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 graphâs 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 functionâs domain 99
Describing a functionâs 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 âaâ in the standard form 116
Following up with âbâ and âcâ 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: Itâs 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 âInterestâ 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 Cramerâs Rule to Defeat Unwieldy Fractions 238
Setting up the linear system for Cramer 239
Applying Cramerâs 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 thatâs 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
