Algebra II For Dummies

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 1About This Book 1Foolish Assumptions 2Icons Used in This Book 3Beyond the Book 4Where to Go from Here 4Part 1: Homing In On Basic Solutions 5Chapter 1: Going Beyond Beginning Algebra 7Outlining Algebraic Properties 8Keeping order with the commutative property 8Maintaining group harmony with the associative property 9Distributing a wealth of values 9Checking out an algebraic ID 10Singing along in-verses 11Ordering Your Operations 11Zeroing in on the Multiplication Property of Zero 12Expounding on Exponential Rules 13Multiplying and dividing exponents 13Getting to the roots of exponents 14Raising or lowering the roof with exponents 14Making nice with negative exponents 15Implementing Factoring Techniques 15Factoring two terms 16Taking on three terms 17Factoring four or more terms by grouping 19Chapter 2: Toeing the Straight Line: Linear Equations 21Linear Equations: Handling the First Degree 21Tackling basic linear equations 22Clearing out fractions 23Isolating different unknowns 24Linear Inequalities: Algebraic Relationship Therapy 25Solving linear inequalities 26Introducing interval notation 27Compounding inequality issues 28Absolute Value: Keeping Everything in Line 30Solving absolute value equations 31Seeing through absolute value inequality 31Chapter 3: Conquering Quadratic Equations 35Implementing the Square Root Rule 36Dismantling Quadratic Equations into Factors 37Factoring binomials 37Factoring trinomials 39Factoring by grouping 40Resorting to the Quadratic Formula 41Finding rational solutions 42Straightening out irrational solutions 42Formulating huge quadratic results 43Completing the Square: Warming Up for Conics 43Squaring up a quadratic equation 44Completing the square twice over 45Tackling Higher-Powered Polynomials 46Handling the sum or difference of cubes 47Tackling quadratic-like trinomials 48Solving Quadratic Inequalities 49Keeping inequality strictly quadratic 50Signing up for fractions 52Increasing the number of factors 53Considering absolute value inequalities 53Chapter 4: Rooting Out the Rational, Radical, and Negative 55Acting Rationally with Fraction-Filled Equations 56Systematically solving rational equations 56Solving rational equations with proportions 60Ridding Yourself of a Radical 61Squaring both sides of a radical equation 62Calming two radicals 63Changing Negative Attitudes about Exponents 65Flipping negative exponents out of the picture 65Factoring out negatives to solve equations 66Fooling Around with Fractional Exponents 68Combining terms with fractional exponents 69Factoring fractional exponents 69Solving equations by working with fractional exponents 70Chapter 5: Graphing Your Way to the Good Life 73Coordinating Your Graphing Efforts 74Identifying the parts of the coordinate plane 74Plotting from dot to dot 75Streamlining the Graphing Process with Intercepts and Symmetry 76Finding x- and y-intercepts 77Reflecting on a graph’s symmetry 78Graphing Lines 80Finding the slope of a line 81Facing two types of equations for lines 82Identifying parallel and perpendicular lines 85Looking at 10 Basic Forms 86Lines and quadratics 86Cubics and quartics 87Radicals and rationals 87Exponential and logarithmic curves 88Absolute values and circles 89Solving Problems with a Graphing Calculator 89Entering equations into graphing calculators correctly 90Looking through the graphing window 92Part 2: Facing Off With Functions 95Chapter 6: Formulating Function Facts 97Defining Functions 98Introducing function notation 98Evaluating functions 98Homing In on Domain and Range 99Determining a function’s domain 99Describing a function’s range 100Betting on Even or Odd Functions 102Recognizing even and odd functions 102Applying even and odd functions to graphs 103Facing One-to-One Confrontations 104Defining one-to-one functions 104Eliminating one-to-one violators 105Going to Pieces with Piecewise Functions 106Doing piecework 107Applying piecewise functions 108Composing Yourself and Functions 110Performing compositions 110Simplifying the difference quotient 111Singing Along with Inverse Functions 112Determining if functions are inverses 