Math 10 - Unit 5 Final Review - Polynomials

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1 Class: Date: Math 10 - Unit 5 Final Review - Polynomials Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Factor the binomial 44a + 99a 2. a. a( a) c. 11a(4 + 9a) b. 11(4a + 9a 2 ) d. 22a(2 + 9a) 2. Factor the binomial 15y 2 48y. a. 3(5y 2 16y) c. y(15y 48) b. 3y(5y 16y) d. 3y(5y 16) 3. Factor the trinomial 4 8n + 12n 2. a. 4( 2n + 3n 2 ) c. 2(2 4n + 6n 2 ) b. 4(1 2n + 3n 2 ) d. 4(1 + 2n + 3n 2 ) 4. Factor the trinomial 33b b a. 11(3b 2 9b + 7) c. 11(3b 2 9b 7) b. 33(b 2 3b 7) d. 33( b b + 7) 5. Factor the trinomial 24c 3 d 40c 2 d 2 32cd 3. a. 8cd(3c 2 5cd 4d 2 ) c. 8cd( 3c 2 + 5cd + 4d 2 ) b. 8cd(3c 2 + 5cd + 4d 2 ) d. 8cd(3c 2 + 5cd + 4d 2 ) 6. Factor the trinomial 42x 5 y 6 24x 4 y 5 54x 3 y 7. a. 6x 4 y 5 ( 7xy 4 9y 2 ) c. 3x 3 y 5 (14x 2 y + 8x + 18y 2 ) b. 6x 3 y 5 (7x 2 y + 4x + 9y 2 ) d. 6x 3 (7x 2 y 6 + 4xy 5 + 9y 7 ) 7. Factor the binomial 10m 2 40m 4. a. 10m 2 (1 + 4m 2 ) c. 10(m 2 + 4m 4 ) b. 10m 2 (4m 2 ) d. 5m 2 (2 + 8m 2 ) 8. Simplify the expression y 2 + 8y 6 9y 2 24y 26, then factor. a. 8(y 2 2y 4) c. 4(2y 2 + 4y + 8) b. 8(y 2 + 2y + 4) d. 4(2y 2 + 4y + 1) 1

2 9. Which expression represents the area of the shaded region? a. 2r(2r π) b. r 2 (1 π) c. r 2 (4 π) d. r(r 2π) 10. Expand and simplify: (p + 3)(p 7) a. p 2 4p 21 c. p p 21 b. p 2 10p 21 d. p 2 + 4p Expand and simplify: (4 r)(7 r) a r + r 2 c r + r 2 b. 28 3r + r 2 d r + r Factor: t 2 + 9t 36 a. (t 2)(t + 18) c. (t + 12)(t 3) b. (t + 2)(t 18) d. (t 12)(t + 3) 13. Factor: v 2 13v + 36 a. (v + 3)(v + 12) c. (v 4)(v 9) b. (v 3)(v 12) d. (v + 4)(v + 9) 14. Factor: 24 2x + x 2 a. (6 + x)( 4 + x) c. ( 3 + x)(8 + x) b. (3 + x)( 8 + x) d. ( 6 + x)(4 + x) 15. Factor: z + z 2 a. (42 + z)( 2 + z) c. ( 42 + z)(2 + z) b. ( 6 + z)(14 + z) d. (6 + z)( 14 + z) 16. Factor: 3b b + 18 a. 3(b 2)(b + 3) c. 3(b 1)(b + 6) b. 3(b + 2)(b 3) d. 3(b + 1)(b 6) 17. Factor: 4d 2 28d a. 4(d + 3)(d 20) c. 4(d 3)(d + 20) b. 4(d + 5)(d 12) d. 4(d 5)(d + 12) 2

