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Algebra and Trigonometry, 9th Edition Solution Manual

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    C H A P T E R P Prerequisites Section P.1 Review of Real Numbers and Their Properties..................................... 2 Section P.2 Exponents and Radicals ......................................................................... 5 Section P.3 Polynomials and Special Products....................................................... 10 Section P.4 Factoring Polynomials.......................................................................... 16 Section P.5 Rational Expressions ............................................................................ 24 Section P.6 The Rectangular Coordinate System and Graphs ............................... 32 Review Exercises .......................................................................................................... 38 Problem Solving ........................................................................................................... 45 Practice Test ............................................................................................................... 49

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    2 C H A P T E R P Prerequisites Section P.1 Review of Real Numbers and Their Properties 1. irrational 2. origin 3. absolute value 4. composite 5. terms 6. Zero-Factor Property 7. 7 2 2 3 9, , 5, , 2, 0, 1, 4, 2, 11 − − − − (a) Natural numbers: 5, 1, 2 (b) Whole numbers: 0, 5, 1, 2 (c) Integers: 9, 5, 0, 1, 4, 2, 11 − − − (d) Rational numbers: 7 2 2 3 9, , 5, , 0, 1, 4, 2, 11 − − − − (e) Irrational numbers: 2 8. 7 5 3 4 5, 7, , 0, 3.12, , 3, 12, 5 − − − (a) Natural numbers: 12, 5 (b) Whole numbers: 0, 12, 5 (c) Integers: 7, 0, 3, 12, 5 − − (d) Rational numbers: 7 5 3 4 7, , 0, 3.12, , 3, 12, 5 − − − (e) Irrational numbers: 5 9. 2.01, 0.666 . . ., 13, 0.010110111 . . ., 1, 6 − − (a) Natural numbers: 1 (b) Whole numbers: 1 (c) Integers: 13, 1, 6 − − (d) Rational numbers: 2.01, 0.666 . . ., 13, 1, 6 − − (e) Irrational numbers: 0.010110111 . . . 10. 12 1 5 2 25, 17, , 9, 3.12, , 7, 11.1, 13 π − − − (a) Natural numbers: 25, 9, 7, 13 (b) Whole numbers: 25, 9, 7, 13 (c) Integers: 25, 17, 9, 7, 13 − (d) Rational numbers: 12 5 25, 17, , 9, 3.12, 7, 11.1, 13 − − − (e) Irrational numbers: 1 2 π 11. (a) (b) (c) (d) 12. (a) (b) (c) (d) 13. 4 8 − > − 14. 16 3 1 < 15. 5 2 6 3 > 16. 8 3 7 7 − < − 17. (a) The inequality 5 x ≤ denotes the set of all real numbers less than or equal to 5. (b) (c) The interval is unbounded.

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    Section P.1 Review of Real Numbers and Their Properties 3 18. (a) The inequality 0 x < denotes the set of all real numbers less than zero. (b) (c) The interval is unbounded. 19. (a) The interval [ ) 4, ∞ denotes the set of all real numbers greater than or equal to 4. (b) (c) The interval is unbounded. 20. (a) ( ) , 2 −∞ denotes the set of all real numbers less than 2. (b) (c) The interval is unbounded. 21. (a) The inequality 2 2 x − < < denotes the set of all real numbers greater than 2 − and less than 2. (b) (c) The interval is bounded. 22. (a) The inequality 0 6 x < ≤ denotes the set of all real numbers greater than zero and less than or equal to 6. (b) (c) The interval is bounded. 23. (a) The interval [ ) 5, 2 − denotes the set of all real numbers greater than or equal to 5 − and less than 2. (b) (c) The interval is bounded. 24. (a) The interval ( ] 1, 2 − denotes the set of all real numbers greater than 1 − and less than or equal to 2. (b) (c) The interval is bounded. 25. [ ) 0; 0, y ≥ ∞ 26. ( ] 25; , 25 y ≤ −∞ 27. [ ] 10 22; 10, 22 t ≤ ≤ 28. [ ) 3 5; 3, 5 k − ≤ < − 29. ( ) 65; 65, W > ∞ 30. [ ] 2.5% 5%; 2.5%, 5% r ≤ ≤ 31. ( ) 10 10 10 − = − − = 32. 0 0 = 33. ( ) 3 8 5 5 5 − = − = − − = 34. 4 1 3 3 − = = 35. 1 2 1 2 1 − − − = − = − 36. ( ) 3 3 3 3 6 − − − = − − = − 37. ( ) 5 5 5 1 5 5 5 − − − = = = − − − − 38. ( ) 3 3 3 3 9 − − = − = − 39. If 2, x < − then 2 x + is negative. So, ( ) 2 2 1. 2 2 x x x x + − + = = − + + 40. If 1, x > then 1 x − is positive. So, 1 1 1. 1 1 x x x x − − = = − − 41. 4 4 − = because 4 4 − = and 4 4. = 42. 5 5 − = − since 5 5. − = − 43. 6 6 − − < − because 6 6 − = and ( ) 6 6 6. − − = − = − 44. 2 2 − − = − because 2 2. − = − 45. ( ) 126, 75 75 126 51 d = − = 46. ( ) ( ) 126, 75 75 126 51 d − − = − − = 47. ( ) ( ) 5 5 5 2 2 2 , 0 0 d − = − − = 48. ( ) 5 1 11 11 1 4 4 4 4 2 , d = − = 49. ( ) 16 16 128 112 112 5 75 75 5 75 , d = − = 50. ( ) 9.34, 5.65 5.65 9.34 14.99 d − = − − = 51. ( ) , 5 5 d x x = − and ( ) , 5 3, d x ≤ so 5 3. x − ≤ i ection P.1 Review of Real Numbers and The

