Symbols Is equal to Is not equal to L Is approximately equal to Is greater than $ Is greater than or equal to , Is less than # Is less than or equal to a , x , b 0.34 The repeating decimal 0.343434 . LCD Least common denominator {a, b} {x x $ 2} a P B a is an element of set B a B a is not an element of set B A # B Set A is a subset of set B A B Set A is not a subset of set B A > B Set intersection A < B Set union 0x [ bn n "a a is less than x and x is less than b The set whose elements are a and b The set of all x such that x is greater than or equal to Null set The absolute value of x nth power of b nth root of a "a a bi (a, b) f, g, h, etc f (x) Functional value at x f ° g The composition of functions f and g f logb x ln x log x n! i 21 Principal square root of a Imaginary unit Complex number Plus or minus Ordered pair: first component is a and second component is b Names of functions The inverse of the function f Logarithm, to the base b, of x Natural logarithm (base e) Common logarithm (base 10) n factorial Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it area A perimeter P length l width w surface area S altitude (height) h Rectangle A lw P 2l 2w base b circumference C radius r Triangle A5 Square bh A s 2 P 4s h w l volume V area of base B slant height s s s s s b Parallelogram A bh Trapezoid h(b b2) A5 b1 r b2 b 30°–60° Right Triangle Right Triangle a2 b2 c x 60° A pr 2 C 2pr h h Circle 2x Isosceles Right Triangle c 30° x√2 x b x√3 a x Right Circular Cylinder Sphere Right Circular Cone V pr 2h S 2pr 2prh S 4pr 2 V pr 3 r V pr 2h S pr prs r h s h r Pyramid V5 Bh Prism V Bh h h Base Base Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it 1.1 • Numerical and Algebraic Expressions Tenth Edition i Intermediate Algebra Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Tenth Edition Intermediate Algebra Jerome E Kaufmann Karen L Schwitters Seminole State College of Florida Australia • Brazil • Mexico • Singapore • United Kingdom • United States Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it This is an electronic version of the print textbook Due to electronic rights restrictions, some third party content may be suppressed Editorial review has deemed that any suppressed content does not materially affect the overall learning experience The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Intermediate Algebra, Tenth Edition Jerome E Kaufmann and Karen L Schwitters © 2015, 2011, 2006 Cengage Learning General Manager, Developmental Studies: Gareth Hancock ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means, graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 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Researcher: PreMediaGlobal ISBN-13: 978-1-285-19572-8 Copy Editor: Graphic World Inc ISBN-10: 1-285-19572-8 Illustrators: Network Graphics; Graphic World Inc Text Designer: Diane Beasley Cover Designer: Irene Morris Cover Image: antishock/fotolia Cengage Learning 200 First Stamford Place, 4th Floor Stamford, CT 06902 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at www.cengage.com/global Cengage Learning products are represented in Canada by Nelson Education, Ltd To learn more about Cengage Learning Solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.cengagebrain.com Printed in the United States of America 1 2 3 4 5 6 7 17 16 15 14 13 Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it Contents 1.1 • Numerical and Algebraic Expressions v Basic Concepts and Properties 1.0 Review of Fractions 1.1 Sets, Real