3 18. Complete: (a + 6)(a ) = a 2 + a 12 a. (a + 6)(a 4) = a 2 + 4a 12 c. (a + 6)(a 2) = a 2 + 2a 12 b. (a + 6)(a 2) = a 2 + 4a 12 d. (a + 6)(a 4) = a 2 + 2a Factor: c 2 4c 117 a. (c 9)(c + 13) c. (c + 9)(c 13) b. (c 3)(c + 39) d. (c + 3)(c 39) 20. Factor: 12 4g g 2 a. (4 g)(3 + g) c. (6 g)(2 + g) b. (6 + g)(2 g) d. (4 + g)(3 g) 21. Expand and simplify: (h 6)(h + 11) a. h 2 5h 66 c. h h 66 b. h 2 + 5h 66 d. h 2 17h Factor: 5m m + 60 a. 5(m + 2)(m 6) c. 5(m 4)(m + 3) b. 5(m 2)(m + 6) d. 5(m + 4)(m 3) 23. Factor: 7n 2 14n 105 a. 7(n + 3)(n 5) c. 7(n 15)(n + 1) b. 7(n + 15)(n 1) d. 7(n 3)(n + 5) 24. Which multiplication sentence does this set of algebra tiles represent? a. (2x 2)(2x + 2) c. (2x 2 + 2x)(2x 2 + 2x) b. (2x 2 + 2)(2x 2 + 2) d. (2x + 2)(2x + 2) 3

4 25. Which set of algebra tiles represents 3x 2 + x + 4? a. c. b. d. 26. Expand and simplify: (6p + 3)(5p 6) a. 30p p 18 c. 30p p 18 b. 30p 2 21p 18 d. 30p 2 51p Expand and simplify: (8g 3)(7 3g) a. 24g g 21 c. 24g g 21 b. 24g 2 65g 21 d. 24g g Factor: 25x x + 16 a. (25x + 4)(x + 4) c. (5x + 4)(5x + 4) b. (25x + 8)(x + 2) d. (5x + 8)(5x + 2) 29. Factor: 16s 2 137s 63 a. (4s 7)(4s + 9) c. (16s + 7)(s 9) b. (4s + 7)(4s 9) d. (16s 7)(s + 9) 30. Factor: 48y 2 116y + 60 a. (16y 12)(3y 5) c. 4(4y 5)(3y 3) b. 4(4y 3)(3y 5) d. 4(4y + 3)(3y + 5) 31. Factor: 24b b 14 a. 2(4b 1)(3b + 7) c. 2(4b 7)(3b + 1) b. 2(4b + 7)(3b + 1) d. 2(4b + 1)(3b 7) 32. Expand and simplify: 3(1 2t)(9 + 4t) a. 24t t + 27 c. 72t 2 126t + 81 b. 24t t + 27 d. 24t 2 42t Factor: 7n n 15 a. (7n 1)(n + 15) c. (7n + 15)(n 1) b. (7n + 1)(n 15) d. (7n 15)(n + 1) 4

5 34. Factor: 4 9z 13z 2 a. (2 13z)(2 + z) c. (2 + 13z)(2 z) b. (4 13z)(1 + z) d. (4 + 13z)(1 z) 35. Factor: a + 30a 2 a. 5(4 + 3a)(9 + 2a) c. 5(4 3a)(9 2a) b. (20 15a)(9 2a) d. 10(18 1a)(1 3a) 36. Factor: 96w w 42 a. 6(8w + 1)(2w 7) c. 6(8w 7)(2w + 1) b. 6(8w + 7)(2w 1) d. 6(8w 1)(2w + 7) 37. Expand and simplify: (8h + 3)(7h 2 4h + 1) a. 56h 3 53h 2 20h + 3 c. 56h 3 11h 2 4h + 3 b. 56h h 2 12h + 3 d. 56h 3 32h 2 + 8h Expand and simplify: (5m 3n) 2 a. 25m 2 9n 2 c. 25m 2 30mn + 9n 2 b. 25m 2 15mn + 9n 2 d. 25m 2 + 9n Expand and simplify: (7m 2n) 2 a. 49m 2 4n 2 c. 49m 2 28mn + 4n 2 b. 49m 2 14mn + 4n 2 d. 49m 2 + 4n Expand and simplify: (7m 3n) 2 a. 49m 2 9n 2 c. 49m 2 42mn + 9n 2 b. 49m 2 21mn + 9n 2 d. 49m 2 + 9n Expand and simplify: (4s + 9t)(5s 4t 3) a. 20s st 12s 36t 2 27t c. 20s st + 12s 36t t b. 20s st 12s + 36t 2 27t d. 20s st 12s 36t 2 27t 42. Expand and simplify: (10v 13w)(10v + 13w) a. 100v vw + 169w 2 c. 100v 2 169w 2 b. 100v w 2 d. 100v 2 260vw + 169w Expand and simplify: (4d 1)(5d d 3) a. 20d d c. 20d d 2 24d + 3 b. 20d d 2 12d + 3 d. 20d d Expand and simplify: (f + 5g)(2f 5g + 7) a. 2f 2 + 5fg + 7f + 25g g c. 2f 2 + 5fg + 7f 25g g b. 2f 2 15fg + 7f 25g g d. 2f 2 5fg + 7f 25g g 5