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    4 Chapter P Prerequisites 52. ( ) , 10 10 , d x x − = + and ( ) , 10 6, d x − ≥ so 10 6. x + ≥ 53. ( ) , d y a y a = − and ( ) , 2, d y a ≤ so 2. y a − ≤ 54. 60 23 37 F − = ° Receipts, R Expenditures, E − R E 55. $1880.1 $2292.8 1880.1 2292.8 $412.7 billion − = 56. $2406.9 $2655.1 2406.9 2655.1 $248.2 billion − = 57. $2524.0 $2982.5 2524.0 2982.5 $458.5 billion − = 58. $2162.7 $3456.2 2162.7 3456.2 $1293.5 billion − = 59. 7 4 x + Terms: 7 , 4 x Coefficient: 7 60. 3 6 5 x x − Terms: 3 6 , 5 x x − Coefficient: 6, 5 − 61. 3 4 5 2 x x + − Terms: 3 4 , , 5 2 x x − Coefficients: 1 4, 2 62. 2 3 3 1 x + Terms: 2 3 3 , 1 x Coefficients: 3 3 63. 4 6 x − (a) ( ) 4 1 6 4 6 10 − − = − − = − (b) ( ) 4 0 6 0 6 6 − = − = − 64. 9 7 x − (a) ( ) 9 7 3 9 21 30 − − = + = (b) ( ) 9 7 3 9 21 12 − = − = − 65. 2 5 4 x x − + − (a) ( ) ( ) 2 1 5 1 4 1 5 4 10 − − + − − = − − − = − (b) ( ) ( ) 2 1 5 1 4 1 5 4 0 − + − = − + − = 66. 1 1 x x + − (a) 1 1 2 1 1 0 + = − Division by zero is undefined. (b) 1 1 0 0 1 1 2 − + = = − − − 67. ( ) ( ) 1 6 1, 6 6 h h h + = ≠ − + Multiplicative Inverse Property 68. ( ) ( ) 3 3 0 x x + − + = Additive Inverse Property 69. ( ) 2 3 2 2 3 x x + = ⋅ + ⋅ Distributive Property 70. ( ) 2 0 2 z z − + = − Additive Identity Property 71. ( ) ( ) ( ) 3 3 Associative Property of Multiplication 3 Commutative Property of Multiplication x y x y x y = ⋅ = 72. ( ) ( ) 1 1 7 7 7 12 7 12 Associative Property of Multiplication 1 12 Multiplicative Inverse Property 12 Multiplicative Identity Property ⋅ = ⋅ = ⋅ = r equisites requisites

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    Section P.2 Exponents and Radicals 5 73. 5 5 15 10 9 3 1 4 8 12 6 24 24 24 24 8 − + = − + = = 74. ( ) 4 1 8 2 6 6 3 − ⋅ = − ⋅ = − 75. 2 8 3 5 3 4 12 12 12 x x x x x − = − = 76. 5 2 5 1 5 6 9 3 9 27 x x x ⋅ = ⋅ = 77. (a) Because 0, 0. A A > − < The expression is negative. (b) Because , 0. B A B A < − < The expression is negative. (c) Because 0, 0. C C < − > The expression is positive. (d) Because , 0. A C A C > − > The expression is positive. 78. The types of real numbers that may be used in a range of prices are nonnegative rational numbers because a price is neither negative nor irrational. For example, a price of 1.80 − or $ 2 would not be reasonable. The types of real numbers to describe a range of lengths are nonnegative and rational because lengths are often not in whole unit amounts but in parts, such as 2.3 centimeters or 3 4 inch. 79. False. Because 0 is nonnegative but not positive, not every nonnegative number is positive. 80. False. Two numbers with different signs will always have a product less than zero. 81. (a) (b) (i) As n approaches 0, the value of 5 n increases without bound (approaches infinity). (ii) As n increases without bound (approaches infinity), the value of 5 n approaches 0. Section P.2 Exponents and Radicals 1. exponent; base 2. scientific notation 3. square root 4. index; radicand 5. like radicals 6. conjugates 7. rationalizing 8. power; index 9. (a) 3 4 3 3 3 81 ⋅ = = (b) 2 2 4 2 3 1 1 3 3 3 9 − = = = 10. (a) ( ) 0 3 3 1 = (b) 2 3 9 − = − 11. (a) ( ) 2 3 2 3 2 2 2 6 4 2 3 2 3 2 3 64 81 5184 ⋅ ⋅ ⋅ = ⋅ = ⋅ = ⋅ = (b) ( ) 3 2 3 2 3 3 2 2 3 3 2 1 3 5 3 5 1 1 3 5 5 3 5 3 3 3 5 5 − − − § · § · − = − ⋅ = − ⋅ ⋅ ¨ ¸ ¨ ¸ © ¹ © ¹ = − ⋅ = − 12. (a) 4 5 4 3 3 3 3 243 3 − = ⋅ = = (b) ( ) ( ) 3 3 48 48 3 48 4 64 4 4 − − = = − = − − 13. (a) ( ) 2 2 1 2 1 2 1 4 3 16 4 2 3 4 4 3 2 3 3 − − − − − − − ⋅ = ⋅ ⋅ = ⋅ ⋅ = ⋅ (b) ( ) 0 2 1 − = 14. (a) 1 2 3 7 1 1 4 3 4 12 12 12 3 2 − − + = + = + = (b) ( ) 2 2 4 1 1 4 81 3 3 3 − − = = = 15. When 2, x = ( ) 3 3 3 3 2 24. x − = − = − 16. When 4, x = ( ) 2 2 2 7 7 7 7 4 . 4 16 x − − = = = 17. When 10, x = ( ) ( ) 0 0 6 6 10 6 1 6. x = = = 18. When 3, x = − ( ) ( ) 3 3 2 2 3 2 27 54. x = − = − = − n 0.0001 0.01 1 100 10,000 5 n 50,000 500 5 0.05 0.0005 ection P.2 Exponents a Section P.2 Exponents 3 5 5 5 ( ) ( ) ) (

