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Algebra II
Algebra II
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CliffsQuickReview™
Algebra II
By Edward Kohn, MS and David Alan Herzog
An International Data Group Company
Cleveland, OH • Indianapolis, IN • New York, NY
6371-8 FM.F 4/19/01 8:47 AM Page i
CliffsQuickReview™Algebra II
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About the Authors
Ed Kohn is an outstanding educator and author,
with over 33 years’ experience teaching mathe-
matics. Currently, he is the testing coordinator and
math department chairman at the Sherman Oaks
Center for Enriched Studies. Formerly an instruc-
tor at Fairleigh Dickinson University, David Alan
Herzog is the author of over 100 audio visual titles
and computer programs and has written and
edited several books in mathematics and science.
Publisher’s Acknowledgments
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6371-8 FM.F 4/19/01 8:47 AM Page ii
Table of Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
Why You Need This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
How to Use This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
Visit Our Web Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
Chapter 1: Linear Sentences in One Variable . . . . . . . . . . . . . . . . . . . . .5
Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7
Absolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
Compound Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
Chapter 2: Segments, Lines, and InequalIties . . . . . . . . . . . . . . . . . . . .22
Rectangular Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
Distance Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
Midpoint Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
Slope of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26
Slope of Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . .28
Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
Graphs of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
Chapter 3: Linear Sentences in Two Variables . . . . . . . . . . . . . . . . . . .41
Linear Equations: Solutions Using Graphing . . . . . . . . . . . . . . . . . . .41
Linear Equations: Solutions Using Substitution . . . . . . . . . . . . . . . . .43
Linear Equations: Solutions Using Eliminations . . . . . . . . . . . . . . . .44
Linear Equations: Solutions Using Matrices . . . . . . . . . . . . . . . . . . . .47
Linear Equations: Solutions Using Determinants . . . . . . . . . . . . . . . .48
Linear Inequalities: Solutions Using Graphing . . . . . . . . . . . . . . . . . .51
Chapter 4: Linear Equations in Three Variables . . . . . . . . . . . . . . . . . .57
Linear Equations: Solutions Using Elimination . . . . . . . . . . . . . . . . .57
Linear Equations: Solutions Using Matrices . . . . . . . . . . . . . . . . . . . .61
Linear Equations: Solutions Using Determinants . . . . . . . . . . . . . . . .64
Chapter 5: Polynomial Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70
Adding and Subtracting Polynomials . . . . . . . . . . . . . . . . . . . . . . . . .70
Monomial, binomial, and trinomial . . . . . . . . . . . . . . . . . . . . . . .70
Like terms or similar terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71
Ascending and descending order . . . . . . . . . . . . . . . . . . . . . . . . . .72
6371-8 FM.F 4/19/01 8:47 AM Page iii
Multiplying Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73
Special Products of Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75
Dividing Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77
Synthetic Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81
Chapter 6: Factoring Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84
Greatest Common Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .84
Difference of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
Sum or Difference of Cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86
Trinomials of the Form x2+ bx + c . . . . . . . . . . . . . . . . . . . . . . . . . . .88
Trinomials of the Form ax2+ bx +c . . . . . . . . . . . . . . . . . . . . . . . . . .89
Square Trinomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91
Factoring by Regrouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91
Summary of Factoring Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . .92
Solving Equations by Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93
Chapter 7: Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .97
Examples of Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . .97
Simplifying Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . .98
Multiplying Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . .99
Dividing Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100
Adding and Subtracting Rational Expressions . . . . . . . . . . . . . . . . .101
Complex Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
Solving Rational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .107
Proportion, Direct Variation, Inverse Variation, Joint Variation . . . .109
Proportion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
Direct variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110
Inverse variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
Joint variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112
Graphing Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .113
Chapter 8: Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Domain and range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .121
Function Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
Compositions of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124
Algebra of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .125
Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126
iv CliffsQuickReview Algebra II
6371-8 FM.F 4/19/01 8:47 AM Page iv
Table of Contents v
Chapter 9: Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
Polynomial Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132
Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133
Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .134
Zeros of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135
Rational Zero Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135
Graphing Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .137
Chapter 10: Radicals and Complex Numbers . . . . . . . . . . . . . . . . . . .141
Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .141
Simplifying Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144