Numbers, and Numerical Expressions 13 1.2 Operations with Real Numbers 22 1.3 Properties of Real Numbers and the Use of Exponents 31 1.4 Algebraic Expressions 38 Chapter Summary 48 Chapter Review Problem Set 51 Chapter Test 53 Equations, Inequalities, and Problem Solving 55 2.1 Solving First-Degree Equations 56 2.2 Equations Involving Fractional Forms 64 2.3 Equations Involving Decimals and Problem Solving 71 2.4 Formulas 79 2.5 Inequalities 89 2.6 More on Inequalities and Problem Solving 96 2.7 Equations and Inequalities Involving Absolute Value 105 Chapter Summary 112 Chapter Review Problem Set 118 Chapter Test 120 Chapters – Cumulative Review Problem Set 121 Polynomials 123 3.1 Polynomials: Sums and Differences 124 3.2 Products and Quotients of Monomials 130 3.3 Multiplying Polynomials 136 3.4 Factoring: Greatest Common Factor and Common Binomial Factor 144 3.5 Factoring: Difference of Two Squares and Sum or Difference of Two Cubes 152 3.6 Factoring Trinomials 158 3.7 Equations and Problem Solving 166 Chapter Summary 173 Chapter Review Problem Set 179 Chapter Test 181 v Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Answer Section 37. 6.8 hours 39. 1.5 hours 41. 34.7 years 43. 6.7 45. Approximately times 31. f(x) Chapter 11 Sample Problems (page 588) e f 3. (a) log2 3 (b) 24 16 4. 24 5. 2.1461 6. 5.4638 7. 3.19 f(x) (4, 2) x (1, 0) 9. logb x logb y logb z 10. 536 11. 5216 12. 52.106 13. 13,466 14 $2391.24 15 $2394.43 16 155.4 milligrams 17 5.48 hours 18 7.5 Chapter 11 Review Problem Set (page 594) x x (1, −) 33. logb x logb y logb z 34. logb c logb d 35. 51.18826 36. 52639.46 37. 50.01322896 38. 50.0771976 39. 546 40. 596 41. e f 42. 586 (7, 3) (1, 1) x −1) (−, f(x) (3, 2) (2, 0) (1, −1) (−2, 4) 8. 32. (8, 2) (4, 1) f(x) (0, 1) 643 45 15 f 45. 556 46. e f 47. 53.406 48. 51.956 49. 55.306 50. 55.616 51. (a) $933.05 (b) $305.96 (c) $98.54 52. (a) $7910 (b) $14,766 (c) $15,000 53. $19,632.75 54. $40,305.65 55. Approximately 8.8 yr 56. Approximately 22.9 yr 57 $5656.26 58. Approximately 19.8 yr 59. 133 g 60. 46.4 hr 61. 61,070; 67,493; 74,591 62. Approximately 4.8 yr 63. 8.1 64. Approximately times as intense 43. 556 44. e Chapter 11 Test (page 597) 1. 2. 1 3. 1 4. 21 5. 5236 6. e f 7. e f 2 8. 52436 9. 526 10. e f 11. 4.1919 12. 0.2031 13. 0.7325 14. 5.4538 15. 55.176 16. 510.296 x 3y 17. 4.0069 18. logb a b 19. $6342.08 z 20. 13.5 years 21. 7.8 hours 22. 4813 grams 23 24 3. 5246 4. e f 5. 536 6. e f 7. log8 64 1 8. loga c b 9. log3 a b 22 10. log10 a b 21 10 11. 53 125 12. x z y 13. 223 25 14. 10 10,000 15. 7 16. 3 17. 4 18. 23 19. 2 20. 13 21. 0 22. 22 23. 1.8642 24. 22.6289 25. 24.2687 26. 4.7380 27. 4.164 28. 2.548 29. 23.129 30. 28.118 Unless otherwise noted, all content on this page is © Cengage Learning Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net 644 Answer Section Appendix A Practice Exercises (page 601) 1. x 8x 7y 28x 6y2 56x 5y3 70x 4y4 56x 3y5 28x 2y6 8xy7 y8 2. x 7x 6y 21x 5y2 35x 4y3 35x 3y4 21x 2y5 7xy6 y7 3. 81x 108x 3y 54x 2y2 12xy3 y4 4. x 8x 3y 24x 2y2 32xy3 16y4 5. x 5x 4y 10x 3y2 10x 2y3 5xy4 y5 6. x 4x 3y 6x 2y2 4xy3 y4 7. x 10 10x 9y 45x 8y2 120x 7y3 210x 6y4 252x 5y5 210x 4y6 120x 3y7 45x 2y8 10xy9 y10 8. x 9x 8y 36x 7y2 84x 6y3 126x 5y4 126x 4y5 84x 3y6 36x 2y7 9xy8 y9 9. 64x 192x 5y 240x 4y2 160x 3y3 60x 2y4 12xy5 y6 10. x 15x 4y 90x 3y2 270x 2y3 405xy4 243y5 11. x 15x 4y 90x 3y2 270x 2y3 405xy4 243y5 12. 64x 192x 5y 240x 4y2 160x 3y3 60x 2y4 12xy5 y6 13. 243a5 810a4b 1080a3b2 720a2b3 240ab4 32b5 14. 