Algebra II Essentials For Dummies

Mary Jane Sterling taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of Algebra II For Dummies and Algebra II Workbook For Dummies.

Introduction 1

About This Book 1

Conventions Used in This Book 2

Foolish Assumptions 2

Icons Used in This Book 2

Where to Go from Here 3

Chapter 1: Making Advances in Algebra 5

Bringing Out the Best in Algebraic Properties 5

Making short work of the basic properties 6

Organizing your operations 7

Enumerating Exponential Rules 8

Multiplying and dividing exponents 8

Rooting out exponents 9

Powering up exponents 10

Working with negative exponents 10

Assigning Factoring Techniques 10

Making two terms factor 11

Factoring three terms 12

Factoring four or more terms by grouping 13

Chapter 2: Lining Up Linear Equations 15

Getting the First Degree: Linear Equations 15

Solving basic linear equations 16

Eliminating fractions 16

Lining Up Linear Inequalities 17

Solving basic inequalities 18

Introducing interval notation 19

Absolute Value: Keeping Everything in Line 20

Solving absolute value equations 20

Seeing through absolute value inequality 21

Chapter 3: Making Quick Work of Quadratic Equations 23

Using the Square Root Rule When Possible 24

Solving Quadratic Equations by Factoring 24

Factoring quadratic binomials 25

Factoring quadratic trinomials 26

The Quadratic Formula to the Rescue 27

Realizing rational solutions 27

Investigating irrational solutions 27

Promoting Quadratic-like Equations 28

Solving Quadratic Inequalities 29

Keeping it strictly quadratic 30

Signing up for fractions 31

Increasing the number of factors 33

Chapter 4: Rolling Along with Rational and Radical Equations 35

Rounding Up Rational Equations and Eliminating Fractions 35

Making your least common denominator work for you 36

Proposing proportions for solving rational equations 38

Reasoning with Radicals 39

Squaring both sides of the equation 39

Taking on two radicals 40

Dealing with Negative Exponents 42

Factoring out a negative exponent as a greatest common factor 42

Solving quadratic-like trinomials 43

Fiddling with Fractional Exponents 44

Solving equations by factoring fractional exponents 44

Promoting techniques for working with fractional exponents 44

Chapter 5: Forging Function Facts 47

Describing Function Characteristics 47

Denoting function notation 48

Using function notation to evaluate functions 48

Determining Domain and Range 49

Delving into domain 49

Wrangling with range 50

Counting on Even and Odd Functions 51

Determining whether even or odd 52

Using even and odd functions in graphs 53

Taking on Functions One-to-One 53

Defining which functions are one-to-one 54

Testing for one-to-one functions 54

Composing Functions 55

Composing yourself with functions 55

Composing with the difference quotient 56

Getting into Inverse Functions 57

Finding which functions are inverses 58

Finding an inverse of a function 59

Chapter 6: Graphing Linear and Quadratic Functions 61

Identifying Some Graphing Techniques 61

Finding x- and y-intercepts 62

Reflecting on a graph’s symmetry 62

Mastering the Graphs of Lines 64

Determining the slope of a line 64

Describing two line equations 65

Identifying parallel and perpendicular lines 67

Coming to Terms with the Standard Form of a Quadratic 67

Starting with “a” in the standard form 68

Following “a” with “b” and “c” 69

Eyeing a Quadratic’s Intercepts 69

Finding the one and only y-intercept 69

Getting at the x-intercepts 70

Finding the Vertex of a Parabola 71

Computing vertex coordinates 71

Linking up with the axis of symmetry 72

Sketching a Graph from the Available Information 72

Chapter 7: Pondering Polynomials 75

Sizing Up a Polynomial Equation 75

Identifying Intercepts and Turning Points 76

Interpreting relative value and absolute value 76

Dealing with intercepts and turning points 77

Solving for y-intercepts and x-intercepts 78

Determining When a Polynomial is Positive or Negative 79

Incorporating a sign line 79

Recognizing a sign change rule 80

Solving Polynomial Equations 81

Factoring for roots 81

Taking sane steps with the rational root theorem 82

Putting Descartes in charge of signs 84

Finding Roots Synthetically 86

Using synthetic division when searching for roots 86

Synthetically dividing by a binomial 88

Chapter 8: Being Respectful of Rational Functions 91

Examining Rational Functions 91

Deliberating on domain 92

Investigating intercepts 92

Assigning Roles to Asymptotes 93

Validating vertical asymptotes 93

Finding equations for horizontal asymptotes 94

Taking vertical and horizontal asymptotes to graphs 94

Getting the scoop on oblique (slant) asymptotes 96

Discounting Removable Discontinuities 97

Finding removable discontinuities by factoring 97

Evaluating the removals 98

Looking at Limits of Rational Functions 99

Determining limits at function discontinuities 100

Finding infinity 102

Looking at infinity 104

Chapter 9: Examining Exponential and Logarithmic Functions 107

Computing Exponentially 107

Getting to the Base of Exponential Functions 108

Classifying bases 108

Introducing the more frequently used bases: 10 and e 110

Exponential Equation Solutions 110

Creating matching bases 111

Quelling quadratic patterns 111

Looking into Logarithmic Functions 113

Presenting the properties of logarithms 113

Doing more with logs than sawing 115

Solving Equations Containing Logs 117

Seeing all logs created equal 117

Solving log equations by changing to exponentials 118

Chapter 10: Getting Creative with Conics 121

Posing with Parabolas 122

Generalizing the form of a parabola’s equation 123

Making short work of a parabola’s sketch 124

Changing a parabola’s equation to the standard form 125

Circling around a Conic 126

Getting Eclipsed by Ellipses 127

Determining the shape 129

Finding the foci 130

Getting Hyped for Hyperbolas 130

Including the asymptotes 131

Graphing hyperbolas 132

Chapter 11: Solving Systems of Equations 135

Looking at Solutions Using the Standard Linear-Systems Form 136

Solving Linear Systems by Graphing 136

Interpreting an intersection 137

Tackling the same line 137

Putting up with parallel lines 137

Using Elimination (Addition) to Solve Systems of Equations 138

Finding Substitution to Be a Satisfactory Substitute 139

Variable substituting made easy 139

Writing solutions for coexisting lines 140

Taking on Systems of Three Linear Equations 141

Finding the solution of a system of three linear equations 141

Generalizing with a system solution 143

Increasing the Number of Equations 144

Intersecting Parabolas and Lines 146

Determining if and where lines and parabolas cross paths 147

Determining that there’s no solution 149

Crossing Parabolas with Circles 150

Finding multiple intersections 150

Sifting through the possibilities for solutions 151

Chapter 12: Taking the Complexity Out of Complex Numbers 155

Simplifying Powers of i 156

Getting More Complex with Complex Numbers 157

Performing complex operations 157

Performing complex division by multiplying by the conjugate 158

Simplifying reluctant radicals 159

Unraveling Complex Solutions in Quadratic Equations 160

Investigating Polynomials with Complex Roots 160

Classifying conjugate pairs 161

Making use of complex zeros 161

Chapter 13: Ten (or So) Special Formulas 163

Using Multiplication to Add 163

Factoring in Factorial 164

Picking Out Permutations 164

Collecting Combinations 164

Adding n Integers 165

Adding n Squared Integers 165

Adding Odd Numbers 165

Going for the Geometric 166

Calculating Compound Interest 166

Index 167