Index

Symbols

  • ! (factorial), 342–343
  • ∑ (epsilon), 330
  • ≥ (greater than or equal to), 25
  • ≤ (less than or equal to), 25
  • < (less than), 25
  • > (greater than), 25
  • || (absolute value), 30
  • ^ (caret), 91

A

  • absolute value inequalities, 53–54
  • absolute values, 30–34
    • defined, 30
    • graphs of, 89
    • inequalities, 31–33
    • solving equations, 31
  • abundant numbers, 368
  • addition (adding). See also summing sequences
    • ancient symbols, 29
    • associative property of, 9
    • casting out 9s, 357–358
    • commutative property of, 8
    • complex numbers, 276–277
    • matrices, 290–291
    • order of operations, 11–12
  • additive identity, applying, 10
  • additive inverses
    • defined, 11
    • determining, 299
  • algebra
    • basics of, 7
    • properties of, 8–11
      • associative property, 9
      • commutative property, 8
      • distributive property, 9–10
      • identities, 10
      • inverses, 11
  • Algebra For Dummies, 3, 30
  • Algebra I For Dummies, 39
  • Algebra I Workbook For Dummies, 150, 163
  • Algebra Rules icon, 3
  • alternating sequential patterns, 309–310
  • amicable numbers, 367
  • ancient symbols, 29
  • arithmetic sequences
    • overview, 313
    • recursive rule for, 317
    • summing, 319–320
  • associative property, 9
  • asymptotes
    • defined, 160
    • graphing
      • oblique asymptotes, 163
      • overview, 160
      • vertical and horizontal asymptotes, 161–163
    • horizontal, 161
    • of hyperbolas, 223–224
    • oblique, 163
    • vertical, 160
  • axis of symmetry, of quadratic functions, 124–125

B

  • bases, of exponential functions, 179–182
  • basketballs, shooting, quadratic function example, 128–130
  • binomials
    • defined, 2
    • factoring into the product of two, 71
    • synthetic division by, 153–154
  • blocks, stacking, 323
  • bouncing a ball, 324–326

C

  • calculators, graphing
    • entering equations into, 90–93
    • entering exponents, 91
    • entering fractions, 90–91
    • finding irrational roots of polynomials, 145
    • negating or subtracting in, 92
    • calculators, graphing (continued)
    • radicals in, 91–92
    • sketching a parabola, 126
    • solving problems with, 89–93
    • window of, 92–93
  • Calculus For Dummies (Ryan), 112
  • candles, quadratic function example, 127–128
  • caret (^), 91
  • Cartesian coordinate system, 74
  • circles
    • with center at the origin, 217
    • circumference of, 217
    • defined, 215
    • graphs of, 89
    • parabolas intersecting with
      • finding common solutions, 258–259
      • multiple intersections, 256–258
    • standard form for the equation of, 215
    • standardizing, 215–216
    • unit, 217
  • circumference of circles, 217
  • coefficients
    • defined, 2
    • quadratic functions, 116
  • coexisting lines, systems of linear equations and
    • identifying, 237–238
    • recognizing, 235–236
  • columns of matrices, 289
  • combinations
    • defined, 348
    • mixing up sets with, 348–350
    • tree diagrams for, 352
  • combining terms with fractional exponents, 68
  • common denominator
    • finding, 361
    • least (LCD)
      • finding, 56–57
      • solving rational equations, 56–59
  • commutative property, 8
  • complements of sets, 335
  • completing the square
    • conic sections, 43–46
    • twice over, 45–46
  • complex numbers
    • adding, 276
    • applications of, 276
    • examples of, 276
    • multiplying, 277
    • multiplying by the conjugate to perform division, 277–279
    • operations on, 276–277
    • standard form, 275
    • subtracting, 277
  • complex solutions
    • solving polynomials with, 282–285
    • solving quadratic equations with, 280–282
  • complex zeros, 283–285
  • composition of functions, 110
  • compound inequality, 28
  • compound interest formula, 186–188
  • compounding, continuous, 188–189
  • conic sections. See also parabolas
    • completing the square, 43–46
    • definition and features, 206
    • identifying from their equations, 227–228
    • overview, 205–206
  • conjugate, multiplying by to perform division, 277–279
  • conjugate axis, 222
  • conjugate pairs, complex zeros in, 283
  • constant, defined, 2
  • constant terms
    • polynomials with, Rational Root Theorem and, 146–147
    • polynomials without, Rational Root Theorem and, 147
  • continuous compounding, 188–189
  • coordinate plane, 74–75
  • coordinates, defined, 75
  • counting off the slope, 83
  • Cramer, Gabriel, 229, 241
  • Cramer’s Rule, 238–241
  • crossing polynomials, 261–262
  • cross-products, solving rational equations with, 61
  • cube roots, solving cubes by taking, 48
  • cubics, 87
  • curves
    • cubic, 87
    • disconnecting, on graphing calculators, 93
    • exponential, 88–89
    • logarithmic, 88–89
    • quadratic, 86
    • quartic, 87
    • radical, 88–89
    • rational, 88–89