16a4 96a3b 216a2b2 216ab3 81b4 15. x 6x 5y3 15x 4y6 20x 3y9 15x 2y12 6xy15 y18 16. x 10 5x 8y 10x 6y2 10x 4y3 5x 2y4 y5 17. x 14x 84x 280x 560x 672x 448x 128 18. x 18x 135x 540x 1215x 1458x 729 19. x 12x 54x 2 108x 81 20. x 9x 36x 84x 126x 126x 84x 36x 9x 21. x 15 15x 14y 105x 13y2 455x 12y3 22. x 12 12x 11y 66x 10y2 220x 9y3 23. a13 26a12b 312a11b2 2288a10b3 24. x 20 20x 19y 190x 18y2 1140x 17y3 25. 462x 5y6 26. 56x 5y3 27. 2160x 3y3 28. 126x 5y4 29. 2000x 3y2 30. 189a2b5 Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Index Abscissa, 350 Absolute value, 23 equations involving, 105–106 functions involving, 462, 465–466 inequalities involving, 106–109 Addition, 13 of complex numbers, 297–298 distributive property and, 34 English phrases for, 42 of fractions, 8–10, 195–197 order of operations and, 11, 18 of polynomials, 125–127 properties of, 31–33 of radicals, 260 of rational expressions, 196–200, 202–204 of rational numbers, 195 of real numbers, 24 Addition property of equality, 57 Addition property of inequality, 91 Additive inverse, 23, 33 Additive inverse property, 33 Algebraic equations, 56 See also Equations Algebraic expressions, 13 compared to equations, 66 evaluating, 40–42 translating English phrases into, 42–43 See also Simplifying algebraic expressions Algebraic inequalities, 89 See also Inequalities Analytic geometry, 350 Angles, 68 Associative property of addition, 32 of multiplication, 32 Asymptotes of hyperbola, 424 Axes, coordinate, 349 Axis of symmetry of ellipse, 420 of parabola, 452 Base in exponential notation, 35, 553 of logarithm, 565, 576–577, 584–585 Binary operations, 32 Binomial(s), 124 cube of, 140–141, 599 division of polynomial by, 211–215 factoring out of polynomial, 146, 147 multiplication of, 138–140 square of, 138–139, 599 Binomial expansions, 599–602 Calculators logarithms with, 576–577 scientific notation with, 283–284 See also Graphing utilities Cartesian coordinate system, 350 Change-of-base formula, 585 Circle(s), 404, 414–417 Closure property for addition, 31, 32 for multiplication, 31–32 Coefficient, numerical, 38, 124 Combining similar terms, 39, 125 Common denominator, See also Least common denominator (LCD) Commutative property of addition, 32 of multiplication, 32 Complementary angles, 68 Completely factored form, 145 Completing the square, 311–316 Complex fractions, 205–208 Complex numbers, 296–301 as solutions of quadratic equations, 319–320 as solutions of system of equations, 534–535 Composite function, 470 Composite number, Composition of functions, 469–473 of inverse functions, 477 Compound interest, 326, 556–557, 559, 581–583 compounded continuously, 560, 582–583 Compound statements, 98–101 Conic sections, 404 See also Circle(s); Ellipse(s); Hyperbola(s); Parabola(s) Conjugates, 267 complex, 300 Conjunctions, 98–101 Consistent system of equations, 505 Constant function, 449 Constant of proportionality, 482 Constant of variation, 482 Contradiction, 59 Coordinate axes, 349 Coordinate geometry, 350 Coordinates of a point, 22, 350 Cost, 73–75, 355–356 Counting numbers, 14 Critical numbers, 332 Cross-multiplication property of proportions, 219 Cube(s) of a binomial, 140–141, 599 of a number, 35–36, 251 sum and difference of, 154 Cube root, 251 Decimals, 15–16 equations containing, 71–72 inequalities containing, 98 in scientific notation, 281–284 Declaring the variable, 60 Decreasing function, 554 Degree of monomial, 124 of polynomial, 125 Denominator common, least common (LCD), 10, 64–66, 196–200 rationalizing, 255–256, 267–268 