6 45. Which polynomial, written in simplified form, represents the area of this rectangle? a. 8x 2 36xy 20y 2 c. 16x xy 40y 2 b. 8x xy 20y 2 d. 8x xy 20y Expand and simplify: (2x 2 + 5x 6)(5x 2 2x + 3) a. 10x x 3 34x x 18 c. 10x x 3 24x x + 18 b. 10x x 3 34x 2 3x + 18 d. 10x 4 29x 3 34x x Expand and simplify: (n 2 2n + 3)( 2n 2 + 7n + 8) a. 2n n 3 12n 2 + 5n + 24 c. 2n 4 3n n + 24 b. 2n n n + 24 d. 2n 4 3n 3 12n 2 + 5n Expand and simplify: (6p + 3)(6p 7) (7p 4)(p 2) a. 29p 2 42p 13 c. 29p 2 6p 29 b. 29p 2 6p 13 d. 29p 2 42p Expand and simplify: (6x y)(3x + 8y) (2x 3y) 2 a. 14x xy 17y 2 c. 14x xy + 1y 2 b. 14x xy + 1y 2 d. 14x xy 17y Expand and simplify: (3c + 2)(2c 7) + 3( 2c + 1)(7c 5) a. 36c 2 + 8c 29 c. 36c 2 8c 19 b. 36c c 29 d. 36c 2 8c Expand and simplify: (4a b 2)(3a 7) (3a + 4b) 2 a. 3a 2 34a 3ab + 7b b 2 c. 3a 2 22a 27ab + 7b b 2 b. 3a 2 34a 27ab + 7b b 2 d. 3a 2 34a + 21ab + 7b b 2 6

7 52. Each shape is a rectangle. Write a polynomial, in simplified form, to represent the area of the shaded region. a. 5x x + 66 c. 5x x + 30 b. 5x x + 30 d. 5x x Factor: 121a a + 25 a. (11a + 5)(11a 5) c. (11a 5) 2 b. (121a + 5)(a + 5) d. (11a + 5) Factor: 36 60r + 25r 2 a. (9 5r)(4a 5r) c. (6 + 5r) 2 b. (6 5r)(6 + 5r) d. (6 5r) Factor: 16p 2 81q 2 a. (4p 9q) 2 c. (16p 9q)(p 9q) b. (4p + 9q) 2 d. (4p + 9q)(4p 9q) 56. Find an integer to replace so that this trinomial is a perfect square. x xy + 9y 2 a. 7 c. 49 b. 14 d Find an integer to replace so that this trinomial is a perfect square. 64v 2 vw + 81w 2 a. 144 c. 72 b. 648 d Factor: 49s 2 112st + 64t 2 a. (7s 8t) 2 c. (7s t)(7s 64t) b. (7s + 8t) 2 d. (7s 8t)(7s + 8t) 59. Identify this polynomial as a perfect square trinomial, a difference of squares, or neither. 9a 2 + 9a + 36 a. Difference of squares c. Neither b. Perfect square trinomial 7