D

  • Dantzig, George, 296
  • decomposing fractions, 248–250
  • deficient numbers, 368
  • derivatives, difference quotient and, 111
  • Descartes, Rene, 74
    • Rule of Signs, 148–150
  • diagrams
    • tree
      • for combinations, 352
      • for permutations, 351
    • Venn
      • adding sets to, 339–341
      • applying, 337–338
      • overview, 336
      • with set operations, 338–339
  • difference between terms, in sequential patterns, 311
  • difference of cubes, quadratic equations, 47–48
  • difference of two perfect squares, factoring two terms, 16
  • difference of two squares, factoring, quadratic equations, 38–39
  • difference quotient, 111–112
  • Diophantes, 29
  • directrix, defined, 206
  • disconnecting curves, on graphing calculators, 93
  • discontinuities
    • defined, 108, 160
    • rational functions and, 160
    • removable, rational functions and
      • determining an existent limit without tables, 169–170
      • determining which functions have limits, 170
      • evaluating limits at discontinuities, 168–169
      • evaluating the removal restrictions, 165
      • overview, 164–166
      • showing on a graph, 165–166
  • distributive property, 9–10
  • divide/average method, for finding square roots, 279
  • division (dividing)
    • complex numbers, 278–279
    • exponents, 13–14
    • matrices, by using inverses, 304
    • order of operations, 11–12
    • synthetic
      • by binomials, 153–154
      • Remainder Theorem, 154–155
      • to test for roots, 150–153
  • divisors, determining, 362
  • domains
    • of functions, 99–100
    • of rational functions, 158
  • double roots, trinomials, 40

E

  • effective rates, 187–188
  • elimination, solving systems of two linear equations by using, 233–236
  • ellipses
    • defined, 218
    • determining the shape of, 219–220
    • finding the foci of, 220–221
    • overview, 218
    • sketching graph of, 221–222
    • standard equation for, 218–219
    • three dots following a short list of terms, 308
  • empty (null) sets, 331
  • epsilon (∑), 330
  • equations, entering into graphing calculators, 89–93
  • even functions
    • applying to graphs, 103–104
    • overview, 102–104
  • exponential curves, 88–89
  • exponential expressions, evaluating, 178
  • exponential functions (exponential equations)
    • base of, 179–182
    • evaluating, 178
    • financial applications, 186
      • compound interest formula, 186–188
      • effective rates, 187–188
      • overview, 186
      • target sums, 186–187
    • graphing
      • identifying a rise or fall, 197
      • overview, 196
    • grouping according to their bases, 179–180
    • grouping according to their exponents, 180
    • intersecting, 263–265
    • inverses of, 200–201
    • rewriting log equations as, 195–196
    • solving exponential equations
      • making bases match, 182–184
      • recognizing and using quadratic patterns, 184–185
  • exponential rules, 13
  • exponents
    • entering into graphing calculators, 91
    • fractional
      • combining negative exponents and, 71–72
      • combining terms with, 69
      • factoring, 69
      • factoring out variables with, 70–71
      • introduction to, 14
      • overview, 68
      • solving equations with, 70–72
    • multiplying and dividing, 13–14
    • negative
      • combining fractional exponents and, 71–72
      • factoring out to solve equations, 66–68
      • flipping to positive exponents, 65–66
      • rational equations with, 65–68
      • use of, 15
    • notation of, 13–14
  • expression, defined, 2