Dependent system of equations, 505 Difference, 13, 43 of two cubes, 154 of two squares, 152–154 Difference quotient, 443–444 Direct variation, 482–483, 485–486 Discount, 72–73 Discriminant, 319–320 Disjunctions, 98–101 Distance absolute value as, 23, 105, 109 between points in the plane, 365–368 in rate-time word problems, 84–85, 225–227, 327 units of, 44 Distance formula, 367 Distributive property, 34, 35, 39, 144–145 in simplifying radical expressions, 260–261, 266–267 Dividend, in polynomial division, 212 Division, 13 of both sides of equation, 57 of both sides of inequality, 92 of complex numbers, 300–301 English phrases for, 13, 43 of fractions, 5–8, 191–192 of monomials, 133–134 notations for, 13 order of operations and, 11, 18 of polynomials, 211–215 of radical expressions, 267–268, 278–279 ratio as, 219 of rational expressions, 192–193 of rational numbers, 191–192 of real numbers, 27 synthetic, 213–215 Divisor, in polynomial division, 212 Domain of composite function, 470–472 of a function, 441, 442, 443, 460 I-1 Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net I-2 Index Domain (continued) of inverse function, 476 of a relation, 440 e, 559, 577 Elements of a set, 14 Elimination-by-addition method, 515–518, 526–529 Ellipse(s), 404, 419–421 English phrases, translating into algebraic expressions, 42–43 English system of measurement, 44 Equality, properties of, 17, 57, 271 Equal sets, 14 Equations absolute value, 105–106 algebraic, 56 checking the solution, 57–58 of circles, 415 contradiction, 59 decimals in, 71–72 equivalent, 56 exponential, 551–552, 566 expressions compared to, 66 first-degree in one variable, 56–60 fractions in, 64–66, 217–221, 223–229 function determined by, 447 identity, 59 linear in two variables, 348–349, 352–356, 375–382 linear in three variables, 524 linear systems in two variables, 504–506, 508, 510–513, 515–521 linear systems in three variables, 524–530 literal, 82 logarithmic, 566–567, 570–571 nonlinear, 386–391 nonlinear systems, 532–535 quadratic, 303–308, 311–321, 323–328 radical, 271–274 raising both sides to a power, 271 relation described by, 440 solution of, 56 solving by factoring, 148–149, 154–156, 166–168, 304 translating sentences into, 60–61 Equivalent equations, 56 Equivalent fractions, 9, 196 Evaluating algebraic expressions, 40–42 Evaluating functions, 442–443 Exponent(s), 35–36 integer, 244–248 properties of, 245, 551 rational, 276–278 real, 551 in scientific notation, 281–284 zero as, 244 See also Power(s) Exponential decay, 558–561 Exponential equations, 551–552, 566, 580 Exponential form, 565–566 Exponential functions, 552–554, 556–562 Exponential growth, 560–561, 583 Extraneous solutions, 272 Factor(s) literal, 38, 124 of a number, Factorial notation, 600 Factoring polynomials common binomial factor, 146, 147 common monomial factor, 145–146 completely, 145 difference of two squares, 152–154 by grouping, 146–147 solving equations by, 148–149, 154–156, 166–170, 304 sum and difference of cubes, 154 summary of techniques, 164 trinomials, 158–164 First-degree equations in one variable, 56–60 FOIL, 138 Formulas, 79–84, 225 Fractional equations, 64–66, 217–221, 223–229 Fractional exponents, 276–278 Fractions addition of, 8–10, 195–197 complex, 205–208 division of, 7–8, 191–192 equivalent, 9, 196 formulas involving, 225 inequalities containing, 96–97 multiplication of, 6–7, 190 negative sign in, 184–185 reducing, 5–6, 185–186 simplest form of, 6, 185 simplifying numerical expressions containing, 11 subtraction