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    6 Chapter P Prerequisites 19. When 2, x = − ( ) ( ) 4 4 3 3 2 3 16 48. x − = − − = − = − 20. When 1 3 , x = − ( ) ( ) ( ) 3 3 1 1 4 3 27 9 12 12 12 . x − = = = 21. (a) ( ) ( ) 3 3 3 3 5 5 125 z z z − = − = − (b) ( ) 4 2 4 2 6 5 5 5 x x x x + = = 22. (a) ( ) 2 2 2 2 3 3 9 x x x = = (b) ( ) 0 3 0 0 4 4 1, 0 x x x = ⋅ = ≠ 23. (a) ( ) ( ) ( ) 2 2 2 0 2 2 2 6 2 6 2 1 6 4 24 y y y y y = ⋅ = = (b) ( ) ( ) ( ) ( ) 3 3 4 3 4 3 4 7 3 1 3 1 3 3 z z z z z z + − = − = − ⋅ ⋅ = − 24. (a) 2 2 3 1 3 7 7 7 7 x x x x x − − = = = (b) ( ) ( ) ( ) ( ) 3 3 1 2 12 4 4 9 3 3 x y x y x y x y − + = + = + + 25. (a) 3 4 3 4 3 4 3 4 7 4 3 4 3 64 81 5184 y y y y y y + § · § · ⋅ = ⋅ = = ¨ ¸ ¨ ¸ © ¹ © ¹ (b) 2 2 2 2 2 2 2 1, 0, 0 b b a b a b a a b a − − § · § ·§ · § · = = ≠ ≠ ¨ ¸ ¨ ¸¨ ¸ ¨ ¸ © ¹ © ¹ © ¹© ¹ 26. (a) ( ) ( ) 1 1 1 2 2 2 2 2 2 2 2 x y x y x y x y − − − − − − ª º = « » ¬ ¼ = = (b) ( ) ( ) ( )( )( )( )( )( ) ( )( ) ( )( )( ) 3 3 2 6 2 6 3 6 18 3 6 18 0 0 0 5 5 5 5 5 1 1 1 1 x z x z x z x z x z − − − − = = = = 27. (a) ( ) 0 5 1, 5 x x + = ≠ − (b) ( ) ( ) 2 2 2 4 2 1 1 2 4 2 x x x − = = 28. (a) ( )( ) ( )( ) ( ) 2 4 4 2 2 4 8 4 8 32 y y y y − − = = (b) ( ) ( ) ( ) ( ) 3 1 4 4 1 2 2 2 2 z z z z − − − + + = + = + 29. (a) 3 3 3 4 3 4 4 12 5 125 5 x y x x y y − − § · § · = = ¨ ¸ ¨ ¸ © ¹ © ¹ (b) 3 2 2 3 5 2 2 3 5 a b b b b b a a a a − − § · § ·§ · § · = = ¨ ¸ ¨ ¸¨ ¸ ¨ ¸ © ¹ © ¹ © ¹© ¹ 30. (a) 2 2 3 3 3 3 3 n n n n n + ⋅ = = (b) 2 2 2 3 1 3 3 1 n n n n n n x x x x x x x x x + + − − − + ⋅ = = = = ⋅ 31. 4 10,250.4 1.02504 10 = × 32. 4 0.000125 1.25 10 − − = − × 33. 5 0.00003937 3.937 10 inch − = × 34. 7 57,300,000 5.73 10 square miles = × 35. 5 1.801 10 180,100 − × = − 36. 4 3.14 10 0.000314 − × = 37. 12 9.46 10 = 9,460,000,000,000 kilometers × 38. 5 9.0 10 0.00009 meter − × = 39. (a) ( )( ) 9 4 5 2.0 10 3.4 10 6.8 10 − × × = × (b) ( )( ) 7 3 4 1.2 10 5.0 10 6.0 10 − × × = × 40. (a) 8 11 3 6.0 10 2.0 10 3.0 10 − × = × × (b) 3 5 6 2 2.5 10 0.5 10 5.0 10 5.0 10 − − − × = × = × × 41. (a) 9 3 = (b) 3 3 3 27 27 3 8 2 8 = = 42. (a) 3 27 3 = (b) ( ) 3 3 36 (6) 216 = = 43. (a) ( ) 5 5 5/5 1 2 2 2 2 = = = (b) 5 5 5 5 32 (2 ) 2 x x x = = r equisites requisites

