examview - unit5review polynomials - digital...
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Name: ________________________ Class: ___________________ Date: __________ ID: A
1
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 + 99a2.
a. a(44 + 99a) c. 11a(4 + 9a)
b. 11(4a + 9a2) d. 22a(2 + 9a)
____ 2. Factor the binomial 15y2
− 48y.
a. 3(5y2
− 16y) c. y(15y − 48)
b. 3y(5y − 16y) d. 3y(5y − 16)
____ 3. Factor the trinomial 4 − 8n + 12n2.
a. 4(−2n + 3n2) c. 2(2 − 4n + 6n
2)
b. 4(1 − 2n + 3n2) d. 4(1 + 2n + 3n
2)
____ 4. Factor the trinomial −33b2
+ 99b + 77.
a. −11(3b2
− 9b + 7) c. −11(3b2
− 9b − 7)
b. −33(b2
− 3b − 7) d. 33(−b2
+ 27b + 7)
____ 5. Factor the trinomial −24c3d − 40c
2d
2− 32cd
3.
a. −8cd(3c2
− 5cd − 4d2) c. 8cd(−3c
2+ 5cd + 4d
2)
b. 8cd(3c2
+ 5cd + 4d2) d. −8cd(3c
2+ 5cd + 4d
2)
____ 6. Factor the trinomial −42x5y
6− 24x
4y
5− 54x
3y
7.
a. 6x4y
5(−7xy − 4 − 9y
2) c. −3x
3y
5(14x
2y + 8x + 18y
2)
b. −6x3y
5(7x
2y + 4x + 9y
2) d. −6x
3(7x
2y
6+ 4xy
5+ 9y
7)
____ 7. Factor the binomial −10m2
− 40m4.
a. −10m2(1 + 4m
2) c. −10(m
2+ 4m
4)
b. −10m2(4m
2) d. −5m
2(2 + 8m
2)
____ 8. Simplify the expression y2
+ 8y − 6 − 9y2
− 24y − 26, then factor.
a. −8(y2
− 2y − 4) c. −4(2y2
+ 4y + 8)
b. −8(y2
+ 2y + 4) d. −4(2y2
+ 4y + 1)
Name: ________________________ ID: A
2
____ 9. Which expression represents the area of the shaded region?
a. 2r(2r − π) b. r2(1 − π) c. r
2(4 − π) d. r(r − 2π)
____ 10. Expand and simplify: (p + 3)(p − 7)
a. p2
− 4p − 21 c. p2
+ 10p − 21
b. p2
− 10p − 21 d. p2
+ 4p − 21
____ 11. Expand and simplify: (4 − r)(7 − r)
a. 28 − 11r + r2
c. 28 + 3r + r2
b. 28 − 3r + r2
d. 28 + 11r + r2
____ 12. Factor: t2
+ 9t − 36
a. (t − 2)(t + 18) c. (t + 12)(t − 3)
b. (t + 2)(t − 18) d. (t − 12)(t + 3)
____ 13. Factor: v2
− 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 + x2
a. (6 + x)(−4 + x) c. (−3 + x)(8 + x)
b. (3 + x)(−8 + x) d. (−6 + x)(4 + x)
____ 15. Factor: −84 + 8z + z2
a. (42 + z)(−2 + z) c. (−42 + z)(2 + z)
b. (−6 + z)(14 + z) d. (6 + z)(−14 + z)
____ 16. Factor: −3b2
+ 15b + 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: −4d2
− 28d + 240
a. −4(d + 3)(d − 20) c. −4(d − 3)(d + 20)
b. −4(d + 5)(d − 12) d. −4(d − 5)(d + 12)
Name: ________________________ ID: A
3
____ 18. Complete: (a + 6)(a − ) = a2
+ a − 12
a. (a + 6)(a − 4) = a2
+ 4a − 12 c. (a + 6)(a − 2) = a2
+ 2a − 12
b. (a + 6)(a − 2) = a2
+ 4a − 12 d. (a + 6)(a − 4) = a2
+ 2a − 12
____ 19. Factor: c2
− 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 − g2
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. h2
− 5h − 66 c. h2
+ 17h − 66
b. h2
+ 5h − 66 d. h2
− 17h − 66
____ 22. Factor: −5m2
+ 20m + 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: 7n2
− 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. (2x2
+ 2x)(2x2
+ 2x)
b. (2x2
+ 2)(2x2
+ 2) d. (2x + 2)(2x + 2)
Name: ________________________ ID: A
4
____ 25. Which set of algebra tiles represents 3x2
+ x + 4?
a. c.
b. d.