8 60. Identify this polynomial as a perfect square trinomial, a difference of squares, or neither. 25g 2 9h 2 a. Perfect square trinomial c. Neither b. Difference of squares 61. Factor: 9c 2 12c + 4 a. (3c 2) 2 c. (6c 4) 2 b. (3c 2)(3c + 2) d. (6c 4)(6c + 4) 62. Factor: 8y 2 58yz + 60z 2 a. 4(2y 3z)(y 5z) c. 2(4y 5z)(y 6z) b. 2(4y + 5z)(y 6z) d. 2(4y 5z)(y + 6z) 63. Factor: 3z 4 768z 2 a. 3z 2 (z + 16)(z 16) c. z 2 (z + 48)(z 16) b. 3z 2 (z + 16) 2 d. 3z 2 (z 16) Factor: 48b bc 3c 2 a. (2b + 3c)(24b c) c. (16b + c)(3b 3c) b. (48b 3c)(b c) d. (2b 3c)(24b + c) 65. Factor: 8m 2 34mn + 33n 2 a. (4m 11n)(2m 3n) c. (4m + 11n)(2m + 3n) b. (8m 33n)(m n) d. (4m 11n)(2m + 3n) 66. Factor: 162 2w 4 a. (9 w 2 )(18 w 2 ) c. 2(9 w 2 ) 2 b. 2(9 + w 2 )(3 + w)(3 w) d. 2(9 + w 2 ) Determine the area of the shaded region in factored form. a. 4(x + 12) c. (3x + 12)(x + 2) b. (3x + 2)(x + 12) d. (3x 2)(x 12) 8

9 68. From the list, which terms are like 7x? 7x 2, 6x, 5, 8x, 7x, 8x 2, 7 a. 7x 2, 6x, 8x, 7x, 8x 2 c. 7x b. 6x, 8x, 7x d. 7x 2, 7x, These algebra tiles may be used in the following question. x 2 x 2 x x 1 1 Which pair of tiles represents a zero pair? i) ii) iii) iv) a. i b. ii c. iii d. iv 70. Combine like terms. 6x + 5x 2 + 4x + 2x 2 a. 5x 2 b. 2x 2 + 7x 4 c. 2x + 7x 2 d. 5x 71. These algebra tiles may be used in the following question. x 2 x 2 x x 1 1 Write the polynomial sum modelled by this set of tiles. a. ( x 2 3x 4) + ( x 2 + x + 4) c. ( 2) + ( 2) + 0 b. (x 2 + 3x + 4) + (x 2 + x + 4) d Add. ( 5 6x 2 ) + (8x 2 + 7) a. 4x 2 b x 4 c. 4 d x 2 9

10 73. Add. ( 5 + 5r + 5r 2 ) + ( 2 8r 2 7r) a. 7 2r 2 3r 4 c. 7 2r 3r 2 b. 2 3r 2r 2 d. 2r 2 3r These algebra tiles may be used in the following question. x 2 x 2 x x 1 1 Write the subtraction sentence that these algebra tiles represent. a. (x 2 + 3x + 2) (x 2 + x + 2) = 2x b. ( x 2 3x + 2) ( x 2 x + 2) = 2x c. ( x 2 x + 2) ( x 2 3x + 2) = 2x d. (x 2 + x + 2) (x 2 + 3x + 2) = 2x 75. Subtract. ( 3n 2 + 8) (5 7n 2 ) a. 10n c. 10n b. 8n d. 4n Subtract. ( 4b 6b 2 2) (8b + 8 7b 2 ) a. 12b 13b 2 10 c. b 14b b. 12b 13b d. 12b + b

11 77. These algebra tiles may be used in the following question. x 2 x 2 x x 1 1 What product is modelled by this set of algebra tiles? a. 3(x 2 4x 3) c. 3(x 2 + 4x + 3) b. 3( x 2 4x + 3) d. 3(x 2 4x + 3) 78. Write the multiplication sentence modelled by this rectangle. a. 2(4x + 7) = 8x + 14 c. 2(4x) + 7 = 8x + 7 b. 2(4x 7) = 8x 14 d. 2(4x + 7) = 8x Multiply: 9(5x 2 4x) a. 45x 2 + 5x b. 45x 2 4x c. 14x 2 5x d. 45x 2 36x 80. Multiply: 4(7c 2 5c 3) a. 28c 2 5c 3 c. 28c c + 12 b. 3c 2 9c 7 d. 28c 2 20c Write the multiplication sentence modelled by this set of algebra tiles. a. 2z(4z 2 + 8z) = 2z + 4 c. z(2z + 4) = 2z 2 + 4z b. 2z(2z + 4) = 4z 2 + 8z d. 2(2z 2 + 4) = 4z