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    Section P.2 Exponents and Radicals 7 44. (a) 12 3 36 6 ⋅ = = (b) ( ) 4 4 2 4 8 4 2 3 3 3 x x x = = 45. (a) 20 4 5 4 5 2 5 = ⋅ = = (b) 3 3 3 3 3 128 64 2 64 2 4 2 = ⋅ = = 46. (a) 3 3 3 3 3 3 16 2 2 2 2 27 3 3 ⋅ = = (b) 2 2 75 5 3 5 3 4 2 2 ⋅ = = 47. (a) 3 2 72 36 2 6 2 x x x x x = ⋅ = (b) 2 2 3 2 2 18 18 18 18 z z z z z z z = = = ⋅ 48. (a) ( ) 2 4 2 2 2 54 6 3 3 6 xy x y y x = ⋅ ⋅ ⋅ = (b) ( ) ( ) 2 2 2 2 4 2 2 2 2 2 32 4 2 a a b b a b ⋅ ⋅ = = 49. (a) 3 3 5 3 2 3 2 16 8 2 2 2 x x x x x = ⋅ = (b) 2 2 4 4 2 4 2 75 75 25 3 5 3 x x y y x y x y − = ⋅ = = 50. (a) ( ) 1 4 4 2 4 2 4 1 4 1 2 4 3 3 3 3 x y x y x y x y = = = (b) ( ) ( ) ( ) 1 5 5 8 4 8 4 1 5 5 5 3 4 1 5 3 4 5 3 4 160 160 2 5 2 5 2 5 x z x z x x z x x z x x z = = ⋅ ⋅ ⋅ = = 51. (a) ( ) ( ) 10 32 6 18 10 16 2 6 9 2 10 4 2 6 3 2 40 2 18 2 22 2 − = ⋅ − ⋅ = − = − = (b) ( ) 3 3 3 3 3 3 3 3 3 3 3 16 3 54 2 2 3 2 3 2 2 3 3 2 2 2 9 2 11 2 + = ⋅ + ⋅ = + ⋅ = + = 52. (a) 5 3 2 x x x − = (b) ( ) 2 9 10 2 3 10 6 10 4 y y y y y y y − + = − + = − + = 53. (a) ( ) ( ) 2 2 2 2 2 2 3 48 7 75 3 3 4 7 3 5 3 4 3 7 5 3 12 3 35 3 23 3 x x x x x x x x x − + = − ⋅ ⋅ + ⋅ ⋅ = − ⋅ + ⋅ = − + = (b) ( ) ( ) 7 80 2 125 7 16 5 2 25 5 7 4 5 2 5 5 28 5 10 5 18 5 x x x x x x x x x − = ⋅ − ⋅ = − = − = 54. (a) 3 3 3 7 5 7 4 7 x x x − − + − = − (b) ( ) ( ) 3 3 2 2 2 2 11 245 9 45 11 5 7 9 5 3 11 7 5 9 3 5 77 5 27 5 50 5 x x x x x x x x x x x x x x x x − = ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ = ⋅ − ⋅ = − = S ection P.2 Exponents a Section P.2 Exponents C 50 5 0 5 x x 5 5 = =

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    8 Chapter P Prerequisites 55. 1 1 3 3 3 3 3 3 = ⋅ = 56. 3 3 3 2 3 3 3 3 3 2 8 8 2 8 4 8 4 4 4 2 2 2 8 2 = ⋅ = = = 57. ( ) ( ) ( ) ( ) ( ) 2 2 5 14 2 5 14 2 5 14 2 5 5 14 2 14 2 14 4 10 2 14 2 14 2 14 2 14 2 + + + + + = ⋅ = = = = − − − + − 58. ( ) ( ) ( ) ( ) 3 5 6 3 5 6 3 3 5 6 3 5 6 3 6 5 5 6 1 5 6 5 6 5 6 − − − = ⋅ = = = − − = − − − + + − 59. 8 4 2 2 2 2 2 2 2 2 2 1 2 2 ⋅ = = = ⋅ = 60. 2 2 2 2 3 3 2 3 2 = ⋅ = 61. ( ) ( ) 5 3 5 3 5 3 5 3 2 3 3 5 3 3 5 3 3 5 3 + + − − = ⋅ = = − − − 62. ( ) ( ) ( ) 7 3 7 3 7 3 7 9 2 1 4 4 7 3 4 7 3 4 7 3 2 7 3 − − + − − = ⋅ = = = − + + + + 63. 3 64 4, = Given 1 3 64 4, = Answer 64. 4 3 3 3 4 (2 ) 8 , ab a b = Answer 3 4 (2 ) , ab Given Radical Form Rational Exponent Form 65. 3 2 3 , 0 x x ≠ 2 3 2 3 3 3 x x − = 66. 2 , x x Given 5 1 2 2 2 x x x = , Answer 67. 3 , x xy Given 3 1 1 1 1 1 2 2 2 2 2 2 3 3 x x y x y ⋅ = , Answer 68. (a) 2 3 2 1 2 4 3 2 3 1 2 4 3 2 1 2 3 4 1 (2 ) 2 2 2 2 2 2 x x x x x x x − − − − − − − ⋅ ⋅ = ⋅ ⋅ ⋅ = = ⋅ = 69. (a) 5 3 3 5 3 5 3 1 1 1 1 32 32 (2) 8 ( 32) − = = = = (b) 3 3 4 3 4 3 4 16 81 81 3 27 81 16 16 2 8 − § · § · § · § · = = = = ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ © ¹ © ¹ © ¹ 70. (a) ( ) 3 3 2 3 1 100 100 10 1000 − − − = = = (b) 1 2 1 2 9 4 4 2 4 9 3 9 − § · § · = = = ¨ ¸ ¨ ¸ © ¹ © ¹ r equisites requisites