____ 26. Expand and simplify: (6p + 3)(5p − 6)
a. 30p2
+ 21p − 18 c. 30p2
+ 51p − 18
b. 30p2
− 21p − 18 d. 30p2
− 51p − 18
____ 27. Expand and simplify: (8g − 3)(7 − 3g)
a. −24g2
+ 65g − 21 c. −24g2
+ 47g − 21
b. −24g2
− 65g − 21 d. 24g2
+ 65g − 21
____ 28. Factor: 25x2
+ 58x + 16
a. (25x + 4)(x + 4) c. (5x + 4)(5x + 4)
b. (25x + 8)(x + 2) d. (5x + 8)(5x + 2)
____ 29. Factor: 16s2
− 137s − 63
a. (4s − 7)(4s + 9) c. (16s + 7)(s − 9)
b. (4s + 7)(4s − 9) d. (16s − 7)(s + 9)
____ 30. Factor: 48y2
− 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: 24b2
+ 50b − 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. −24t2
+ 42t + 27 c. −72t2
− 126t + 81
b. −24t2
+ 66t + 27 d. −24t2
− 42t + 27
____ 33. Factor: 7n2
+ 104n − 15
a. (7n − 1)(n + 15) c. (7n + 15)(n − 1)
b. (7n + 1)(n − 15) d. (7n − 15)(n + 1)
Name: ________________________ ID: A
5
____ 34. Factor: 4 − 9z − 13z2
a. (2 − 13z)(2 + z) c. (2 + 13z)(2 − z)
b. (4 − 13z)(1 + z) d. (4 + 13z)(1 − z)
____ 35. Factor: 180 − 175a + 30a2
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: 96w2
+ 324w − 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)(7h2
− 4h + 1)
a. 56h3
− 53h2
− 20h + 3 c. 56h3
− 11h2
− 4h + 3
b. 56h3
+ 11h2
− 12h + 3 d. 56h3
− 32h2
+ 8h + 3
____ 38. Expand and simplify: (5m − 3n)2
a. 25m2
− 9n2
c. 25m2
− 30mn + 9n2
b. 25m2
− 15mn + 9n2
d. 25m2
+ 9n2
____ 39. Expand and simplify: (7m − 2n)2
a. 49m2
− 4n2
c. 49m2
− 28mn + 4n2
b. 49m2
− 14mn + 4n2
d. 49m2
+ 4n2
____ 40. Expand and simplify: (7m − 3n)2
a. 49m2
− 9n2
c. 49m2
− 42mn + 9n2
b. 49m2
− 21mn + 9n2
d. 49m2
+ 9n2
____ 41. Expand and simplify: (4s + 9t)(5s − 4t − 3)
a. 20s2
+ 29st − 12s − 36t2
− 27t c. 20s2
+ 29st + 12s − 36t2
+ 27t
b. 20s2
+ 29st − 12s + 36t2
− 27t d. 20s2
+ 61st − 12s − 36t2
− 27t
____ 42. Expand and simplify: (10v − 13w)(10v + 13w)
a. 100v2
+ 260vw + 169w2
c. 100v2
− 169w2
b. 100v2
+ 169w2
d. 100v2
− 260vw + 169w2
____ 43. Expand and simplify: (4d − 1)(5d2
+ 12d − 3)
