Table of Contents
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 8
Enumerating Exponential Rules 8
Multiplying and dividing exponents 9
Rooting out exponents 9
Powering up exponents 10
Working with negative exponents 10
Assigning Factoring Techniques 11
Making two terms factor 11
Factoring three terms 12
Factoring four or more terms by grouping 14
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 25
Factoring quadratic binomials 25
Factoring quadratic trinomials 26
The Quadratic Formula to the Rescue 27
Realizing rational solutions 27
Investigating irrational solutions 28
Promoting Quadratic-like Equations 28
Solving Quadratic Inequalities 29
Keeping it strictly quadratic 30
Signing up for fractions 32
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 40
Taking on two radicals 41
Dealing with Negative Exponents 42
Factoring out a negative exponent as a greatest common factor 43
Solving quadratic-like trinomials 44
Fiddling with Fractional Exponents 45
Solving equations by factoring fractional exponents 45
Promoting techniques for working with fractional exponents 45
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 56
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 68
Starting with "a" in the standard form 68
Following "a" with "b" and "c" 69
Eyeing a Quadratic's Intercepts 70
Finding the one and only y-intercept 70
Getting at the x-intercepts 71
Finding the Vertex of a Parabola 72
Computing vertex coordinates 72
Linking up with the axis of symmetry 73
Sketching a Graph from the Available Information 73
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 85
Finding Roots Synthetically 86
Using synthetic division when searching for roots 87
Synthetically dividing by a binomial 89
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 95
Getting the scoop on oblique(slant) asymptotes 96
Discounting Removable Discontinuities 97
Finding removable discontinuities by factoring 98
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 137
Interpreting an intersection 137
Tackling the same line 137
Putting up with parallel lines 138
Using Elimination (Addition) to Solve Systems of Equations 138
Finding Substitution to Be a Satisfactory Substitute 140
Variable substituting made easy 140
Writing solutions for coexisting lines 141
Taking on Systems of Three Linear Equations 142
Finding the solution of a system of three linear equations 142
Generalizing with a system solution 144
Increasing the Number of Equations 145
Intersecting Parabolas and Lines 148
Determining if and where lines and parabolas cross paths 148
Determining that there's no solution 150
Crossing Parabolas with Circles 151
Finding multiple intersections 152
Sifting through the possibilities for solutions 153
Chapter 12 Taking the Complexity Out of Complex Numbers 157
Simplifying Powers of i 158
Getting More Complex with Complex Numbers 159
Performing complex operations 159
Performing complex division by multiplying by the conjugate 160
Simplifying reluctant radicals 161
Unraveling Complex Solutions in Quadratic Equations 161
Investigating Polynomials with Complex Roots 162
Classifying conjugate pairs 163
Making use of complex zeros 163
Chapter 13 Ten (Or So) Special Formulas 165
Using Multiplication to Add 165
Factoring in Factorial 166
Picking Out Permutations 166
Collecting Combinations 166
Adding n Integers 167
Adding n Squared Integers 167
Adding Odd Numbers 167
Going for the Geometric 168
Calculating Compound Interest 168
Index 171