F

  • factor (noun), defined, 2
  • factor (verb), defined, 2
  • factorials
    • overview, 342–343
    • in sequences, 309
    • symbol for (!), 342–343
  • factoring
    • binomials, 37–39
    • four or more terms by grouping, 19
    • fractional exponents, 69
    • by grouping, quadratic equations, 40
    • introduction to, 15
    • for polynomial roots
      • overview, 143
      • patterns and groupings, 143–144
      • unfactorable equations, 144–145
    • into the product of two binomials, 71
    • removal of discontinuities by, 164–165
    • solving cubes by, 47
    • three terms, 17
    • trinomials, 39–40
    • two terms, 16–17
    • variables with fractional exponents, 70–71
  • factors, Rational Root Theorem and changing from roots to, 147–148
  • first difference, in sequential patterns, 311
  • Fit button, on graphing calculators, 93
  • flattening curves, complex roots and, 284
  • focus of parabolas, defined, 206
  • FOILing
    • multiplying complex numbers, 277
    • overview, 17, 18
  • four-color problem, 349
  • fractional exponents
    • combining negative exponents and, 71–72
    • combining terms with, 69
    • factoring, 69
    • factoring out variables with, 70–71
    • introduction to, 14
    • overview, 68
    • solving equations with, 70–72
  • fractions. See also rational functions
    • decomposing, 248–250
    • determining divisors, 362
    • entering into graphing calculators, 90–91
    • linear equations and, 23
    • rational equations, 56
    • solving quadratic equations, 52
    • vulgar, 166
    • zeros in the denominators, 99
  • functions

G

  • Gallimard, 29
  • GCF (greatest common factor)
    • factoring out
      • for quadratic binomials, 38
      • to solve rational equations, 66–67
    • factoring three terms, 17
    • factoring two terms, 16, 17
    • solving exponential functions and, 185
  • geometric sequences
    • general formula or rule for, 315–316
    • recursive rule for, 317
    • summing, 320–322
  • graphing calculators
    • entering equations into, 90–93
    • entering exponents, 91
    • entering fractions, 90–91
    • finding irrational roots of polynomials, 145
    • negating or subtracting in, 92
    • radicals in, 91–92
    • sketching a parabola, 126
    • solving problems with, 89–93
    • window of, 92–93
  • graphs (graphing; sketching)
    • absolute values, 89
    • applying even and odd functions to, 103–104
    • asymptotes
      • oblique asymptotes, 163
      • overview, 160
      • vertical and horizontal asymptotes, 161–163
    • basic forms of, 86–89
    • with calculators. See graphing calculators
    • Cartesian coordinate system, 74
    • circles, 89
    • connecting the points in order, 75–76
    • coordinate plane, 74–75
    • cubics and quartics, 87
    • ellipses, 221–222
    • horizontal line test, 105, 106
    • hyperbolas, 224–226
    • inequalities, 251, 268–269
    • intercepts, 76–77
    • lines, 81–85
    • overview, 73–74
    • piecewise functions, 107–108
    • radicals and rationals, 88–89
    • rational functions, 173–175
    • removable discontinuities, 165–166
    • solutions of linear systems, 230–233
    • symmetry, 76, 78–80
    • vertical line test, 105
  • greater than (>), 25
  • greater than or equal to (≥), 25
    • interval notation, 28
  • greatest common divisor (GCD), finding, 362
  • greatest common factor (GCF)
    • factoring out
      • for quadratic binomials, 38
      • to solve rational equations, 66–67
    • factoring three terms, 17
    • factoring two terms, 16, 17
    • solving exponential functions and, 185
  • grouping
    • factoring by, 19, 40
    • finding quadratic factors in a, 41

H

  • happy numbers, 368
  • hexagonal numbers, 366–367
  • horizontal asymptotes, 161
  • hyperbolas
    • asymptotes of, 223–224
    • axes of, 222–223
    • basic equations for, 223
    • defined, 222–223
    • foci of, 222
    • graphing, 224–226