of, 8–10, 195–197 See also Denominator; Rational expressions Function notation, 442 Functions, 440–441 applications of, 444–445 basic graphs of, 460–462 composition of, 469–473 constant, 449 decreasing, 554 difference quotient of, 443–444 domain of, 441–443, 460 evaluating, 442–443 exponential, 552–554 identity, 448 increasing, 554 inverse, 476–479 linear, 447–451 logarithmic, 574–575 notations for, 441–442 one-to-one, 476 quadratic, 451–457 variation represented by, 482 vertical-line test for, 475 Fundamental principle of fractions, 5–6, 185 Graph(s) of basic functions, 460–462 of circles, 415–417 of ellipses, 419–421 of exponential functions, 553–554 of a function, 447 of hyperbolas, 423–428 of inverse function, 476 of line, given point and slope, 370–371 of line, given slope and intercept, 379 of linear equation in two variables, 351–358 of linear equation in three variables, 524 of linear functions, 447–449 of linear inequalities, 90, 361–364 of logarithmic functions, 574–575, 585 of nonlinear equations, 386–391 of nonlinear inequalities, 331–334 of parabolas, 386–388, 404–410, 412–414, 452, 454, 460 of system of inequalities, 506–508 of system of linear equations, 504–506, 508, 525 of system of nonlinear equations, 532–535 transformations of, 460–467 Graphing techniques, 389, 460–467 Graphing utilities, 357–358, 390–391 Half-life, 558 Horizontal lines, 355, 369, 449 Horizontal-line test, 475–476 Horizontal translation, 463 Hyperbola(s), 404, 423–428 Hypotenuse, 169 i, 296, 298 Identity, 59 Identity function, 448 Identity property of addition, 32 Identity property of multiplication, 33 Imaginary numbers, 297 Imaginary part, 296 Inconsistent system of equations, 505, 520, 529 Increasing function, 554 Index of a radical, 251 Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Index Inequalities, 22, 89–94 absolute value, 106–109 checking the solution, 93 compound statements involving, 98–101 decimals in, 98 fractions in, 96–97 linear in two variables, 361–364 nonlinear, 331–334 systems of, 506–508 word problems involving, 101–103 Infinity symbol, 91 Integer exponents, 244–248 Integers, 14, 16, 17 Intercepts, 352–354 in finding equation of a line, 378–380 Interest compound, 326, 556–557, 559, 560, 581–583 simple, 79–80, 83 Intersection of sets, 99, 101 Interval notation, 90–91, 101 Inverse additive, 23, 33 multiplicative, 33 Inverse functions, 476–479 Inverse variation, 482–486 Investments, 76, 79–80, 102, 326, 529–530, 557, 560 Irrational numbers, 16, 17 Isosceles right triangle, 307 Joint variation, 485 Law of decay, 560 Least common denominator (LCD), 10, 64–66, 196–200 Least common multiple, 3–5, 196 Like terms, 39, 125 Linear equations in two variables, 348–349, 351–358 systems of, 504–506, 508, 510–513, 515–521 Linear equations in three variables, 524 systems of, 524–530 Linear functions, 447–451 Linear inequalities in two variables, 361–364 systems of, 506–508 Lines finding equations of, 375–382 graphing, 370–371, 379 slopes of, 368–372, 377–380 Literal equations, 82 Literal factors, 38, 124 Logarithm(s), 565 applications of, 581–584 change-of-base formula, 585 common, 576–577 natural, 577 properties of, 567–570 Logarithmic equations, 566–567, 570–571, 580–581 Logarithmic form, 565–566 Logarithmic functions, 574–575, 577, 585 Lowest terms, 6, 185 Major axis of ellipse, 420 Maximum value of a function, 454–456 Measurement, units of, 44 Members of a set, 14 Metric system, 44 Minimum value of a function, 454, 455 Minor axis of ellipse, 420 Mixture