Algebra II For Dummies

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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

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.

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