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    Section P.2 Exponents and Radicals 9 71. (a) ( ) ( ) 3 2 3 2 2 3 2 2 1 2 4 1 2 4 3 3 2 3 4 1 3 2 1 2 1 1 2 4 2 2 2 2 2 2 2 2 2 x x x x x x x x x − − − = = = = = (b) ( ) 4 3 2 3 4 3 2 3 3 3 1 3 1 3 1 3 1 3 1 3 x y x y x y xy x y xy = = = 72. (a) 3 1 2 1 2 1 3 2 1 3 2 3 1 2 1 3 2 3 3 3 1 , 0 x x x x x x x x x x x x − − + − − − ⋅ ⋅ = ⋅ ⋅ = = = > (b) ( ) 1 2 5 2 1 2 5 2 3 2 3 2 3 2 1 2 5 2 1 3 2 3 2 5 5 5 5 5 5 5 5 , 0 5 5 x x x x x x x x x − − − ⋅ ⋅ = = = = > 73. (a) 4 2 2 4 1 2 3 3 3 3 = = = (b) ( ) ( ) ( ) ( ) 4 4 6 2 3 2 6 3 1 1 1 1 x x x x + = + = + = + 74. (a) 6 3 3 6 1 2 x x x x = = = (b) ( ) 4 2 2 4 3 3 x x = 75. (a) ( ) 1 2 1 2 4 4 4 1 4 32 32 32 32 16 2 2 2 = = = = ⋅ = (b) ( ) ( ) ( ) 1 2 1 4 1 8 8 4 2 2 2 2 x x x x = = = 76. (a) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 1 4 4 4 4 243 1 243 1 243 1 243 1 3 81 1 3 3 1 x x x x x x ª + = + º « ¼ ¬ = + = + = ⋅ + = + (b) ( ) ( ) ( ) 1 2 1 3 3 7 7 1 6 7 6 6 6 10 10 10 10 10 a b a b a b a a b a ab = = = ⋅ ⋅ = 77. (a) 1/ 3 2 / 3 3 / 3 ( 1) ( 1) ( 1) ( 1) x x x x − − = − = − (b) 1/ 3 4 / 3 3 / 3 1 1 ( 1) ( 1) ( 1) ( 1) 1 x x x x x − − − − − = − = − = − 78. (a) 5 / 2 5 / 3 15 / 6 10 / 6 5 / 6 (4 3) (4 3) (4 3) (4 3) 3 (4 3) , 4 x x x x x x − − + + = + + = + ≠ − (b) 5 / 2 1/ 2 15 / 6 3 / 6 12 / 6 2 2 (4 3) (4 3) (4 3) (4 3) (4 3) (4 3) 1 3 , (4 3) 4 x x x x x x x x − − − − + + = + + = + = + = ≠ − + S ection P.2 Exponents a Section P.2 Exponents

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    10 Chapter P Prerequisites 79. ( ) 5 2 5 2 0.03 12 12 , 0 12 t h h ª º = − − ≤ ≤ ¬ ¼ (a) (b) As h approaches 12, t approaches ( ) 5 2 0.03 12 8.64 3 14.96 seconds. = = 80. 3 500 x = 3 3 1 3 250 2 2 x x § · = = ¨ ¸ © ¹ So, twice the length of a side of package B is 3 2 3 1 3 2 2 4 . 2 x x x § · = = ¨ ¸ © ¹ Because 3 4 , x x < the length of a side of package A is less than twice the length of a side of package B. 81. True. When dividing variables, you subtract exponents. 82. False. When a power is raised to a power, you multiply the exponents: ( ) . k n nk a a = 83. False. When a sum is raised to a power, you multiply the sum by itself using the Distributive Property. 2 2 2 2 2 ( ) 2 a b a ab b a b + = + + ≠ + 84 . False. To rationalize the denominator, multiply the fraction by 1. b b = This does not change the value of the expression, but raising the numerator and denominator to the second power does. Section P.3 Polynomials and Special Products 1. 0 ; ; n n a a 2. monomial; binomial; trinomial 3. like terms 4. First terms; Outer terms; Inner terms; Last terms 5. (a) Standard form: 5 1 2 14 x x − + (b) Degree: 5 Leading coefficient: 1 2 − (c) Binomial 6. (a) Standard form: 7 x (b) Degree: 1 Leading coefficient: 7 (c) Monomial 7. (a) Standard form: 6 3 x − + (b) Degree: 6 Leading coefficient: í 1 (c) Binomial 8. (a) Standard form: 2 25 1 y y − + (b) Degree: 2 Leading coefficient: 25 (c) Trinomial 9. (a) Standard form: 3 (b) Degree: 0 Leading coefficient: 3 (c) Monomial 10. (a) Standard form: 2 8 t − (b) Degree: 2 Leading coefficient: 1 (c) Binomial 11. (a) Standard form: 5 4 4 6 1 x x − + + (b) Degree: 5 Leading coefficient: í 4 (c) Trinomial 12. (a) Standard form: 2 3 x + (b) Degree: 1 Leading coefficient: 2 (c) Binomial h (in centimeters) t (in seconds) 0 0 1 2.93 2 5.48 3 7.67 4 9.53 5 11.08 6 12.32 7 13.29 8 14.00 9 14.50 10 14.80 11 14.93 12 14.96 r equisites requisites c) Binomial c) Binomial