I

  • icons used in this book, 3–4
  • identities, defined, 10
  • identity matrices
    • characteristics of, 289–290
    • determining multiplicative inverses and, 299–304
  • imaginary numbers. See also complex numbers
    • defined, 273
    • simplifying radicals, 279–280
  • income tax, piecewise functions, 109–110
  • inequalities
    • absolute value, 31–33, 53–54
    • expressing ranges in, 100
    • expressing the domain of functions with, 99
    • graphing, 251, 268–269
    • intersecting to share a portion of the plane, 269
    • linear, 25–29
    • moving in opposite directions, 32–33
    • overview, 268
    • quadratic, 49–53
    • rational, 52
    • sandwiching the values in, 32
  • inequality notation, interval notation and, 27–28
  • infinity
    • adding all the terms of geometric sequences to, 321–322
    • rational functions and, 170–173
    • rational limits at, 172–173
  • intercepts. See also x-intercepts; y-intercepts
    • graphing log functions with, 199
    • of polynomials
      • counting, 135–138
      • interpreting relative value and absolute value, 135
      • overview, 134
      • solving for, 138–139
    • in quadratics, 118–119
    • of rational functions, 159
    • streamlining the graphing process with, 76–80
  • interval notation
    • expressing the domain of functions with, 99
    • overview, 27–28
  • intervals of polynomials, positive and negative
    • overview, 139–140
    • sign-line method, 140
  • inverse functions, 112–114
  • inverse matrices
    • dividing matrices by using, 304
    • identifying, for any size square matrix, 301–303
    • multiplicative inverses, 299–304
    • quick-and-slick rule for, 303–304
  • inverses
    • additive
      • defined, 11
      • determining, 299
    • defined, 11
    • of exponential functions, 200–201
    • multiplicative
      • defined, 11
      • determining, 299–304
    • rational functions as, 267–268
  • irrational numbers
    • quadratic equations, 42–43
    • rational numbers distinguished from, 146
  • irrational roots, unfactorable equations, 144–145
  • irrational solutions, straightening out, solving quadratic equations, 42–43

L

  • lead coefficient of the standard form, 116, 117
  • least common denominator (LCD)
    • finding, 56–57
    • solving rational equations, 56–59
  • Leibniz, 29
  • less than (<), 25
  • less than or equal to (≤), 25
  • linear, defined, 3
  • linear equations
    • absolute value and, 30–34
    • clearing out fractions, 23
    • isolating different unknowns, 24–25
    • overview, 21–22
    • solving basic, 22–23
    • systems of
      • Cramer’s Rule for solving, 238–241
      • decomposing fractions, 248–250
      • defined, 229
      • elimination method for solving, 233–236
      • graphing, 230–233
      • matrices for finding solutions for, 305–306
      • with more than three linear equations, 244–247
      • overview, 229
      • parallel lines, 232–233
      • pinpointing the intersection, 231–232
      • real-world applications, 247–248
      • recognizing solutions indicating parallel or coexisting lines, 235–236
      • standard form for, 230
      • substitution method for solving, 235–236
      • with three linear equations, 241–247
      • two equations representing the same line, 232
  • linear inequalities
    • compound inequality, 28–29
    • overview, 25–26
    • solving, 26–27
  • lines
    • basic forms of graphs, 86
    • coexisting
      • identifying, 237–238
      • recognizing, 235–236
    • combined with parabolas
      • finding two solutions, 253
      • no-answer answer, 254–255
      • overview, 252
      • point(s) where a line and parabola cross paths, 253–254
      • settling for one solution, 253–254
    • finding the slope of, 81
    • graphs of, overview, 81–84
    • identifying parallel and perpendicular, 85
    • intersections of a rational functions and, 265–266
    • mixing polynomials and, 260–261
    • parallel. See parallel lines
    • slope of
      • counting off, 83
      • finding, 81–82
      • formulating the value of, 81–82
      • identifying characteristics of, 81
    • slope-intercept form, equation for
      • changing from standard form to, 84
      • overview, 83–84
    • standard form, equation for
      • changing from slope-intercept form to, 84
      • overview, 82
  • logarithmic curves, 88–89
  • logarithmic functions (logarithmic equations)
    • expanding with log notation, 192
    • graphing
      • with intercepts, 199
      • overview, 196–197
      • sketching exponential graphs, 197–198
    • overview, 189
    • properties of, 190–191
    • rewriting for compactness, 192–193
    • rewriting log equations as exponentials, 195–196
    • setting log equal to log, 194–195
    • solving, 193–196
    • uses of, 191–192