a. 20d3
+ 53d2
+ 3 c. 20d3
+ 43d2
− 24d + 3
b. 20d3
+ 48d2
− 12d + 3 d. 20d3
+ 43d2
+ 3
____ 44. Expand and simplify: (f + 5g)(2f − 5g + 7)
a. 2f2
+ 5fg + 7f + 25g2
+ 35g c. 2f2
+ 5fg + 7f − 25g2
+ 35g
b. 2f2
− 15fg + 7f − 25g2
+ 35g d. 2f2
− 5fg + 7f − 25g2
+ 35g
Name: ________________________ ID: A
6
____ 45. Which polynomial, written in simplified form, represents the area of this rectangle?
a. 8x2
− 36xy − 20y2
c. 16x2
+ 72xy − 40y2
b. 8x2
+ 22xy − 20y2
d. 8x2
+ 36xy − 20y2
____ 46. Expand and simplify: (2x2
+ 5x − 6)(5x2
− 2x + 3)
a. 10x4
+ 21x3
− 34x2
+ 27x − 18 c. 10x4
+ 21x3
− 24x2
+ 27x + 18
b. 10x4
+ 21x3
− 34x2
− 3x + 18 d. 10x4
− 29x3
− 34x2
+ 27x − 18
____ 47. Expand and simplify: (n2
− 2n + 3)(−2n2
+ 7n + 8)
a. −2n4
+ 11n3
− 12n2
+ 5n + 24 c. −2n4
− 3n3
+ 37n + 24
b. −2n4
+ 11n3
+ 37n + 24 d. −2n4
− 3n3
− 12n2
+ 5n + 24
____ 48. Expand and simplify: (6p + 3)(6p − 7) − (7p − 4)(p − 2)
a. 29p2
− 42p − 13 c. 29p2
− 6p − 29
b. 29p2
− 6p − 13 d. 29p2
− 42p − 29
____ 49. Expand and simplify: (6x − y)(3x + 8y) − (2x − 3y)2
a. 14x2
+ 51xy − 17y2
c. 14x2
+ 57xy + 1y2
b. 14x2
+ 33xy + 1y2
d. 14x2
+ 57xy − 17y2
____ 50. Expand and simplify: (3c + 2)(2c − 7) + 3(−2c + 1)(7c − 5)
a. −36c2
+ 8c − 29 c. −36c2
− 8c − 19
b. −36c2
+ 34c − 29 d. −36c2
− 8c − 29
____ 51. Expand and simplify: (4a − b − 2)(3a − 7) − (3a + 4b)2
a. 3a2
− 34a − 3ab + 7b + 14 − 16b2
c. 3a2
− 22a − 27ab + 7b + 14 − 16b2
b. 3a2
− 34a − 27ab + 7b + 14 − 16b2
d. 3a2
− 34a + 21ab + 7b + 14 + 16b2
Name: ________________________ ID: A
7
____ 52. Each shape is a rectangle. Write a polynomial, in simplified form, to represent the area of the shaded region.
a. 5x2
+ 31x + 66 c. 5x2
+ 31x + 30
b. 5x2
+ 37x + 30 d. 5x2
+ 37x + 66
____ 53. Factor: 121a2
+ 110a + 25
a. (11a + 5)(11a − 5) c. (11a − 5)2
b. (121a + 5)(a + 5) d. (11a + 5)2
____ 54. Factor: 36 − 60r + 25r2
a. (9 − 5r)(4a − 5r) c. (6 + 5r)2
b. (6 − 5r)(6 + 5r) d. (6 − 5r)2
____ 55. Factor: 16p2
− 81q2
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.
x2
+ 42xy + 9y2
a. 7 c. 49
b. 14 d. 196
____ 57. Find an integer to replace � so that this trinomial is a perfect square.
64v2
− vw + 81w2
a. 144 c. 72
b. 648 d. 18
____ 58. Factor: 49s2
− 112st + 64t2
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.