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    Section P.3 Polynomials and Special Products 11 13. (a) Standard form: 3 4 x y (b) Degree: 4 (add the exponents on x and y ) Leading coefficient: 4 (c) Monomial 14. (a) Standard form: 5 2 2 4 2 x y x y xy − + + (b) Degree: 6 Leading coefficient: í 1 (c) Trinomial 15. 3 2 3 8 x x − + is a polynomial. Standard form: 3 3 2 8 x x − + + 16. 4 2 2 5 2 x x x − − + is not a polynomial because it includes a term with a negative exponent. 17. 1 3 4 4 3 3 4 x x x x − + = + = + is not a polynomial because it includes a term with a negative exponent. 18. 2 2 3 2 x x + − is a polynomial. Standard form: 2 1 3 2 2 x x + − 19. 2 4 3 y y y − + is a polynomial. Standard form: 4 3 2 y y y − + + 20. 4 y y − is not a polynomial because it includes a term with a square root. 21. ( ) ( ) ( ) ( ) 6 5 8 15 6 5 8 15 6 8 5 15 2 10 x x x x x x x + − + = + − − = − + − = − − 22. ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 2 1 1 2 x x x x x x x x x x x + − − + = + − + − = − + + − = + 23. ( ) ( ) ( ) 3 3 3 3 3 3 3 1 6 5 1 6 5 6 5 1 5 5 1 t t t t t t t t t t t − − + − = − + + − = − + − + = − + 24. ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 5 1 3 5 5 1 3 5 5 3 1 5 2 4 x x x x x x x − − − − + = − + + − = − + + − = − − 25. ( ) ( ) ( ) ( ) 2 3 2 2 3 2 3 2 2 3 2 15 6 8.3 14.7 17 15 6 8.3 14.7 17 8.3 15 14.7 6 17 8.3 29.7 11 x x x x x x x x x x x − − − − − = − + + + = + + + − + = + + 26. ( ) ( ) ( ) ( ) ( ) 4 4 4 4 4 4 4 15.6 14 17.4 16.9 9.2 13 15.6 14 17.4 16.9 9.2 13 15.6 16.9 14 9.2 17.4 13 1.3 4.8 30.4 w w w w w w w w w w w w w w − − − − + = − − − + − = − + − + + − − = − − − 27. ( ) ( ) ( ) 5 3 10 8 5 3 10 8 5 3 10 8 5 3 10 8 12 8 z z z z z z z z z z z z z − ª − + º = − − − ¬ ¼ = − + + = − + + = + 28. ( ) ( ) ( ) ( ) ( ) ( ) 3 2 3 2 3 2 3 2 3 2 1 1 3 7 1 1 3 7 1 1 3 7 3 1 1 7 3 7 y y y y y y y y y y y y y y y ª º + − + + − = + − + − − ¬ ¼ = + − − − + = − − + − + = − − + 29. ( ) ( ) ( ) ( ) 2 2 3 2 3 2 1 3 3 2 3 1 3 6 3 x x x x x x x x x x x − + = + − + = − + 30. ( ) ( ) ( ) ( ) 2 2 2 2 2 2 4 3 2 4 2 3 4 2 3 4 2 3 y y y y y y y y y y y + − = + + − = + − 31. ( ) ( ) ( )( ) 2 5 3 1 5 3 5 1 15 5 z z z z z z z − − = − + − − = − + 32. ( )( ) ( ) ( )( ) 2 3 5 2 3 5 3 2 15 6 x x x x x x x − + = − + − = − − S ection P.3 Polynomials and Spec Section P.3 Polynomials and Spe

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    12 Chapter P Prerequisites 33. ( )( ) ( ) ( ) 3 3 4 4 1 4 1 4 4 4 4 4 4 x x x x x x x x x − = − = − = − + 34. ( ) ( ) ( )( ) 3 3 4 4 4 3 4 3 4 12 4 4 12 x x x x x x x x x − − = − + − − = − + = − 35. ( )( ) ( )( ) ( )( ) 2 2 3 1.5 5 3 1.5 3 5 3 4.5 15 t t t t t t t + − = − + − = − − 36. ( )( ) ( ) ( )( ) 3 3 3 3 4 4 3 2 3.5 2 2 2 3.5 2 4 7 7 4 y y y y y y y y y − = + − = − = − + 37. ( ) ( )( ) ( )( ) 2 2 0.1 17 2 0.1 2 17 0.2 34 x x x x x x x − + = − + − = − − 38. ( ) ( )( ) ( )( ) 2 2 3 3 8 8 9 9 4 4 6 5 6 5 6 30 30 y y y y y y y y y − = − = − = − + 39. ( ) ( ) ( ) ( ) ( ) 3 2 2 3 3 2 3 2 7 2 8 3 4 7 3 2 8 4 4 2 4 x x x x x x x x − + + − − = − + − + − = − + 40. ( ) ( ) ( ) ( ) ( ) 5 3 3 5 3 3 5 3 2 3 2 3 4 6 2 3 4 2 3 6 2 3 3 x x x x x x x x x x x x x − + + + + − = + − + + + + − = + + − 41. ( ) ( ) ( ) ( ) 2 2 2 2 5 3 8 3 5 3 8 3 5 3 8 3 5 4 11 x x x x x x x x x x x − + − − = − + − + = + − − + + = − + 42. ( ) ( ) ( ) ( ) 4 2 4 2 4 2 4 2 4 4 2 2 4 2 0.6 2 0.5 5.6 0.6 2 0.5 5.6 0.6 2 0.5 5.6 1.6 2.5 5.6 t t t t t t t t t t t t t − − − + − = − + − + = + + − − + = − + 43. ( )( ) 2 2 7 2 3 2 3 14 21 FOIL 2 17 21 x x x x x x x + + = + + + = + + 44. ( ) ( ) 2 4 2 1 x x x x + − − + Multiply: 2 2 4 3 2 3 2 2 4 3 2 4 2 1 4 2 2 8 4 5 9 4 x x x x x x x x x x x x x x x x + − − + + − − − + + − − − + − 45. ( )( ) 2 2 3 4 4 3 12 FOIL 7 12 x x x x x x x + + = + + + = + + 46. ( )( ) 2 2 5 10 10 5 50 FOIL 5 50 x x x x x x x − + = + − − = + − 47. ( )( ) 2 2 3 5 2 1 6 3 10 5 FOIL 6 7 5 x x x x x x x − + = + − − = − − 48. ( )( ) 2 2 7 2 4 3 28 21 8 6 FOIL 28 29 6 x x x x x x x − − = − − + = − + 49. ( )( ) 2 2 1 1 x x x x − + + + Multiply: 2 2 4 3 2 3 2 2 4 3 2 4 2 1 1 1 0 0 1 1 x x x x x x x x x x x x x x x x x x − + + + − + − + − + − + + + = + + 50. ( )( ) 2 2 2 4 3 2 x x x x − + + + Multiply: 2 2 4 3 2 3 2 2 4 3 2 2 4 3 2 2 4 6 3 12 4 2 8 2 5 5 10 8 x x x x x x x x x x x x x x x x − + + + − + − + − + + + + + r equisites requisites