M

  • matrices
    • adding and subtracting, 290–291
    • applications of
      • determining how many of each item was sold, 294–295
      • determining how to increase sales, 296–297
      • determining sales by salesperson, 295–296
      • overview, 293–294
    • determining dimensions of, 292
    • finding solutions for systems of equations and, 305–306
    • identity, 289–290
    • inverse
      • dividing matrices by using, 304
      • identifying, for any size square matrix, 301–303
      • multiplicative inverses, 299–304
      • quick-and-slick rule for, 303–304
    • multiplying
      • by scalars, 291
      • two matrices, 291–292
    • overview, 287
    • quick-and-slick rule for, 303–304
    • row operations, 297–298
    • rows and columns of, 289
    • simplex method, 296
    • square, 289
    • zero, 289
  • monomial, defined, 3
  • mortgage, paying off your, 25
  • multiples, sequential patterns, 312–313
  • multiplication (multiplying)
    • ancient symbols, 29
    • associative property of,9
    • commutative property of, 8
    • complex numbers, 277
    • distributing, 9
    • exponents, 13–14
    • matrices
      • by scalars, 291
      • two matrices, 291–292
    • multiplication (multiplying) (continued)
    • order of operations, 11–12
    • property of zero, 12
    • time-saving tricks
      • casting out 9s, 358–359
      • finding the next perfect square, 356
      • multiplying by 5, 360–361
      • multiplying by 11, 359–360
      • multiplying two-digit numbers, 362–363
      • recognizing the pattern in multiples of 9 and 11, 357
      • squaring numbers that end in 5, 355–356
  • multiplication principle, applied to sets, 344–345
  • multiplicative identity, use of, 10
  • multiplicative inverses
    • defined, 11
    • determining, 299–304

N

  • narcissistic numbers, 368–369
  • negative button, in graphing calculators, 92
  • negative exponents
    • combining fractional exponents and, 71–72
    • factoring out to solve equations, 66–68
    • flipping to positive exponents, 65–66
    • rational equations with, 65–68
    • use of, 15
  • negative numbers
    • origin of, 13
    • solving linear inequalities, 26–27
  • negative roots of polynomials, 149–150
  • negatives
    • factoring out to solve rational equations, 66–68
    • order of operations on graphing calculators, 92
  • nonlinear equations
    • combined parabolas and lines, 252
    • crossing parabolas with lines, 252
  • notation
    • of exponents, 13
    • function, 98
    • interval, overview, 27–28
    • for inverse functions, 112
    • for limits of rational functions, 167
    • sequence, 308
    • set
      • overview, 329–330
      • set builder notation, 330
      • subsets, 331–333
      • universal and empty (null) sets, 331
    • summation, 318–319
  • numbers
    • abundant, 368
    • amicable, 367
    • complex
      • adding, 276
      • applications of, 276
      • examples of, 276
      • multiplying, 277
      • multiplying by the conjugate to perform division, 277–279
      • multiplying numbers by FOILing, 277
      • operations on, 276–277
      • standard form, 275
      • subtracting, 277
    • deficient, 368
    • happy, 368
    • hexagonal, 366–367
    • irrational
      • quadratic equations, 42–43
      • rational numbers distinguished from, 146
    • ISBN, 30
    • narcissistic, 368–369
    • palindromic, rational equations and, 64
    • perfect, 367
    • prime, 369
    • square, 366
    • triangular, 365–366

O

  • oblique asymptotes, 163
  • odd functions
    • applying to graphs, 103–104
    • overview, 102–104
  • one-sided limits, rational functions, 171–173
  • one-to-one functions
    • defining, 104–105
    • eliminating violators, 105–106
    • overview, 104
  • order of operations
    • basic description of, 11–12
    • on graphing calculators, 92
  • origin, symmetry with respect to the, 80