9a2
+ 9a + 36a. Difference of squares c. Neither
b. Perfect square trinomial
Name: ________________________ ID: A
8
____ 60. Identify this polynomial as a perfect square trinomial, a difference of squares, or neither.
25g2
− 9h2
a. Perfect square trinomial c. Neither
b. Difference of squares
____ 61. Factor: 9c2
− 12c + 4
a. (3c − 2)2
c. (6c − 4)2
b. (3c − 2)(3c + 2) d. (6c − 4)(6c + 4)
____ 62. Factor: 8y2
− 58yz + 60z2
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: 3z4
− 768z2
a. 3z2(z + 16)(z − 16) c. z
2(z + 48)(z − 16)
b. 3z2(z + 16)
2d. 3z
2(z − 16)
2
____ 64. Factor: 48b2
+ 70bc − 3c2
a. (2b + 3c)(24b − c) c. (16b + c)(3b − 3c)
b. (48b − 3c)(b − c) d. (2b − 3c)(24b + c)
____ 65. Factor: 8m2
− 34mn + 33n2
a. (4m − 11n)(2m − 3n) c. (4m + 11n)(2m + 3n)
b. (8m − 33n)(m − n) d. (4m − 11n)(2m + 3n)
____ 66. Factor: 162 − 2w4
a. (9 − w2)(18 − w
2) c. 2(9 − w
2)
2
b. 2(9 + w2)(3 + w)(3 − w) d. 2(9 + w
2)
2
____ 67. 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)
Name: ________________________ ID: A
9
____ 68. From the list, which terms are like 7x?
7x2, 6x, 5, −8x, −7x, 8x
2, 7
a. 7x2, 6x, −8x, −7x, 8x
2c. −7x
b. 6x, −8x, −7x d. 7x2, −7x, 7
____ 69. These algebra tiles may be used in the following question.
x2 –x2 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 + 5x2
+ 4x + 2x2
a. 5x2
b. −2x2
+ 7x4
c. −2x + 7x2
d. 5x
____ 71. These algebra tiles may be used in the following question.
x2 –x2 x –x 1 –1
Write the polynomial sum modelled by this set of tiles.
a. (−x2
− 3x − 4) + (−x2
+ x + 4) c. (−2) + (−2) + 0
b. (x2
+ 3x + 4) + (x2
+ x + 4) d. 8 + 6
____ 72. Add.
(−5 − 6x2) + (8x
2+ 7)
a. 4x2
b. 2 + 2x4
c. 4 d. 2 + 2x2
Name: ________________________ ID: A
10
____ 73. Add.
(−5 + 5r + 5r2) + (−2 − 8r
2− 7r)
a. −7 − 2r2
− 3r4
c. −7 − 2r − 3r2
b. −2 − 3r − 2r2
d. −2r2
− 3r − 2
____ 74. These algebra tiles may be used in the following question.
x2 –x2 x –x 1 –1
Write the subtraction sentence that these algebra tiles represent.
a. (x2
+ 3x + 2) − (x2
+ x + 2) = 2x
b. (−x2
− 3x + 2) − (−x2
− x + 2) = −2x
c. (−x2
− x + 2) − (−x2
− 3x + 2) = −2x
d. (x2
+ x + 2) − (x2
+ 3x + 2) = 2x
____ 75. Subtract.
(−3n2
+ 8) − (5 − 7n2)
a. −10n2
+ 15 c. −10n2
+ 3
b. −8n2
+ 15 d. 4n2
+ 3
____ 76. Subtract.
(−4b − 6b2
− 2) − (8b + 8 − 7b2)
a. −12b − 13b2
− 10 c. b − 14b2
+ 5
b. −12b − 13b2
+ 6 d. −12b + b2
− 10
Name: ________________________ ID: A
11
____ 77. These algebra tiles may be used in the following question.
x2 –x2 x –x 1 –1
What product is modelled by this set of algebra tiles?
a. 3(x2
− 4x − 3) c. 3(x2
+ 4x + 3)
b. 3(−x2
− 4x + 3) d. 3(x2
− 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 + 9
____ 79. Multiply: 9(5x2
− 4x)
a. 45x2
+ 5x b. 45x2
− 4x c. 14x2
− 5x d. 45x2
− 36x
____ 80. Multiply: −4(7c2
− 5c − 3)