Adding and Subtracting Radical Expressions . . . . . . . . . . . . . . . . . .144
Multiplying Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . .145
Dividing Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .146
Rationalizing the denominator . . . . . . . . . . . . . . . . . . . . . . . . . .147
Conjugates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148
Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148
Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149
Chapter 11: Quadratics in One Variable . . . . . . . . . . . . . . . . . . . . . . . .153
Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .153
Solving Quadratics by Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . .154
Solving Quadratics by the Square Root Property . . . . . . . . . . . . . . .154
Solving Quadratics by Completing the Square . . . . . . . . . . . . . . . . .155
Solving Quadratics by the Quadratic Formula . . . . . . . . . . . . . . . . .157
Solving Equations in Quadratic Form . . . . . . . . . . . . . . . . . . . . . . .159
Solving Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160
Solving Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165
Chapter 12: Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171
The Four Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .171
Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .172
Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174
Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .179
Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .185
Chapter 13: Quadratic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .192
Systems of Equations Solved Algebraically . . . . . . . . . . . . . . . . . . . .192
Systems of Equations Solved Graphically . . . . . . . . . . . . . . . . . . . . .194
Systems of Inequalities Solved Graphically . . . . . . . . . . . . . . . . . . . .196
6371-8 FM.F 4/19/01 8:47 AM Page v
Chapter 14: Exponential and Logarithmic Functions . . . . . . . . . . . . .200
Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .200
Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .204
Properties of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .207
Exponential and Logarithmic Equations . . . . . . . . . . . . . . . . . . . . .209
Chapter 15: Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . .215
Definition and Examples of Sequences . . . . . . . . . . . . . . . . . . . . . . .215
Arithmetic Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .217
Arithmetic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .218
Geometric Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .219
Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .221
Summation Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .223
Chapter 16: Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226
Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226
Binomial Coefficients and the Binomial Theorem . . . . . . . . . . . . . .227
Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .229
Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .231
Chapter 17: Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233
General Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233
Simple Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .234
Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235
Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236
Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236
Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .239
Arithmetic/Geometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241
CQR Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .244
CQR Resource Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .250
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .253
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .260
vi CliffsQuickReview Algebra II
6371-8 FM.F 4/19/01 8:47 AM Page vi
INTRODUCTION
CliffsQuickReview Algebra II is a comprehensive study guide to the
many topics of a second course in algebra, including information on
subjects ranging from linear equations in one, two, and three variables,
through inequalities, complex numbers, conic sections, quadratic equa-
tions, and logarithms. Whether you are looking for an in-depth treatment
of the entire subject matter of the course or occasional reinforcement of
one or more aspects of algebra, this is the place to find it.
For the purpose of this review, your knowledge of the following funda-
mental ideas is assumed:
■Natural numbers
■Whole numbers
■Integers
■Rational numbers
■Irrational numbers
■Real numbers
■Rules for positive and negative numbers
■Exponents
■Grouping symbols
■Order of operations
■Scientific notation
■Absolute value
If you feel that you are weak in any of these topics, refer to
CliffsQuickReview Algebra I.
6371-8 Intro.F 4/19/01 8:47 AM Page 1
Why You Need This Book
Can you answer yes to any of these questions?
■Do you need to review the fundamentals of Algebra II fast?
■Do you need a course supplement to Algebra II?
■Do you need to prepare for your Algebra II test?
■Do you need a concise, comprehensive reference for Algebra II?
■Do you need a no-nonsense approach to Algebra II that gets you the
results you need?
If so, then CliffsQuickReview Algebra II is for you!
How to Use This Book
You’re in charge here. You get to decide how to use this book. You can read
it straight through (not usually a good idea with a math book) or just look
for the information that you want and then put the book back on the shelf
for later use. Here are a few of the recommended ways to search for infor-
mation about a particular topic:
■Use the Pocket Guide to find essential information, such as impor-
tant algebraic formulas and concepts.
■Look for areas of interest in the book’s Table of Contents or use the
index to find specific topics.
■Flip through the book, looking for subject areas at the top of each page.
■Get a glimpse of what you’ll gain from a chapter by reading through
the “Chapter Check-In” at the beginning of each chapter.
■Use the “Chapter Checkout” at the end of each chapter to gauge your
grasp of the important information you need to know.
■Test your knowledge more completely in the CQR Review and look
for additional sources of information in the CQR Resource Center.
■Look in the Glossary for important terms and definitions. If a word
is boldfaced in the text, you can find a more complete definition in
the book’s glossary.
2CliffsQuickReview Algebra II
6371-8 Intro.F 4/19/01 8:47 AM Page 2
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