112Solving for the inverse of a function 113Chapter 7: Sketching and Interpreting Quadratic Functions 115Interpreting the Standard Form of Quadratics 116Starting with “a” in the standard form 116Following up with “b” and “c” 117Investigating Intercepts in Quadratics 118Finding the one and only y-intercept 119Finding the x-intercepts 120Going to the Extreme: Finding the Vertex 123Lining Up along the Axis of Symmetry 124Sketching a Graph from the Available Information 125Applying Quadratics to the Real World 127Selling candles 127Shooting basketballs 128Launching a water balloon 130Chapter 8: Staying Ahead of the Curves: Polynomials 133Taking a Look at the Standard Polynomial Form 134Exploring Polynomial Intercepts and Turning Points 134Interpreting relative value and absolute value 135Counting intercepts and turning points 135Solving for polynomial intercepts 138Determining Positive and Negative Intervals 139Using a sign-line 140Interpreting the rule 141Finding the Roots of a Polynomial 143Factoring for polynomial roots 143Saving your sanity: The Rational Root Theorem 145Letting Descartes make a ruling on signs 148Synthesizing Root Findings 150Using synthetic division to test for roots 150Synthetically dividing by a binomial 153Wringing out the Remainder (Theorem) 154Chapter 9: Reasoning with Rational Functions 157Exploring Rational Functions 158Sizing up domain 158Introducing intercepts 159Adding Asymptotes to the Rational Pot 160Determining the equations of vertical asymptotes 160Determining the equations of horizontal asymptotes 161Graphing vertical and horizontal asymptotes 161Crunching the numbers and graphing oblique asymptotes 163Accounting for Removable Discontinuities 164Removal by factoring 164Evaluating the removal restrictions 165Showing removable discontinuities on a graph 165Pushing the Limits of Rational Functions 167Evaluating limits at discontinuities 168Going to infinity 170Catching rational limits at infinity 172Putting It All Together: Sketching Rational Graphs from Clues 173Chapter 10: Exposing Exponential and Logarithmic Functions 177Evaluating Exponential Expressions 178Exponential Functions: It’s All about the Base, Baby 179Observing the trends in bases 179Meeting the most frequently used bases: 10 and e 180Solving Exponential Equations 182Making bases match 182Recognizing and using quadratic patterns 184Showing an “Interest” in Exponential Functions 186Applying the compound interest formula 186Looking at continuous compounding 188Logging On to Logarithmic Functions 189Meeting the properties of logarithms 190Putting your logs to work 191Solving Logarithmic Equations 193Setting log equal to log 194Rewriting log equations as exponentials 195Graphing Exponential and Logarithmic Functions 196Expounding on the exponential 196Not seeing the logs for the trees 198Part 3: Conquering Conics And Systems Of Equations 203Chapter 11: Cutting Up Conic Sections 205Cutting Up a Cone 206Opening Every Which Way with Parabolas 206Looking at parabolas with vertices at the origin 207Observing the general form of parabola equations 210Sketching the graphs of parabolas 211Converting parabolic equations to the standard form 214Going Round and Round in Conic Circles 215Standardizing the circle 215Specializing in circles 217Preparing Your Eyes for Solar Ellipses 218Raising the standards of an ellipse 218Sketching an elliptical path 221Feeling Hyper about Hyperbolas 222Including the asymptotes 223Graphing hyperbolas 224Identifying Conics from Their Equations, Standard or Not 227Chapter 12: Solving Systems of Linear Equations 229Looking at the Standard Linear-Systems Form and Its Possible Solutions 230Graphing Solutions of Linear Systems 230Pinpointing the intersection 231Toeing the same line twice 232Dealing with parallel lines 232Solving Systems of Two Linear Equations by Using Elimination 233Getting to the point with elimination 234Recognizing solutions indicating parallel or coexisting lines 235Making Substitution the Choice 236Variable substituting made easy 236Identifying parallel and coexisting lines 237Using Cramer’s Rule to Defeat Unwieldy Fractions 238Setting up the linear system for Cramer 239Applying Cramer’s Rule to a