P

  • palindromic numbers, rational equations and, 64
  • parabolas. See also quadratic functions
    • axis of symmetry of, 207
    • circles intersecting with
      • finding common solutions, 258–259
      • multiple intersections, 256–258
    • combined with lines
      • finding two solutions, 253
      • no-answer answer, 254–255
      • overview, 252
      • point(s) where a line and parabola cross paths, 253–254
      • settling for one solution, 253–254
    • completing the square on the equation of a, 43
    • converting equations to the standard form, 214–215
    • defined, 115, 206–207
    • general form of parabola equations, 210–211
    • graphing, 118–119
    • overview, 86
    • sketching graphs of
      • overview, 211
      • real-world applications, 212–214
      • steps to follow, 212
  • parallel lines
    • identifying, 85
    • systems of linear equations and
      • identifying, 237–238
      • overview, 232–233
      • recognizing, 235–236
  • patterns
    • factoring for polynomial roots and, 143–144
    • sequential
      • difference between terms, 311
      • multiples and powers, 312–313
      • overview, 309–310
  • perfect numbers, 367
  • perfect square trinomials, finding, 17–18
  • perfect squares
    • factoring two terms, 16
    • finding the next, 356
  • permutations
    • of sets, 345–348
    • tree diagrams for, 351
  • perpendicular lines, identifying, 85
  • piecewise functions
    • applying, 108–110
    • example of, 107
    • overview, 106–107
  • polynomial equations, 46–49
  • polynomial roots
    • factoring for roots, 143–145
    • negative roots, 149–150
    • overview, 143
    • positive roots, 149
    • Rule of Signs, 149–150
    • synthesizing root findings, 150–155
    • synthetic division to test for roots, 150–153
  • polynomials, 133–155
    • comparing graphs that have differing sign behaviors, 142
    • with complex solutions, 282–285
    • defined, 3, 133
    • determining positive and negative intervals, 139–142
    • intercepts of
      • counting, 135–138
      • interpreting relative value and absolute value, 135
      • overview, 134
      • solving for, 138–139
    • intersecting, 261
    • mixing lines and, 260–261
    • Rational Root Theorem. See Rational Root Theorem
    • Remainder Theorem, 154–155
    • roots of
      • factoring for roots, 143–145
      • negative roots, 149–150
      • overview, 143
      • positive roots, 149
      • Rule of Signs, 148–150
      • synthesizing root findings, 150–155
      • synthetic division to test for roots, 150–153
    • standard form, 134
    • turning points of
      • counting, 135–138
      • interpreting relative value and absolute value, 135
  • positive integers
    • sum of odd-numbered, 332
    • sum of the first 10 squares of, 332
  • positive roots of polynomials, 149
  • powers
    • of i, 274–275
    • raising to
      • exponents and, 14–15
      • order of operations, 11–12
      • solving equations with fractional exponents, 70–72
    • sequential patterns, 312–313
  • prime numbers, 369
  • Probability For Dummies (Rumsey), 344
  • proportions
    • definition and features of,60
    • solving rational equations with, 60–61
  • Pythagoras, 262

Q

  • quadratic equations
    • absolute value inequalities, 53–54
    • completing the square, 43–46
    • defined, 3
    • doubling up on a trinomial solution, 39–40
    • factoring binomials, 37–39
    • factoring by grouping, 40
    • factoring difference of squares, 38–39
    • factoring out the greatest common factor (GCF), 38
    • finding rational solutions, 42
    • finding two solutions in trinomials, 39
    • graphs of, 86
    • increasing the number of factors, 53
    • overview, 35–36
    • polynomial equations, 46–49
    • quadratic formula, 41–43
    • rational inequalities, 52
    • solving exponential functions as, 185
    • solving quadratic inequalities, 49–53
    • solving with complex solutions, 280–282
    • square root rule, 36–37
    • straightening out irrational solutions, 42–43
  • quadratic formula, 41–43
  • quadratic functions, 115–131. See also parabolas
    • applying to the real world
      • launching a water balloon, 130–131
      • selling candles, 127–128
      • shooting basketballs, 128–130
    • axis of symmetry of, 124–125
    • b and c coefficients, 117–118
    • intercepts in, 118–119
    • lead coefficient of, 116, 117
    • standard form of
      • overview, 116
      • starting with a in, 116–117
    • vertices of. See vertex
    • x-intercepts of, 120–123
    • y-intercept of, 119–120
  • quadratic inequalities
    • keeping inequalities strictly quadratic, 50–51
    • solving, 49–53
  • quadratic trinomials
    • defined, 16
    • factoring, 16–17
    • solving exponential functions, 185
  • quadratic-like trinomials
    • defined, 48
    • overview, 48–49
    • solving, 67–68
  • quartics, 87
  • quick-and-slick rule for matrices, 303–304