12 82. Write the multiplication sentence modelled by this rectangle. a. 2x(4x) + 5 = 8x c. 2x(4x + 5) = 8x x b. 2x(4x + 5) = 6x + 5 d. 4x(2x + 5) = 8x x 83. Multiply: 5w(7w) a. 35w 2 b. 12w 2 c. 35w 2 d. 2w Multiply: 6c(4c 5) a. 2c b. 24c c c. 24c 2 30c d. 24c How many terms are in the polynomial below? 4a 4 a. 4 b. 3 c. 2 d Is the polynomial below a monomial, binomial, or trinomial? 2p a. Monomial c. Trinomial b. Binomial d. None of the above Short Answer 87. Write an expression for the width of this rectangle. 88. Factor: s 2 33s Expand and simplify: (11t + 2)(4t 3) 90. Factor: 22n 2 + n 5 12

13 91. Factor: 14z 2 49z Expand and simplify: (9z 2 2z + 10)(3z + 12) 93. Expand and simplify: (7x 2y)(3x + 7y 9) 94. Expand and simplify: ( x 4) Expand and simplify: ( 2x + 1) Expand and simplify: 3( a + 2) Find and correct the errors in this solution. (11a + b)(2a 13b + 4) = 13a 2 143ab + 44a 2ab 13b 2 + 4b = 13a 2 145ab 13b 2 44a + 4b 98. Factor: 36a ab + 121b Factor: 49s 2 64t Factor fully: 21p 2 r 165pqr 24q 2 r 101. Find an integer to replace so that the trinomial is a perfect square. 121x 2 308xy + y 2 Problem 102. Multiply this pair of binomials. Sketch and label a rectangle to illustrate the product. ( x + 9) ( x 4) 103. Factor. Check by expanding. n 2 + n Factor. Check by expanding. 8z 2 112z

14 105. Find the area of the rectangle Write a polynomial to represent the area of this rectangle. Simplify the polynomial A student says that the expression 10r r 2 93r 90 represents the volume of this right rectangular prism. Is the student correct? How do you know? 108. Factor. Explain your steps. 196x 2 16y 2 14

15 109. A picture and its frame have dimensions as shown. a) Find an expression for the area of the frame, in factored form. b) Determine the area of the frame when s = 15 cm Identify the equivalent polynomials. Justify your answer. i) 1 + 2x x x ii) 6x 2 + 6x 3 3x + 7 4x 2 iii) 3x x x The diagram below shows one rectangle inside another. a) Determine the area of each rectangle. b) Determine the area of the shaded region. 15

16 Math 10 - Unit 5 Final Review - Polynomials Answer Section MULTIPLE CHOICE 1. C 2. D 3. B 4. C 5. D 6. B 7. A 8. B 9. C 10. A 11. A 12. C 13. C 14. D 15. B 16. D 17. D 18. B 19. C 20. B 21. B 22. A 23. A 24. D 25. B 26. B 27. A 28. B 29. C 30. B 31. A 32. D 33. A 34. B 35. C 36. D 37. C 38. C 39. C 1

17 40. C 41. A 42. C 43. C 44. C 45. D 46. A 47. A 48. C 49. D 50. B 51. B 52. A 53. D 54. D 55. D 56. C 57. A 58. A 59. C 60. B 61. A 62. C 63. A 64. A 65. A 66. B 67. B 68. B 69. B 70. C 71. A 72. D 73. C 74. B 75. D 76. D 77. D 78. A 79. D 80. C 81. B 82. C 83. A 84. B 2

18 85. C 86. A SHORT ANSWER 87. a + 6b 88. ( s 32) ( s 1) t 2 25t ( 11n 5) ( 2n + 1) 91. 7( 2z 5) ( z 1) z z 2 + 6z x xy 14y 2 63x + 18y 94. x 3 12x x x x 2 + 6x a 3 18a 2 36x (11a + b)(2a 13b + 4) = 22a 2 143ab + 44a + 2ab 13b 2 + 4b = 22a 2 141ab + 44a 13b 2 + 4b 98. ( 6a + 11b) ( 7s + 8t) ( 7s 8t) rÊ Ë Á7p + qˆ Ê Ë Á p 8qˆ PROBLEM 102. ( x + 9) ( x 4) = x 2 + ( 4x) + 9x + ( 36) = (x 2 + 5x 36) 3

