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
Other versions by this Author