problems, 85–86 Monomial(s), 124 degree of, 124 division of, 133–134 division of polynomial by, 211 factoring from polynomial, 145–146 multiplication of, 130–132 multiplication of polynomial by, 136–137 powers of, 132–133 Multiple, least common, 3–5, 196 Multiplication, 13 of binomials, 138–140 of complex numbers, 299–300 English phrases for, 43 of fractions, 6–7, 190 of monomials, 130–132 notations for, 13 order of operations and, 11, 18 of polynomial by monomial, 136–137 of polynomials, 136–141 properties of, 31–34 of radical expressions, 265–267, 278–279 of rational expressions, 190–191 of rational numbers, 190 of real numbers, 26 Multiplication property of equality, 57 Multiplication property of inequality, 91–92 Multiplication property of negative one, 33 Multiplication property of zero, 33 Multiplicative inverse, 33, 192 Multiplicative inverse property, 33 Natural exponential functions, 559–562 Natural logarithms, 577 Natural numbers, 14, 16, 17 Negative exponents, 244–248 Negative integers, 14, 16 Negative sign in fractions, 184–185 as opposite of, 23 I-3 Nonlinear equations, 386–391 systems of, 532–535 Nonlinear inequalities, 331–334 Normal distribution, 562 nth root, 251–252 of a product, 253 of a quotient, 254 rational exponents and, 276–277 Null set, 14 as solution of equation, 59 Number line, 22 absolute value on, 23 inequalities on, 90–91 Numbers absolute value of, 23 complex, 296–301 composite, integers, 14, 16, 17 irrational, 16, 17 natural, 14, 16, 17 prime, rational, 15, 16–17, 184–185 real, 14, 16–17, 24–27, 31–34 whole, 14, 16, 17 Numerical coefficient, 38, 124 Numerical expressions, 13 See also Simplifying numerical expressions One-to-one functions, 476 Open sentence, 56, 89 Opposite of a polynomial, 125–126 of a real number, 22, 23, 33 Ordered pair(s), 348–350, 440–441 as solution of a system, 504, 505 Ordered triples, 524, 525 Order of operations agreement, 11, 18 Ordinate, 350 Origin, 349 Origin symmetry, 388–389, 460 Parabola(s), 386–388, 404–410, 412–414, 452, 454, 460 Parallel lines, 380, 381, 505, 506 Pascal’s triangle, 599–600 Perfect powers, 253 Perfect-square trinomials, 163–164, 311 Perpendicular lines, 380–382 Plane(s), 524–525 Point, coordinates of, 350 Point-slope form, 377–378 Polynomials, 124 addition of, 125–127 degree of, 125 dividing by binomials, 211–215 dividing by monomials, 211 multiplication by monomial, 136–137 multiplication of, 136–141 Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net I-4 Index Polynomials (continued) not factorable using integers, 153, 160, 164 prime, 153 subtraction of, 125–127 See also Factoring polynomials Power(s), 35, 132–133, 245 of binomials, 599–602 of both sides of an equation, 271 perfect, 253 See also Cube(s); Exponent(s); Square(s) Price, 72–75 Prime-factored form, Prime number, Prime polynomial, 153 Principal, 556 Principal square root, 250 Problem solving See Word problems Product, 13, 43 Profit, 73–75, 355–356 Proportionality, constant of, 482 Proportions, 219–221 Pythagorean theorem, 169–170, 307–308 Quadrants, 349 Quadratic equations, 303–304 roots of, 319–321 solving by completing the square, 311–316 solving by factoring, 304 solving by most appropriate method, 323–325 solving by taking square roots, 305–307 solving with quadratic formula, 315–319 standard form of, 304 word problems involving, 307–308, 325–328 Quadratic form, 304, 325 Quadratic formula, 316 Quadratic functions, 451–457 Quadratic inequalities, 331–333 Quotient, 13, 43, 212 Radical(s), 250 addition of, 260 