 2 Test 327
Cumulative Review Exercises (Chapters 1-21 326
6 Matrices
and Determinants 589
6.1 Matrix Solutions to Linear Systems 590
6.2 Inconsistent and Dependent Systems and Their
Applications 602
6.3 Matrix Operations and Their Applications 612
Mid-Chapter Check Point 627
6.4 Multiplicative Inverses of Matrices and Matrix
5 Systems of Equations
Equations 628
and Inequalities 515
6.5 Determinants and Cramer's Rule 642
5.1 Systems of Linear Equations in Two Variables 516
Summary, Review, and Test 654
5.2 Systems of Linear Equations in Three Variables 532
Review Exercises 655
5.3 Partial Fractions 540
Chapter 6 Test 658
5.4 Systems of Nonlinear Equations in Two Variables 551
Cumulative Review Exercises (Chapters 1-6} 658
Mid-Chapter Check Point 561
5.5 Systems of lnequal~ies 562
7 Conic
5.6 Linear Programming 575
Sections 659
Summary, Review, and Test 583
Review Exercises 584 7.1 The Ellipse 660
Chapter 5 Test 586 7.2 The Hype<bola 673
Cumulative Review Exercises (Chapters 1-5} 587 Mid-Chapter Check Point 668
7.3 The Parabola 689
Summary, Review, and Test 701
Review Exercises 702
Chapter 7 Test 703
Cumulative Review Exercises (Chapters 1-7} 704
vi
Contents vii
8 Sequences, Induction, SUmmary, Review, and Test 766
and Probabl!J!y 705 Review Exercises 766
8.1 Sequences and SUmmation Notation 706 Chapter 6 Test 791
8.2 Anthmebc Sequences 717 Cumulative Review Exercises (Chapters 1~ 792
8.3 Geometnc Sequences and Series 727
Appendix: Where Did That Come From? Selected Proofs 795
Mid-Chapter Check Point 743
Answers to Selected Exercises AA1
8.4 Mathernatocallnduction 744
Subject Index 11
s.s The B.nomial Theorem 752
Photo Credits C1
8.6 Counting Principles, Permutations, and
Comblnattons 760
8.7 Probability 771
PREFACE
I've written College Algebra, Sixth Edition. to help 1l1e Concept and Vocabulary Checks
diverse students, with different backgrounds and future appear as separate features preceding
goals. to succeed. The book has three fundamental goals: the Exercise Sets.
1. To help students acquire a solid foundation in algebra, Great Question! This feature takes tbe
preparing them for other courses such as calculus, content of each Study Tip in the
business calculus, and finite mathematics. Fifth Edition and presents it in
the context of a student question.
2. To show students how algebra can model and solve
Answers to questions offer suggestions
authentic real-world problems.
for problem solving, point out common
3. To enable students to develop problem-solving skills,
errors to avoid, and provide informal
while fostering critical thinking, within an interesting
hints and suggestions. 'Great Question!'
setting.
should draw students' attention and
One major obstacle in the way of achieving these goals curiosity more than the ·study Tips.' As
is the fact that very few students actually read their a secondary benefit, this new feature
textbook. This has been a regular source of frustration should help students not to feel anxious
for me and for my colleagues in the classroom. Anecdotal or threatened when asking questions in class.
evidence gathered over years highlights two basic reasons
New Chapter-Opening and Section-Opening Scenarios.
that students do not take advantage of their textbook:
Every chapter and every section open with a scenario
• ''I'll never use this information." based on an application, many of which are unique to
the Sixth Edition. These scenarios are revisited in the
• "'I can't follow the explanations."
course of the chapter or section in one of the book's new
I've written every page of the Sixth Edition with the intent examples, exerc.ises, or discussions. The often humorous
of eliminating these two objections. The ideas and tools tone of these openers is intended to help fearful and
I've used to do so are described for the student in reluctant students overcome their negative perceptions
·•A Brief Guide to Getting the Most from 1l1is Book," about math.
which appears at the front of the book.
New Blitzer Bonuses. The Sixth Edition contains
a variety of new but optional enrichment essays.
What's New in the Sixth Edition?
