Mary Jane Sterling was a lecturer in mathematics for more than 35 years, teaching courses in algebra, calculus, and linear programming. She is the author ofAlgebra I For Dummies, Trigonometry For Dummies, Algebra Workbook For Dummies, and Trigonometry Workbook For Dummies.

Introduction 1

About This Book 1

Foolish Assumptions 2

Icons Used in This Book 3

Beyond the Book 4

Where to Go from Here 4

Part 1: Homing in on Basic Solutions 5

Chapter 1: Going Beyond Beginning Algebra 7

Outlining Algebraic Properties 8

Keeping order with the commutative property 8

Maintaining group harmony with the associative property 9

Distributing a wealth of values 9

Checking out an algebraic ID 10

Singing along in-verses 11

Ordering Your Operations 11

Zeroing in on the Multiplication Property of Zero 12

Expounding on Exponential Rules 13

Multiplying and dividing exponents 13

Getting to the roots of exponents 14

Raising or lowering the roof with exponents 14

Making nice with negative exponents 15

Implementing Factoring Techniques 15

Factoring two terms 16

Taking on three terms 17

Factoring four or more terms by grouping 19

Chapter 2: Toeing the Straight Line: Linear Equations 21

Linear Equations: Handling the First Degree 21

Tackling basic linear equations 22

Clearing out fractions 23

Isolating different unknowns 24

Linear Inequalities: Algebraic Relationship Therapy 25

Solving linear inequalities 26

Introducing interval notation 27

Compounding inequality issues 28

Absolute Value: Keeping Everything in Line 30

Solving absolute value equations 31

Seeing through absolute value inequality 31

Chapter 3: Conquering Quadratic Equations 35

Implementing the Square Root Rule 36

Dismantling Quadratic Equations into Factors 37

Factoring binomials 37

Factoring trinomials 39

Factoring by grouping 40

Resorting to the Quadratic Formula 41

Finding rational solutions 42

Straightening out irrational solutions 42

Formulating huge quadratic results 43

Completing the Square: Warming Up for Conics 43

Squaring up a quadratic equation 44

Completing the square twice over 45

Tackling Higher-Powered Polynomials 46

Handling the sum or difference of cubes 47

Tackling quadratic-like trinomials 48

Solving Quadratic Inequalities 49

Keeping inequality strictly quadratic 50

Signing up for fractions 52

Increasing the number of factors 53

Considering absolute value inequalities 53

Chapter 4: Rooting Out the Rational, Radical, and Negative 55

Acting Rationally with Fraction-Filled Equations 56

Systematically solving rational equations 56

Solving rational equations with proportions 60

Ridding Yourself of a Radical 61

Squaring both sides of a radical equation 62

Calming two radicals 63

Changing Negative Attitudes about Exponents 65

Flipping negative exponents out of the picture 65

Factoring out negatives to solve equations 66

Fooling Around with Fractional Exponents 68

Combining terms with fractional exponents 69

Factoring fractional exponents 69

Solving equations by working with fractional exponents 70

Chapter 5: Graphing Your Way to the Good Life 73

Coordinating Your Graphing Efforts 74

Identifying the parts of the coordinate plane 74

Plotting from dot to dot 75

Streamlining the Graphing Process with Intercepts and Symmetry 76

Finding x- and y-intercepts 77

Reflecting on a graph’s symmetry 78

Graphing Lines 80

Finding the slope of a line 81

Facing two types of equations for lines 82

Identifying parallel and perpendicular lines 85

Looking at 10 Basic Forms 86

Lines and quadratics 86

Cubics and quartics 87

Radicals and rationals 87

Exponential and logarithmic curves 88

Absolute values and circles 89

Solving Problems with a Graphing Calculator 89

Entering equations into graphing calculators correctly 90

Looking through the graphing window 92

Part 2: Facing Off with Functions 95

Chapter 6: Formulating Function Facts 97

Defining Functions 98

Introducing function notation 98

Evaluating functions 98

Homing In on Domain and Range 99

Determining a function’s domain 99

Describing a function’s range 100

Betting on Even or Odd Functions 102

Recognizing even and odd functions 102

Applying even and odd functions to graphs 103

Facing One-to-One Confrontations 104

Defining one-to-one functions 104

Eliminating one-to-one violators 105

Going to Pieces with Piecewise Functions 106

Doing piecework 107

Applying piecewise functions 108

Composing Yourself and Functions 110

Performing compositions 110