applications of, 256–257 negative numbers and, 298–299 simplest form of, 253–255 variables in, 261–262 See also Cube root; Square roots Radical equations, 271–274 Radical expressions division of, 267–268, 278–279 multiplication of, 265–267, 278–279 simplifying, 260–262, 265–268 Radicand, 250 Radioactive decay, 558–559 Radius, of a circle, 414 Range of a function, 441, 442 of inverse function, 476 of a relation, 440 Rate-time word problems, 83–85, 225–229, 327–328 Ratio, 219 word problems involving, 220–221 Rational exponents, 276–278 Rational expressions, 186 addition of, 196–200, 202–204 division of, 192–193 multiplication of, 190–191 simplifying, 186–188 subtraction of, 196–200, 202–204 Rationalizing the denominator, 255–256, 267–268 Rational numbers, 15, 16–17, 184–185 as exponents, 276–278 See also Decimals; Fractions Real number line, 22 See also Number line Real numbers, 14, 16–17 addition of, 24 division of, 27 as exponents, 551 multiplication of, 26 properties of, 31–34 subtraction of, 24–26 Real part, 296 Reciprocal, 6, 7, 33, 192 Rectangular coordinate system, 350 Reducing fractions, 5–6, 185–186 Reflections of graphs, 463–464, 476 Reflexive property of equality, 17 Relation, 440 Remainder, in polynomial division, 212, 213, 215 Repeating decimals, 15 Richter numbers, 583–584 Right triangles, 169–170, 307–308 Rigid transformations, 465 Root of an equation, 56 See also Solution(s) Roots of a number, 250–252 See also nth root Sale price, 72–73 Scientific notation, 281–284 Selling price, 73–75, 355–356 Set builder notation, 14 Sets, 14 intersection of, 99, 101 of real numbers, 16–17 subsets of, 16 union of, 100, 101 Shifts of graphs, 462–463 Shrinking graphs, 465 Similar terms, 39, 125 Simple interest, 79–80, 83 Simplest form of fraction, 6, 185 of radical, 253–255 Simplifying algebraic expressions, 39–40 complex fractions, 205–208 containing exponents, 245–248, 277–278 containing radicals, 261–262 rational expressions, 186–188 Simplifying numerical expressions, 11, 17–19, 27–29, 34–35 containing exponents, 36, 245–247, 277 containing radicals, 253–256 Slope, 368–372 in finding equation of a line, 377–380 Slope-intercept form, 378–380 Solution(s) of equation in one variable, 56 of equation in two variables, 348–349 extraneous, 272 of system of equations, 504 Solution set of an equation, 56 Solution set of an inequality, 89–90 compound, 101 Square(s) of a binomial, 138–139, 599 completing, 311–316 difference of, 152–154 of a number, 35–36 sum of, 153 Square roots, 250–251 of negative numbers, 298–299 in solving quadratic equations, 305–307 Standard form of complex number, 296 of equation of circle, 415 of equation of line, 380 of quadratic equation, 304 Stretching graphs, 467 Subset, 16 Substitution method, for solving systems, 510–512, 527, 528 Substitution property of equality, 17 Subtraction, 13 from both sides of equation, 57 from both sides of inequality, 92 of complex numbers, 297–298 distributive property and, 34 English phrases for, 43 of fractions, 8–10, 195–197 order of operations and, 11, 18 of polynomials, 125–127 of radicals, 260 Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Index of rational expressions, 196–200, 202–204 of rational numbers, 195 of real numbers, 24–26 Sum, 13, 42 of two cubes, 154 of two squares, 153 Supplementary angles, 68 Symbols of inclusion, 18 Symmetric property of equality, 17 Symmetry, 386–390, 460 Symmetry axes of ellipse, 420 of parabola, 386, 452 Synthetic division, 213–215 Systems of linear equations in two