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    Section P.3 Polynomials and Special Products 13 51. ( )( ) 2 2 2 10 10 10 100 x x x x + − = − = − 52. ( )( ) ( ) 2 2 2 2 3 2 3 2 3 4 9 x x x x + − = − = − 53. ( )( ) ( ) 2 2 2 2 2 2 2 4 x y x y x y x y + − = − = − 54. ( )( ) ( ) ( ) 2 2 2 2 4 5 4 5 4 5 16 25 a b a b a b a b + − = − = − 55. ( ) ( ) ( )( ) 2 2 2 2 2 3 2 2 2 3 3 4 12 9 x x x x x + = + + = + + 56. ( ) ( )( )( ) ( ) 2 2 2 2 5 8 5 2 5 8 8 25 80 64 x x x x x − = + − + − = − + 57. ( ) ( ) ( )( ) ( ) 2 2 2 3 3 3 6 3 4 3 4 2 4 3 3 16 24 9 x x x x x − = − + = − + 58. ( ) ( ) ( )( ) 2 2 2 2 8 3 8 2 8 3 3 64 48 9 x x x x x + = + + = + + 59. ( ) ( ) ( ) 3 3 2 2 3 3 2 1 3 1 3 1 1 3 3 1 x x x x x x x + = + + + = + + + 60. ( ) ( ) ( ) 3 2 3 2 3 3 2 2 3 2 3 2 2 6 12 8 x x x x x x x − = − + − = − + − 61. ( ) ( ) ( ) ( ) 3 3 2 2 3 3 2 2 3 2 2 3 2 3 2 8 12 6 x y x x y x y y x x y xy y − = − + − = − + − 62. ( ) ( ) ( ) ( ) ( )( ) ( ) 3 3 2 2 3 3 2 2 3 3 2 3 3 3 2 3 3 2 2 27 54 36 8 x y x x y x y y x x y xy y + = + + + = + + + 63. ( ) ( ) ( ) ( ) 2 2 2 2 2 2 3 3 3 6 9 6 9 m n m n m n m m n m n m ª − + ºª − − º = − − ¬ ¼¬ ¼ = − + − = − − + 64. ( ) ( ) ( ) ( ) 2 2 2 2 2 3 3 3 6 9 x y z x y z x y z x xy y z ª − + ºª − − º = − − ¬ ¼¬ ¼ = − + − 65. ( ) ( ) ( ) 2 2 2 2 2 2 2 3 3 2 3 6 9 2 6 2 6 6 9 x y x y x y x x xy y y x xy y x y ª − + º = − + − + ¬ ¼ = − + + − + = + + − − + 66. ( ) ( ) ( )( ) ( ) 2 2 2 2 2 2 2 1 1 2 1 2 1 2 2 2 2 2 1 x y x x y y x x xy y y x xy y x y ª + − º = + + + − + − ¬ ¼ = + + − − + = − + + − + 67. ( )( ) ( ) ( ) 2 2 2 1 1 1 5 5 5 1 25 3 3 3 9 x x x x − + = − = − 68. ( )( ) ( ) ( ) 2 2 2 1 1 1 6 6 6 1 36 3 3 3 9 x x x x + − = − = − 69. ( )( ) ( ) ( ) 2 2 2 1.5 4 1.5 4 1.5 4 2.25 16 x x x x − + = − = − 70. ( )( ) ( ) ( ) 2 2 3 2 3 2 3 2 6 4 3 4 3 4 3 4 9 16 a b a b a b a b − + = − = − 71. ( ) ( ) ( )( ) ( ) 2 2 2 2 1 1 1 4 4 4 5 1 16 2 5 2 5 5 25 x x x x x − = − + − = − + 72. ( ) ( ) ( )( ) ( ) 2 2 2 2 2.4 3 2.4 2 2.4 3 3 5.76 14.4 9 x x x x x + = + + = + + 73. ( ) ( ) ( ) 2 5 1 3 1 2 1 2 2 x x x x x x x x + − + = + = + 74. ( )( ) ( ) ( )( ) 2 2 2 1 3 3 3 2 2 3 2 6 2 6 FOIL 2 8 6 x x x x x x x x x x − + + + = + + = + + + = + + 75. ( )( )( ) ( )( ) 2 2 2 4 2 2 4 4 4 16 u u u u u u + − + = − + = − 76. ( )( )( ) ( )( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 4 4 x y x y x y x y x y x y x y + − + = − + = − = − 77. ( ) ( ) ( ) ( ) 2 2 x y x y x y x y + − = − = − 78. ( ) ( ) ( ) ( ) 2 2 5 5 5 25 x x x x + − = − = − 79. ( ) ( ) ( ) ( ) 2 2 2 2 5 2 5 5 2 5 5 x x x x x − = − + = − + 80. ( ) ( ) 2 2 2 2 3 2 3 3 2 3 3 x x x x x + = + + = + + S ection P.3 Polynomials and Spec Section P.3 Polynomials and Spe y y 3 3 x 2 3 2 = = x 2 3 2