R

  • radical curves, 88–89
  • radicals
    • calming, in rational equations, 63–64
    • in graphing calculator, 91–92
    • as grouping symbol, 12
    • replacing with fractional exponents, 13
    • ridding rational equations of, 61–64
    • to show roots, 13
    • simplifying, 279–280
  • raising to powers
    • exponents and, 14–15
    • order of operations, 12
    • solving equations with fractional exponents, 70–72
  • range, of functions, 100–102
  • rational curves, 88–89
  • rational equations, 56–61
    • calming two radicals in, 63–64
    • factoring out negatives to solve, 66–68
    • least common denominator (LCD) and solving, 56–59
    • with negative exponents, 65–68
    • overview, 56–57
    • ridding of radicals, 61–64
    • solving by working with fractional exponents, 70–72
    • solving with cross-products, 61
    • solving with proportions, 60–61
    • squaring both sides of, 62–63
    • systematically solving, 56–59
  • rational functions. See also fractions
    • asymptotes of. See asymptotes
    • definition and features of, 157
    • domains of, 158
    • graphs of, 173–175
    • intercepts of, 159
    • intersections of lines and, 265–266
    • as inverses, 267–268
    • limits of
      • at infinity, 170–173
      • notation for, 167
      • overview, 167–168
    • with removable discontinuities
      • determining an existent limit without tables, 169–170
      • determining which functions have limits, 170
      • evaluating limits at discontinuities, 168–169
      • evaluating the removal restrictions, 165
      • overview, 164–166
      • showing on a graph, 165–166
    • rounding up, 265–268
    • sketching rational graphs from clues, 173–175
  • rational inequalities, 52
  • rational numbers, defined, 146
  • Rational Root Theorem
    • changing from roots to factors, 147–148
    • constant terms
      • polynomials with, 146–147
      • polynomials without, 147
    • list of numbers that may be roots of a particular polynomial, 146
  • rational solutions, quadratic equations, 42–43
  • real numbers, defined, 148, 273
  • real roots, of polynomials, 148
  • rectangles, graphing hyperbolas and, 225
  • recursively defining functions, sequences and, 317–318
  • relative maximum or minimum values, of polynomials, 135
  • Remainder Theorem, 154–155
  • Remember icon, 3
  • removable discontinuities
    • determining an existent limit without tables, 169–170
    • determining which functions have limits, 170
    • evaluating limits at discontinuities, 168–169
    • evaluating the removal restrictions, 165
    • overview, 164–166
    • showing on a graph, 165–166
  • roots
    • exponents and, 14
    • finding, order of operations, 11
    • fractional exponents and, 14
    • of polynomials
      • factoring for roots, 143–145
      • negative roots, 149–150
      • overview, 143
      • positive roots, 149
      • Rule of Signs, 148–150
      • synthesizing root findings, 150–155
      • synthetic division to test for roots, 150–153
  • roster notation, for set elements, 330
  • row operations, on matrices, 297–298
  • rows, of matrices, 289
  • Rule of Signs, 148

S

  • scalars, multiplying matrices by, 291
  • second difference, in sequential patterns, 311–312
  • sequences
    • alternating, 309–310
    • arithmetic, 313–315
    • defined, 308
    • factorials in, 309
    • geometric, 315–316
    • notation, 308
    • recursively defining functions and, 317–318
    • series, 318–322
    • summing. See summing sequences
    • terminology, 308
  • sequential patterns
    • difference between terms, 311
    • multiples and powers, 312–313
    • overview, 309–310
  • series, 318–322
  • set builder notation, 330
  • set notation
    • overview, 329–330
    • set builder notation, 330
    • subsets, 331–333
    • universal and empty (null) sets, 331
  • sets
    • complements of, 335
    • counting, 344
    • counting the elements in, 335–336
    • empty (null), 331
    • factorials and, 342–344
    • intersections of two, 334–335
    • listing elements with a roster, 330
    • multiplication principle applied to, 344–345
    • operating on, 333
    • permutations of, 345–348
    • subsets, 331–333
    • union of two, 333–334
    • universal, 331
    • Venn diagrams
      • adding sets to, 339–341
      • applying, 337–338
      • overview, 336
      • with set operations, 338–339
  • sign-line method
    • polynomials, 140–141
    • solving quadratic equations, 50, 52, 53
  • simplex method, 296
  • simplify, 3
  • sketching. See graphs
  • slant asymptotes, 163
  • slope of a line
    • counting off, 83
    • finding, 81–82
    • formulating the value of, 81–82
    • identifying characteristics of, 81
  • slope-intercept form of equation for a line, 83–84
    • changing from standard form to, 84
  • solve, 3
  • square matrices, 289
    • identifying an inverse for any size, 301–303
  • square numbers, 366
  • square root rule, quadratic equations, 36–37
  • square roots, 279
  • squares
    • completing the
      • conic sections, 43–46
      • twice over, 45–46
    • difference of two, factoring quadratic equations, 38
    • difference of two perfect, factoring two terms, 16
    • perfect
      • factoring two terms, 16
      • finding the next, 356
  • squaring, rational equations, 62–63
  • squaring numbers that end in 5, 355–356
  • standard form
    • of equation for lines
      • changing from slope-intercept form to, 84
      • overview, 82–83
    • of parabolas
      • converting to, 214–215
      • overview, 207, 211
    • of polynomials, 134
    • systems of linear equations, 230
  • subsets
    • counting the number of, 332–333
    • notation, 332
    • overview, 331
  • substitution method
    • line/parabola systems of equations, 253
    • for solving systems of linear equations, 235–236
  • subtract button, in graphing calculators, 92
  • subtracting (subtraction)
    • ancient symbols, 29
    • complex numbers, 277
    • matrices, 290–291
    • order of operations, 11–12
  • sum or difference of cubes, 47–48
  • summation notation, 318–319
  • summing sequences
    • arithmetically, 319–320
    • geometrically, 320–322
      • adding all the terms to infinity, 321–322
      • adding the first n terms, 320–321
    • real-world applications for sums of sequences, 323–326
      • bouncing a ball, 324–326
      • negotiating your allowance, 323–324
      • special formulas, 326–327
      • stacking blocks, 323
    • special formulas, 326–327
  • symmetry
    • axis of, of a quadratic function, 124–125
    • of graphs, 76, 78–80
    • with respect to the origin, 80
    • with respect to the x-axis, 78–79
    • with respect to the y-axis, 78
  • synthetic division
    • by binomials, 153–154
    • Remainder Theorem, 154–155
    • to test for roots, 150–153
  • system of equations, defined, 229
  • systems of linear equations
    • Cramer’s Rule for solving, 238–241
    • decomposing fractions, 248–250
    • defined, 229
    • elimination method for solving, 233–236
    • graphing, 230–233
    • matrices for finding solutions for, 305–306
    • with more than three linear equations, 244–247
    • overview, 229
    • parallel lines, 232–233
    • pinpointing the intersection, 231–232
    • real-world applications, 247–248
    • recognizing solutions indicating parallel or coexisting lines, 235–236
    • standard form for, 230
    • substitution method for solving, 235–236
    • with three linear equations, 241–247
    • two equations representing the same line, 232