19 103. Two numbers with a sum of 1 and a product of 42 are 7 and 6. So, n 2 + n 42 = (n + 7)(n 6) Check that the factors are correct. Multiply the factors. (n + 7)(n 6) = n 2 6n + 7n 42 = n 2 + n 42 This trinomial is the same as the original trinomial, so the factors are correct z 2 112z The greatest common factor is 8. 8z 2 112z = 8(z 2 14z + 45) Two numbers with a sum of 14 and a product of 45 are 5 and 9. So, z 2 14z + 45 = (z 5)(z 9) And, 8z 2 112z = 8(z 5)(z 9) Check that the factors are correct. Multiply the factors. 8(z 5)(z 9) = 8(z 2 14z + 45) = 8z 2 112z The trinomial is the same as the original trinomial, so the factors are correct Use the formula for the area, A, of a rectangle. A = l w A = ( 5b 6) ( 3b 2) Use the distributive property. A = 5b(3b 2) + ( 6)(3b 2) A = 15b 2 10b 18b + 12 A = 15b 2 28b + 12 The area of the rectangle is 15b 2 28b + 12 square units. 4

20 106. Use the formula for the area, A, of a rectangle: A = lw A = (4x 5y)(6x + 3y) A = 4x(6x) + 4x(3y) 5y(6x) 5y(3y) A = 24x xy 30xy 15y 2 A = 24x 2 18xy 15y 2 The expression 24x 2 18xy 15y 2 represents the area of this rectangle Use the formula for the volume, V, of a right rectangular prism: V = lwh x 2 16y 2 V = (5r 6)(2r + 3)(r + 5) V = (10r r 12r 18)(r + 5) V = (10r 2 + 3r 18)(r + 5) V = 10r 2 (r) + 10r 2 (5) + 3r(r) + 3r(5) 18(r) 18(5) V = 10r r 2 + 3r 90 Since this expression does not match the student s expression, the student is incorrect. The expression 10r r 2 + 3r 90 represents the volume of the right rectangular prism. As written, each term of the binomial is not a perfect square. But the terms have a common factor 4. Remove this common factor. 196x 2 16y 2 = 4(49x 2 4y 2 ) Write each term in the binomial as a perfect square. È 4(49x 2 4y 2 ) = 4 (7x) 2 (2y) 2 ÎÍ = 4(7x 2y)(7x + 2y) Write these terms in binomial factors. 5

21 109. a) The area, A, of the larger square is: ( 8s) 2 = 64s 2 The area, A, of the smaller square is: (8s 4 4) 2 = ( 8s 8) 2 The area, A, of the frame is: A = 64s 2 (64s 2 128s + 64) A = 64s 2 64s s 64 A = 128s 64 A = 64(2s 1) = 64s 2 128s + 64 b) When s = 15 cm, the area, A square centimetres, of the frame is: A = 64(2s 1) È A = 64 ÎÍ 2( 15) 1 A = 1856 The area of the frame is 1856 cm Simplify each polynomial. i) 1 + 2x x x = 2x x 11x = 2x 2 + 3x + 4 ii) 6x 2 + 6x 3 3x + 7 4x 2 = 6x 2 4x 2 + 6x 3x = 2x 2 + 3x + 4 iii) 3x x x 2 = 8x 2 6x 2 + 3x = 2x 2 + 3x + 7 Polynomials i and ii are equivalent because they both simplify to the same polynomial: 2x 2 + 3x a) The outer rectangle has dimensions 9y and 6y. Its area is: (9y)(6y) = 54y 2 The inner rectangle has dimensions 5y and 3y. Its area is: (5y)(3y) = 15y 2 b) The total area of the shaded region is the difference in the areas of the rectangles: 54y 2 15y 2 = 39y 2 6