Simplifying the difference quotient 111

Singing Along with Inverse Functions 112

Determining if functions are inverses 112

Solving for the inverse of a function 113

Chapter 7: Sketching and Interpreting Quadratic Functions 115

Interpreting the Standard Form of Quadratics 116

Starting with “a” in the standard form 116

Following up with “b” and “c” 117

Investigating Intercepts in Quadratics 118

Finding the one and only y-intercept 119

Finding the x-intercepts 120

Going to the Extreme: Finding the Vertex 123

Lining Up along the Axis of Symmetry 124

Sketching a Graph from the Available Information 125

Applying Quadratics to the Real World 127

Selling candles 127

Shooting basketballs 128

Launching a water balloon 130

Chapter 8: Staying Ahead of the Curves: Polynomials 133

Taking a Look at the Standard Polynomial Form 134

Exploring Polynomial Intercepts and Turning Points 134

Interpreting relative value and absolute value 135

Counting intercepts and turning points 135

Solving for polynomial intercepts 138

Determining Positive and Negative Intervals 139

Using a sign-line 140

Interpreting the rule 141

Finding the Roots of a Polynomial 143

Factoring for polynomial roots 143

Saving your sanity: The Rational Root Theorem 145

Letting Descartes make a ruling on signs 148

Synthesizing Root Findings 150

Using synthetic division to test for roots 150

Synthetically dividing by a binomial 153

Wringing out the Remainder (Theorem) 154

Chapter 9: Reasoning with Rational Functions 157

Exploring Rational Functions 158

Sizing up domain 158

Introducing intercepts 159

Adding Asymptotes to the Rational Pot 160

Determining the equations of vertical asymptotes 160

Determining the equations of horizontal asymptotes 161

Graphing vertical and horizontal asymptotes 161

Crunching the numbers and graphing oblique asymptotes 163

Accounting for Removable Discontinuities 164

Removal by factoring 164

Evaluating the removal restrictions 165

Showing removable discontinuities on a graph 165

Pushing the Limits of Rational Functions 167

Evaluating limits at discontinuities 168

Going to infinity 170

Catching rational limits at infinity 172

Putting It All Together: Sketching Rational Graphs from Clues 173

Chapter 10: Exposing Exponential and Logarithmic Functions 177

Evaluating Exponential Expressions 178

Exponential Functions: It’s All about the Base, Baby 179

Observing the trends in bases 179

Meeting the most frequently used bases: 10 and e 180

Solving Exponential Equations 182

Making bases match 182

Recognizing and using quadratic patterns 184

Showing an “Interest” in Exponential Functions 186

Applying the compound interest formula 186

Looking at continuous compounding 188

Logging On to Logarithmic Functions 189

Meeting the properties of logarithms 190

Putting your logs to work 191

Solving Logarithmic Equations 193

Setting log equal to log 194

Rewriting log equations as exponentials 195

Graphing Exponential and Logarithmic Functions 196

Expounding on the exponential 196

Not seeing the logs for the trees 198

Part 3: Conquering Conics and Systems of Equations 203

Chapter 11: Cutting Up Conic Sections 205

Cutting Up a Cone 206

Opening Every Which Way with Parabolas 206

Looking at parabolas with vertices at the origin 207

Observing the general form of parabola equations 210

Sketching the graphs of parabolas 211

Converting parabolic equations to the standard form 214

Going Round and Round in Conic Circles 215

Standardizing the circle 215

Specializing in circles 217

Preparing Your Eyes for Solar Ellipses 218

Raising the standards of an ellipse 218

Sketching an elliptical path 221

Feeling Hyper about Hyperbolas 222

Including the asymptotes 223

Graphing hyperbolas 224

Identifying Conics from Their Equations, Standard or Not 227

Chapter 12: Solving Systems of Linear Equations 229

Looking at the Standard Linear-Systems Form and Its Possible Solutions 230

Graphing Solutions of Linear Systems 230

Pinpointing the intersection 231

Toeing the same line twice 232

Dealing with parallel lines 232

Solving Systems of Two Linear Equations by Using Elimination 233

Getting to the point with elimination 234

Recognizing solutions indicating parallel or coexisting lines 235

Making Substitution the Choice 236

Variable substituting made easy 236

Identifying parallel and coexisting lines 237

Using Cramer’s Rule to Defeat Unwieldy Fractions 238

Setting up the linear system for Cramer 239