Examples include "Using Algebra to Measure Blood
New Applications and Real-World Data. I'm on a constant Alcohol Concentration'" (Section P.l}, ·'Seven Ways
search for data that can be used to illustrate unique to Spend $1 Trillion" (Section P.2), "Addressing
algebraic applications. I researched hundreds of books, Stress Parabolically" (Section 3.1 ), ''five Things
magazines, newspapers, almanacs, and online sites to Scientists Learned from the Hubble Space Telescope"
prepare the Sixth Edition. Among the "108 worked-out (Section 7.3}, and ··Ponzi Schemes and Geometric
examples and exercises based on new data sets, you 'II Sequences" (Section 8.3).
find applications involving modeling blood-alcohol
Sample Homework Assignments. Within each Exercise
concentration (Section P.I), starting salaries for college
Set, I have chosen odd-numbered problems, primarily
graduates (Sectionl.3}, the world's vanishing
from the Practice Exercises. that can serve as sample
tiger population (Section 3.2), and the
homework assignments. These are indicated by a
year humans become immortal
blue underline in the Annotated Instructor's Edition.
(Section 4."1}.
Based on the goals and objectives of your course, you
Concept and Vocabulary may wish to enrich each sample homework assignment
Checks. The Sixth Edition with additional exercises from the other categories in the
contains 470 new short Exercise Set.
answer exercises~ mainly
New Interactive Figures. These new figures bring
fill-in-the-blank and true/
mathematical concepts to life and are included in
false items, that assess
MyMathLab. Used as a lecture tool, the figures help
students' understanding
engage students more fully and save the time spent
of the definitions and
drawing figures by hand. Questions pertaining to each
concepts presented
figure are assignable in MyMathLab and reinforce
in each section.
Preface ix
active learning and critical thinking. Each figure has an
accompanying Exploratory Exercise that encourages
further study and can be used as a presentation tool or as
an open-ended learning assignment.
What Content and Organizational
Changes Have Been Made to the
Sixth Edition?
• Section P.1 (Algebraic Expressions, Mathematical
Models, and Real Numbers) contains a new essay,
now called a Blitzer Bonus, on using algebra to
measure blood-alcohol concentration. This Blitzer
Bonus should set the stage for the book's engaging • Section 5.3 (Partial Fractions) uses the Great
collection of unique applications. Question! feature to include a discussion on
speeding up the process of finding partial fraction
• Section P.6 (Rational Expressions) presents a new
decompositions.
example on excluding numbers from a rational
expression with a trinomial denominator.
What Familiar Features Have Been
• Section 1.2 (Linear Equations and Rational Equations)
Retained in the Sixth Edition?
contains examples involving both inconsistent
equations and identities. • Detailed Worked-Out Examples. Each worked
• Section 2.5 (Transformations of Functions) has a more example is titled, making clear the purpose of the
example. Examples are clearly written and provide
thoroughly developed discussion of how stretching or
students with detailed step-by-step solutions. No steps
shrinking changes a graph's shape.
are omitled and key steps are thorougllly explained to
• Section 2.6 (Combinations of Functions; Composite the right of the mathematics .
Functions) has a new example on finding the domain
• Explanatory Voice Balloons. Voice balloons are
of a function with a square root in the denominator.
used in a variety of ways to demystify mathematics.
There is also a new example that ties in with the
They translate algebraic ideas into everyday English,
section opener (number of births and deaths in the
help clarify problem-solving procedures, present
United States) and illustrates an application of the
alternative ways of understanding concepts, and
algebra of functions.
connect problem solving to concepts students have
• Section 3.2 (Polynomial Functions and Their already learned .
Graphs) contains a new example on graphing
• Check Point Examples. Each example is followed
J(x) = -2(x - If (x + 2), a polynomial function
by a similar matched problem, called a Check
whose equation is given in factored form.
Point, offering students the opportunity to test their
• Section 3.5 (Rational Functions and Their Graphs) has understanding of the example by working a similar
a variety of exercises where students must factor to exercise. The answers to the Check Points are
find vertical asymptotes or holes. provided in the answer section.
• Section 3.6 (Polynomial and Rational Inequalities) • Extensive and Varied Exercise Sets. An abundant
contains a new example on solving a polynomial collection of exercises is included in an Exercise Set at
inequality with irrational boundary points that the end of each section. Exercises are organized with[n
requires the use of the quadratic formula. eight category types: Practice Exercises, Practice
• Section 4.1 (Exponential Functions) presents an Plus Exercises, Application Exercises, Writing in
Mathematics, Technology Exercises, Critical Thinking
intriguing new Blitzer Bonus on the year humans
Exercises, Group Exercises, and Preview Exercises.
become immortal. The section also contains a new
This format makes it easy to create well-rounded
table clarifying interest plans in which interest is paid
homework assignments. The order of the Practice
more than once a year.
Exercises is exactly the same as tbe order of the
• Section 4.4 (Exponential and logarithmic Equations) section's worked examples. This parallel order enables
has a new discussion (within the context of the Great students to refer to the tilled examples and their
Question! feature) on whether a negative number can detailed explanations to achieve working the
succes.~
belong to the solution set of a logarithmic equation. Practice Exercises.