a. −28c2
− 5c − 3 c. −28c2
+ 20c + 12
b. 3c2
− 9c − 7 d. −28c2
− 20c − 12
____ 81. Write the multiplication sentence modelled by this set of algebra tiles.
a. 2z(4z2
+ 8z) = 2z + 4 c. z(2z + 4) = 2z2
+ 4z
b. 2z(2z + 4) = 4z2
+ 8z d. 2(2z2
+ 4) = 4z2
+ 8
Name: ________________________ ID: A
12
____ 82. Write the multiplication sentence modelled by this rectangle.
a. 2x(4x) + 5 = 8x2
+ 5 c. 2x(4x + 5) = 8x2
+ 10x
b. 2x(4x + 5) = 6x + 5 d. 4x(2x + 5) = 8x2
+ 20x
____ 83. Multiply: −5w(7w)
a. −35w2
b. −12w2
c. 35w2
d. 2w2
____ 84. Multiply: −6c(4c − 5)
a. −2c2
+ 11 b. −24c2
+ 30c c. −24c2
− 30c d. −24c2
− 5
____ 85. How many terms are in the polynomial below?
4a − 4
a. 4 b. 3 c. 2 d. 1
____ 86. 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: s2
− 33s + 32
89. Expand and simplify: (11t + 2)(4t − 3)
90. Factor: 22n2
+ n − 5
Name: ________________________ ID: A
13
91. Factor: 14z2
− 49z + 35
92. Expand and simplify: (9z2
− 2z + 10)(3z + 12)
93. Expand and simplify: (7x − 2y)(3x + 7y − 9)
94. Expand and simplify: x − 4( )3
95. Expand and simplify: 2x + 1( )3
96. Expand and simplify: −3 a + 2( )3
97. Find and correct the errors in this solution.
(11a + b)(2a − 13b + 4)
= 13a2
− 143ab + 44a − 2ab − 13b2
+ 4b
= 13a2
− 145ab − 13b2
− 44a + 4b
98. Factor: 36a2
+ 132ab + 121b2
99. Factor: 49s2
− 64t2
100. Factor fully: 21p2r − 165pqr − 24q
2r
101. Find an integer to replace � so that the trinomial is a perfect square.
121x2
− 308xy + y2
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.
n2
+ n − 42
104. Factor. Check by expanding.
8z2
− 112z + 360
Name: ________________________ ID: A
14
105. Find the area of the rectangle.
106. Write a polynomial to represent the area of this rectangle. Simplify the polynomial.
107. A student says that the expression 10r3
+ 35r2
− 93r − 90 represents the volume of this right rectangular
prism.
Is the student correct? How do you know?
108. Factor. Explain your steps.
196x2
− 16y2
Name: ________________________ ID: A
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.
110. Identify the equivalent polynomials.
Justify your answer.
i) −1 + 2x2
+ 14x + 5 − 11x
ii) 6x2
+ 6x − 3 − 3x + 7 − 4x2
iii) 3x + 1 + 8x2
+ 6 − 6x2
111. The diagram below shows one rectangle inside another.
a) Determine the area of each rectangle.
b) Determine the area of the shaded region.
ID: A
1
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
ID: A
2
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
ID: A
3
85. C
86. A
SHORT ANSWER
87. a + 6b
88. s − 32( ) s − 1( )
89. 44t2
− 25t − 6
90. 11n − 5( ) 2n + 1( )
91. 7 2z − 5( ) z − 1( )
92. 27z3
+ 102z2
+ 6z + 120
93. 21x2
+ 43xy − 14y2
− 63x + 18y
94. x3
− 12x2
+ 48x − 64
95. x3
+ 12x2
+ 6x + 1
96. −3a3
− 18a2
− 36x − 24
97. (11a + b)(2a − 13b + 4)
= 22a2
− 143ab + 44a + 2ab − 13b2
+ 4b
= 22a2
− 141ab + 44a − 13b2
+ 4b
98. 6a + 11b( )2
99. 7s + 8t( ) 7s − 8t( )
100. 3r 7p + qÊËÁÁ ˆ
¯˜̃ p − 8qÊËÁÁ ˆ
¯˜̃
101. 196
PROBLEM
102.
x + 9( ) x − 4( ) = x2
+ (−4x) + 9x + (−36)
= (x2
+ 5x − 36)
ID: A
4
103. Two numbers with a sum of 1 and a product of −42 are 7 and −6.
So, n2
+ n − 42 = (n + 7)(n − 6)
Check that the factors are correct. Multiply the factors.