Applying Cramer’s Rule to a linear system 240

Tackling Linear Systems with Three Linear Equations 241

Solving three-equation systems with algebra 241

Generalizing multiple solutions for linear equations 243

Upping the Ante with Larger Systems 244

Applying Linear Systems to Our 3-D World 247

Using Systems to Decompose Fractions 248

Chapter 13: Solving Systems of Nonlinear Equations and Inequalities 251

Crossing Parabolas with Lines 252

Determining the point(s) where a line and parabola cross paths 253

Dealing with a solution that’s no solution 254

Intertwining Parabolas and Circles 255

Managing multiple intersections 256

Sorting out the solutions 258

Planning Your Attack on Other Systems of Equations 260

Mixing polynomials and lines 260

Crossing polynomials 261

Navigating exponential intersections 263

Rounding up rational functions 265

Playing Fair with Inequalities 268

Drawing and quartering inequalities 268

Graphing areas with curves and lines 269

Part 4: Shifting into High Gear with Advanced Concepts 271

Chapter 14: Simplifying Complex Numbers in a Complex World 273

Using Your Imagination to Simplify Powers of i 274

Understanding the Complexity of Complex Numbers 275

Operating on complex numbers 276

Multiplying by the conjugate to perform division 277

Simplifying radicals 279

Solving Quadratic Equations with Complex Solutions 280

Working Polynomials with Complex Solutions 282

Identifying conjugate pairs 283

Interpreting complex zeros 283

Chapter 15: Making Moves with Matrices 287

Describing the Different Types of Matrices 288

Row and column matrices 289

Square matrices 289

Zero matrices 289

Identity matrices 289

Performing Operations on Matrices 290

Adding and subtracting matrices 290

Multiplying matrices by scalars 291

Multiplying two matrices 291

Applying matrices and operations 293

Defining Row Operations 297

Finding Inverse Matrices 298

Determining additive inverses 299

Determining multiplicative inverses 299

Dividing Matrices by Using Inverses 304

Using Matrices to Find Solutions for Systems of Equations 305

Chapter 16: Making a List: Sequences and Series 307

Understanding Sequence Terminology 308

Using sequence notation 308

No-fear factorials in sequences 309

Alternating sequential patterns 309

Looking for sequential patterns 310

Taking Note of Arithmetic and Geometric Sequences 313

Finding common ground: Arithmetic sequences 313

Taking the multiplicative approach: Geometric sequences 315

Recursively Defining Functions 317

Making a Series of Moves 318

Introducing summation notation 318

Summing arithmetically 319

Summing geometrically 320

Applying Sums of Sequences to the Real World 323

Stacking the blocks 323

Negotiating your allowance 323

Bouncing a ball 324

Highlighting Special Formulas 326

Chapter 17: Everything You Wanted to Know about Sets 329

Revealing Set Notation 329

Listing elements with a roster 330

Building sets from scratch 330

Going for all (universal set) or nothing (empty set) 331

Subbing in with subsets 331

Operating on Sets 333

Celebrating the union of two sets 333

Looking both ways for set intersections 334

Feeling complementary about sets 335

Counting the elements in sets 335

Drawing Venn You Feel Like It 336

Applying the Venn diagram 337

Using Venn diagrams with set operations 338

Adding a set to a Venn diagram 339

Focusing on Factorials 342

Making factorial manageable 342

Simplifying factorials 343

How Do I Love Thee? Let Me Count Up the Ways 344

Applying the multiplication principle to sets 344

Arranging permutations of sets 345

Mixing up sets with combinations 348

Branching Out with Tree Diagrams 350

Picturing a tree diagram for a permutation 351

Drawing a tree diagram for a combination 352

Part 5: The Part of Tens 353

Chapter 18: Ten Multiplication Tricks 355

Squaring Numbers That End in 5 355

Finding the Next Perfect Square 356

Recognizing the Pattern in Multiples of 9 and 11 357

Casting Out 9s 357

Casting Out 9s: The Multiplication Moves 358

Multiplying by 11 359

Multiplying by 5 360

Finding Common Denominators 361

Determining Divisors 362

Multiplying Two-Digit Numbers 362

Chapter 19: Ten Special Types of Numbers 365

Triangular Numbers 365

Square Numbers 366

Hexagonal Numbers 366

Perfect Numbers 367

Amicable Numbers 367

Happy Numbers 368

Abundant Numbers 368

Deficient Numbers 368

Narcissistic Numbers 368

Prime Numbers 369

Index 371

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