variables, 504–506, 510–513, 515–521 in three variables, 524–530 Systems of linear inequalities, 506–508 Systems of nonlinear equations, 532–535 Term, 38, 124 Terminating decimals, 15 Transformations of graphs, 462–467 Transitive property of equality, 17 Translating English phrases into algebraic expressions, 42–43 sentences into equations, 60–61 Translations of graphs, 462–463 Triangle area of, 257 right, 169–170, 307–308 sum of measures of angles, 68 Trinomials, 124 factoring, 158–164 perfect-square, 163–164, 311 Union of sets, 100, 101 Units of measurement, 44 Variables, 13 declaring, 60 Variation, 482–486 Vertex of parabola, 404, 405, 452, 454 Vertical lines, 355, 369 Vertical-line test, 475 Vertical stretching and shrinking, 465–467 Vertical translation, 462 Viewing rectangle, 391 I-5 Whole numbers, 14, 16, 17 Word problems checking the solution, 67 on investments, 76, 79–80, 102, 326, 529–530, 557, 560 on mixtures, 85–86 rate-time problems, 83–85, 225–229, 327–328 suggestions for solving, 66–68, 325 translating into equations, 60–61 using formulas, 79–80, 83–85 x-axis reflection, 463–464 x-axis symmetry, 387–388, 460 x intercept, 352–354, 460 y-axis reflection, 464 y-axis symmetry, 386–387, 460 y intercept, 352–354, 378–380, 460 Zero as exponent, 244 as identity element for addition, 32 multiplication property of, 33 Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Properties of Absolute Value 0a $ 0a 02a 0a b 0b a 0 a2 a a2 Properties of Exponents bn · b m bn1m bn bn2m bm (bn)m bmn Interval Notation (a, q) (2q, b) (a, b) [a, q) (2q, b] (a, b] [a, b) [a, b] Set Notation {x x a} {x x , b} {x a , x , b} {x x $ a} {x x # b} {x a , x # b} {x a # x , b} {x a # x # b} (ab)n anbn a n an a b n b b Multiplication Patterns (a b)2 a2 2ab b2 (a b)2 a2 2ab b2 (a b)(a b) a2 b2 (a b)3 a3 3a2b 3ab2 b3 (a b)3 a3 3a2b 3ab2 b3 Factoring Patterns a2 b2 (a b)(a b) a3 b3 (a b)(a2 ab b2) a3 b3 (a b)(a2 ab b2) Properties of Logarithms logb b logb logb rs logb r logb s r logb a b logbr logbs s logb r p p(logb r) Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.downloadslide.net Formulas Quadratic formula: The roots of ax bx c 0 where a 0, are 2b "b2 4ac 2a d "(x2 x1)2 (y2 y1)2 Distance formula for 2-space: a Midpoint of a line segment: Slope of a line: x1 x2 y1 y2 , b 2 m5 y2 y1 x2 x1 Point-slope form: y y1 m(x x1) Simple interest: i Prt and A P Prt r nt A P a1 b and A Pe rt n Compound interest: Temperature: F5 C 32 and C (F 32) Equations Determining Functions Linear function: f(x) ax b Quadratic function: f(x) ax bx c Polynomial function: f(x) anx n an21 x n21 . a1x a0 where n is a whole number Rational function: f (x) Exponential function: f(x) bx where b and b Logarithmic function: f(x) logb x where b and b g(x) where g and h are polynomials; h(x) ? h(x) Copyright 2014 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s) Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it ... need an algebra course that bridges the gap between elementary algebra and the more advanced courses in precalculus mathematics It covers topics that are usually classified as intermediate algebra. .. basis if necessary, by those who only need a brief review of some basic arithmetic and algebraic concepts • Chapter presents an early introduction to the heart of the intermediate algebra course... additional content at any time if subsequent rights restrictions require it Intermediate Algebra, Tenth Edition Jerome E Kaufmann and Karen L Schwitters © 2015, 2011, 2006 Cengage Learning General