(n + 7)(n − 6) = n2
− 6n + 7n − 42
= n2
+ n − 42
This trinomial is the same as the original trinomial, so the factors are correct.
104. 8z2
− 112z + 360
The greatest common factor is 8.
8z2
− 112z + 360 = 8(z2
− 14z + 45)
Two numbers with a sum of −14 and a product of 45 are −5 and − 9.
So, z2
− 14z + 45 = (z − 5)(z − 9)
And, 8z2
− 112z + 360 = 8(z − 5)(z − 9)
Check that the factors are correct. Multiply the factors.
8(z − 5)(z − 9) = 8(z2
− 14z + 45)
= 8z2
− 112z + 360
The trinomial is the same as the original trinomial, so the factors are correct.
105. 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 = 15b2
− 10b − 18b + 12
A = 15b2
− 28b + 12
The area of the rectangle is 15b2
− 28b + 12 square units.
ID: A
5
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 = 24x2
+ 12xy − 30xy − 15y2
A = 24x2
− 18xy − 15y2
The expression 24x2
− 18xy − 15y2 represents the area of this rectangle.
107. Use the formula for the volume, V, of a right rectangular prism:
V = lwh
V = (5r − 6)(2r + 3)(r + 5)
V = (10r2
+ 15r − 12r − 18)(r + 5)
V = (10r2
+ 3r − 18)(r + 5)
V = 10r2(r) + 10r
2(5) + 3r(r) + 3r(5) − 18(r) − 18(5)
V = 10r3
+ 53r2
+ −3r − 90
Since this expression does not match the student’s expression, the student is incorrect.
The expression 10r3
+ 53r2
+ −3r − 90 represents the volume of the right rectangular prism.
108. 196x2
− 16y2
As written, each term of the binomial is not a perfect square. But the terms have a common factor 4. Remove
this common factor.
196x2
− 16y2
= 4(49x2
− 4y2)
Write each term in the binomial as a perfect square.
4(49x2
− 4y2) = 4 (7x)
2− (2y)
2È
ÎÍÍÍÍÍ
˘
˚˙̇˙̇̇ Write these terms in binomial factors.
= 4(7x − 2y)(7x + 2y)
ID: A
6
109. a) The area, A, of the larger square is: 8s( )2
= 64s2
The area, A, of the smaller square is: (8s − 4 − 4)2
= 8s − 8( )2
= 64s2
− 128s + 64
The area, A, of the frame is:
A = 64s2
− (64s2
− 128s + 64)
A = 64s2
− 64s2
+ 128s − 64
A = 128s − 64
A = 64(2s − 1)
b) When s = 15 cm, the area, A square centimetres, of the frame is:
A = 64(2s − 1)
A = 64 2 15( ) − 1ÈÎÍÍÍ
˘˚˙̇˙
A = 1856
The area of the frame is 1856 cm2.
110. Simplify each polynomial.
i) −1 + 2x2
+ 14x + 5 − 11x
= 2x2
+ 14x − 11x − 1 + 5
= 2x2
+ 3x + 4
ii) 6x2
+ 6x − 3 − 3x + 7 − 4x2
= 6x2
− 4x2
+ 6x − 3x − 3 + 7
= 2x2
+ 3x + 4
iii) 3x + 1 + 8x2
+ 6 − 6x2
= 8x2
− 6x2
+ 3x + 1 + 6
= 2x2
+ 3x + 7
Polynomials i and ii are equivalent because they both simplify to the same polynomial: 2x2
+ 3x + 4
111. a) The outer rectangle has dimensions 9y and 6y.
Its area is:
(9y)(6y) = 54y2
The inner rectangle has dimensions 5y and 3y.
Its area is:
(5y)(3y) = 15y2
b) The total area of the shaded region is the difference in the areas of the rectangles:
54y2
− 15y2
= 39y2