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    14 Chapter P Prerequisites 81. (a) ( ) Profit Revenue Cost Profit 95 73 25,000 95 73 25,000 22 25,000 x x x x x = − = − + = − − = − (b) For 5000: x = ( ) Profit 22 5000 25,000 110,000 25,000 $85,000 = − = − = 82. (a) ( ) ( ) ( ) 2 2 2 2 500 1 500 1 500 2 1 500 1000 500 r r r r r r + = + = + + = + + (b) (c) As r increases, the amount increases. 83. ( )( ) 2 2 18 2 14 252 18 28 2 2 46 252 A x x x x x x x = + + = + + + = + + 84. ( )( ) 2 1 4 4 4 x x x x x + + = + + + This illustrates the Distributive Property. 85. Area of shaded region Area of outer rectangle Area of inner rectangle = − ( ) ( ) 2 2 2 2 2 6 4 4 12 4 3 8 A x x x x x x x x x x = + − + = + − − = + 86. Area of shaded region Area of outer triangle Area of inner triangle = − ( )( ) ( )( ) 2 2 2 1 1 2 2 9 12 6 8 54 24 30 A x x x x x x x = − = − = 87. The area of the shaded region is the difference between the area of the larger triangle and the area of the smaller triangle. ( )( ) ( )( ) 2 2 2 1 1 2 2 10 10 4 4 50 8 42 A x x x x x x x = − = − = 88. The area of the shaded region is one-half the area of the rectangle. ( )( ) ( )( ) ( )( ) 2 1 1 2 2 length width 4 2 3 2 1 3 6 3 A x x x x x x = = − = − = − 89. The area of the shaded region is the difference between the area of the larger square and the area of the smaller square. ( ) ( ) ( ) 2 2 2 2 2 4 2 1 16 16 4 2 1 15 18 3 A x x x x x x x x = + − − = + + − − + = + + 90. The area of the shaded region is the difference between the area of the larger triangle and the area of the smaller triangle. ( ) ( ) ( ) ( ) 2 2 2 2 2 2 3 1 1 1 1 2 2 2 2 2 2 8 4 4 32 64 8 16 3 24 48 12 24 A x x x x x x x x x x ª º = + − + = + + − + + = + + = + + ¬ ¼ 91. (a) ( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( ) 3 2 26 2 18 2 2 13 2 9 4 1 13 1 9 4 13 9 4 88 468 V l w h x x x x x x x x x x x x x x x = ⋅ ⋅ = − − = − − = − − − − = − − = − + (b) r 1 2 2 % 3% 4% 1 2 4 % 5% ( ) 2 500 1 r + $525.31 $530.45 $540.80 $546.01 $551.25 ( ) cm x 1 2 3 ( ) 3 cm V 384 616 720 r equisites requisites

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    Section P.3 Polynomials and Special Products 15 92. (a) ( )( ) ( ) ( ) 2 2 2 3 3 2 1 2 1 2 1 2 1 2 Volume length width height 45 3 15 2 45 3 15 2 675 90 45 6 6 135 675 x x x x x x x x x x x x x = × × = − − = − − ª º = − − + ¬ ¼ ª º = − + ¬ ¼ (b) When ( ) ( ) ( ) ( ) [ ] ( ) 3 3 1 1 2 2 1 1 2 2 3: 6 3 135 3 675 3 6 27 135 9 2025 162 1215 2025 972 486 cubic centimeters x V ª º = = − + = ⋅ − ⋅ + ¬ ¼ = − + = = When ( ) ( ) ( ) [ ] [ ] ( ) 3 2 1 1 2 2 1 1 2 2 5: 6 5 135 5 675 5 6 125 135 25 3375 750 3375 3375 750 375 cubic centimeters x V ª º = = − + = ⋅ − ⋅ + ¬ ¼ = − + = = When ( ) ( ) ( ) [ ] [ ] ( ) 3 2 1 1 2 2 1 1 2 2 7: 6 7 135 7 675 7 6 343 135 49 4725 2058 6615 4725 168 84 cubic centimeters x V ª º = = − + = ⋅ − ⋅ + ¬ ¼ = − + = = 93. (a) Estimates will vary. Actual safe loads for 12 : x = ( ) ( ) ( ) 2 2 6 0.06 12 2.42 12 38.71 335.2561 (using a calculator) S = − + = ( ) ( ) ( ) 2 2 8 0.08 12 3.30 12 51.93 568.8225 (using a calculator) S = − + = Difference in safe loads 568.8225 335.2561 233.5664 pounds = − = (b) The difference in safe loads decreases in magnitude as the span increases. 94. (a) ( ) 2 2 1.1 0.0475 0.001 0.23 0.0475 1.099 0.23 T R B x x x x x = + = + − + = + + . (b) (c) Stopping distance increases at an accelerating rate as speed increases 95. False. ( ) ( ) 2 3 2 4 1 3 1 12 4 3 1 x x x x x + + = + + + 96. False. ( ) ( ) 4 3 4 6 4 3 4 6 3 6 9 x x x x + + − + = + − + = + = 97. Because , m n m n x x x + = the degree of the product is . m n + 98. If the degree of one polynomial is m and the degree of the second polynomial is n (and n m > ), the degree of the sum of the polynomials is n . 99. ( ) 2 2 3 9 x x − = + / The student did not remember the middle term when squaring the binomial. The correct method for squaring the binomial is: ( ) ( ) ( )( ) ( ) 2 2 2 2 3 2 3 3 6 9 x x x x x − = − + = − + ( ) cm x 3 5 7 ( ) 3 Volume cm 486 375 84 mi hr x 30 40 55 feet T 75.95 120.19 204.36 S ection P.3 Polynomials and Spec Section P.3 Polynomials and Spe

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