linear system 240Tackling Linear Systems with Three Linear Equations 241Solving three-equation systems with algebra 241Generalizing multiple solutions for linear equations 243Upping the Ante with Larger Systems 244Applying Linear Systems to Our 3-D World 247Using Systems to Decompose Fractions 248Chapter 13: Solving Systems of Nonlinear Equations and Inequalities 251Crossing Parabolas with Lines 252Determining the point(s) where a line and parabola cross paths 253Dealing with a solution that’s no solution 254Intertwining Parabolas and Circles 255Managing multiple intersections 256Sorting out the solutions 258Planning Your Attack on Other Systems of Equations 260Mixing polynomials and lines 260Crossing polynomials 261Navigating exponential intersections 263Rounding up rational functions 265Playing Fair with Inequalities 268Drawing and quartering inequalities 268Graphing areas with curves and lines 269Part 4: Shifting Into High Gear With Advanced Concepts 271Chapter 14: Simplifying Complex Numbers in a Complex World 273Using Your Imagination to Simplify Powers of i 274Understanding the Complexity of Complex Numbers 275Operating on complex numbers 276Multiplying by the conjugate to perform division 277Simplifying radicals 279Solving Quadratic Equations with Complex Solutions 280Working Polynomials with Complex Solutions 282Identifying conjugate pairs 283Interpreting complex zeros 283Chapter 15: Making Moves with Matrices 287Describing the Different Types of Matrices 288Row and column matrices 289Square matrices 289Zero matrices 289Identity matrices 289Performing Operations on Matrices 290Adding and subtracting matrices 290Multiplying matrices by scalars 291Multiplying two matrices 291Applying matrices and operations 293Defining Row Operations 297Finding Inverse Matrices 298Determining additive inverses 299Determining multiplicative inverses 299Dividing Matrices by Using Inverses 304Using Matrices to Find Solutions for Systems of Equations 305Chapter 16: Making a List: Sequences and Series 307Understanding Sequence Terminology 308Using sequence notation 308No-fear factorials in sequences 309Alternating sequential patterns 309Looking for sequential patterns 310Taking Note of Arithmetic and Geometric Sequences 313Finding common ground: Arithmetic sequences 313Taking the multiplicative approach: Geometric sequences 315Recursively Defining Functions 317Making a Series of Moves 318Introducing summation notation 318Summing arithmetically 319Summing geometrically 320Applying Sums of Sequences to the Real World 323Stacking the blocks 323Negotiating your allowance 323Bouncing a ball 324Highlighting Special Formulas 326Chapter 17: Everything You Wanted to Know about Sets 329Revealing Set Notation 329Listing elements with a roster 330Building sets from scratch 330Going for all (universal set) or nothing (empty set) 331Subbing in with subsets 331Operating on Sets 333Celebrating the union of two sets 333Looking both ways for set intersections 334Feeling complementary about sets 335Counting the elements in sets 335Drawing Venn You Feel Like It 336Applying the Venn diagram 337Using Venn diagrams with set operations 338Adding a set to a Venn diagram 339Focusing on Factorials 342Making factorial manageable 342Simplifying factorials 343How Do I Love Thee? Let Me Count Up the Ways 344Applying the multiplication principle to sets 344Arranging permutations of sets 345Mixing up sets with combinations 348Branching Out with Tree Diagrams 350Picturing a tree diagram for a permutation 351Drawing a tree diagram for a combination 352Part 5: The Part Of Tens 353Chapter 18: Ten Multiplication Tricks 355Squaring Numbers That End in 5 355Finding the Next Perfect Square 356Recognizing the Pattern in Multiples of 9 and 11 357Casting Out 9s 357Casting Out 9s: The Multiplication Moves 358Multiplying by 11 359Multiplying by 5 360Finding Common Denominators 361Determining Divisors 362Multiplying Two-Digit Numbers 362Chapter 19: Ten Special Types of Numbers 365Triangular Numbers 365Square Numbers 366Hexagonal Numbers 366Perfect Numbers 367Amicable Numbers 367Happy Numbers 368Abundant Numbers 368Deficient Numbers 368Narcissistic Numbers 368Prime Numbers 369Index 371