T

  • Tartaglia, 29
  • Technical Stuff icon, 4
  • term, 3
  • third difference, in sequential patterns, 312
  • three linear equations, systems of, 241–247
  • Tip icon, 3
  • tree diagrams
    • for combinations, 352
    • for permutations, 351
  • triangular numbers, 365–366
  • trinomials
    • definition of, 3
    • factoring, 17, 39–40
    • finding two solutions in, 39
    • perfect square, finding, 17–18
    • quadratic
      • defined, 16
      • factoring, 16–17
      • solving exponential functions, 185
    • quadratic-like
      • defined, 48
      • overview, 48–49
      • solving, 67–68
    • unFOILing, 18
  • turning points, of polynomials
    • counting, 135–138
    • interpreting relative value and absolute value, 135
  • two-digit numbers, multiplying, 362–363

U

  • unfactorable equations, polynomials, 144–145
  • unFOILing, 17, 18, 184
  • unit circle, 217
  • universal sets, 331
  • utility companies, piecewise functions, 108–110

V

  • variables
    • defined, 3
    • substitution method and, 236–237
  • Venn diagrams
    • adding sets to, 339–341
    • applying, 337–338
    • overview, 336
    • with set operations, 338–339
  • vertex (vertices)
    • defined, 118, 207
    • finding, in quadratic functions
      • lining up along the axis of symmetry, 124–125
      • overview, 123–124
      • sketching a graph, 125–127
    • at the origin
      • opening to the right or left, 207–209
      • opening upward or downward, 209–210
      • overview, 207
  • vertical asymptotes, 160
  • vertical line test, 105
  • vertical lines, slope as non-existent, 81
  • vulgar fractions, 166

W

  • Warning icon, 4
  • water balloons, launching, quadratic function example, 130–131
  • words
    • expressing ranges in, 100
    • expressing the domain of functions with, 99

X

  • x-axis
    • defined, 75
    • symmetry with respect to, 78–79
  • x-intercepts
    • finding, 77
    • on graphing calculator window, 93
    • overview, 77
    • of polynomials
      • overview, 134–135
      • solving for, 138–139
      • unfactorable equations, 144–145
    • of quadratic functions, 120–123
    • of rational functions, 159

Y

  • y-axis
    • defined, 75
    • symmetry with respect to, 78
  • y-intercepts
    • finding, 77
    • overview, 77
    • of quadratic functions, 119–120
    • of rational functions, 159
    • solving for, 138–139

Z

  • zero(s)
    • complex, 283–285
    • finding y-intercepts of rational functions, 159
    • multiplication property of, 12
    • unfactorable equations, 144–145
  • zero factorial, 309
  • zero matrices, 289
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