Exponents and Polynomials Solutions Key ARE YOU READY? 1. F 2. B 3. C 4. D 5. E 6. 4 7 7. 5 2 8. (-10) 4 9. x 3 10. k 5 11. 9 1 12. 3 4 = 3 · 3 · 3 · 3 = 81 13. - 12 2 = -(12 · 12) = -144 14. 5 3 = 5 · 5 · 5 = 125 15. 2 5 = 2 · 2 · 2 · 2 · 2 = 32 16. 4 3 = 4 · 4 · 4 = 64 17. (-1) 6 = (-1)(-1)(-1)(-1)(-1)(-1) = 1 18. 0.06 19. 2,525 20. 15.6 21. 6 + 3p + 14 + 9p = 6 + 14 + 3p + 9p = 20 + 12p 22. 8y - 4x + 2y + 7x - x = 8y + 2y - 4x + 7x - x = 10y + 2x 23. (12 + 3w - 5) + 6w - 3 - 5w = 12 - 5 - 3 + 3w + 6w - 5w = 4 + 4w 24. 6n - 14 + 5n = 6n + 5n - 14 = 11n - 14 25. no 26. yes; √ 81 = √ 9 2 = 9 27. yes; √ 36 = √ 6 2 = 6 28. no 29. yes: √ 100 = √ 10 2 = 10 30. yes; √ 4 = √ 2 2 = 2 31. yes; √ 1 = √ 1 2 = 1 32. no INTEGER EXPONENTS CHECK IT OUT! 1. 5 -3 = 1 __ 5 3 = 1 _______ 5 · 5 · 5 = 1 ____ 125 5 -3 m is equal to 1 ____ 125 m. 2a. 10 -4 = 1 ___ 10 4 = 1 ______________ 10 · 10 · 10 · 10 = 1 ______ 10,000 b. (-2) -4 = 1 _____ (-2) 4 = 1 _______________ (-2)(-2)(-2)(-2) = 1 ___ 16 c. (-2) -5 = 1 _____ (-2) 5 = 1 ___________________ (-2)(-2)(-2)(-2)(-2) = - 1 ___ 32 d. -2 -5 = - 1 __ 2 5 = - 1 ____________ 2 · 2 · 2 · 2 · 2 = - 1 ___ 32 3a. p -3 = 4 -3 = 1 __ 4 3 = 1 _______ 4 · 4 · 4 = 1 ___ 64 b. 8 a -2 b 0 = 8(-2) -2 (6) 0 = 8 · 1 _____ (-2) 2 · 1 = 8 · 1 ________ (-2)(-2) = 8 · 1 __ 4 = 2 4a. 2 r 0 m -3 = 2 · r 0 · m -3 = 2 · 1 · 1 ___ m 3 = 2 ___ m 3 b. r -3 ___ 7 = r -3 · 1 __ 7 = 1 __ r 3 · 1 __ 7 = 1 ___ 7 r 3 c. g 4 ____ h -6 = g 4 · 1 ____ h -6 = g 4 · h 6 = g 4 h 6 THINK AND DISCUSS 1. -2; 0; t 2. Simplifying Expressions with Negative Exponents For a negative exponent in the numerator, move the power to the denominator and change the negative exponent to a positive exponent; possible answer: . For a negative exponent in the denominator, move the power to the numerator and change the negative exponent to a positive exponent; possible answer: . 2 - 3 = 1 __ 2 3 4 ___ x - 5 = 4 x 5 EXERCISES GUIDED PRACTICE 1. 10 -7 = 1 ___ 10 7 = 1 _________________________ 10 · 10 · 10 · 10 · 10 · 10 · 10 = 1 __________ 10,000,000 m 10 -7 m is equal to 1 __________ 10,000,000 m. 2. 6 -2 = 1 __ 6 2 = 1 ____ 6 ·6 = 1 ___ 36 3. 3 0 = 1 4. - 5 -2 = - 1 __ 5 2 = - 1 ____ 5 · 5 = - 1 ___ 25 5. 3 -3 = 1 __ 3 3 = 1 _______ 3 · 3 · 3 = 1 ___ 27 6 CHAPTER 6-1 195 Holt McDougal Algebra 1
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CHAPTER Exponents and Polynomials 6 Solutions Key · 2015-03-06 · Exponents and Polynomials Solutions Key arE you rEady? 1. F 2. B 3. C 4. D 5. E 6. 4 7 7. 5 2 8.(-10) 4 9. x 3
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56. Degree: 3 Terms: 3 c 2 + 7 - 2 c 3 is a cubic trinomial.
57. Degree: 2 Terms: 1 - y 2 is a quadratic monomial.
58. 3.675v + 0.096 v 2 = 3.675(30) + 0.096 (30) 2 = 3.675(30) + 0.096(900) = 110.25 + 86.4 = 196.65 The stopping distance of a car traveling at 30 mi/h is
196.65 ft.
59. always 60. sometimes
61. never 62. sometimes
63a. 4 c 3 - 39 c 2 + 93.5c = 4 (1) 3 - 39 (1) 2 + 93.5(1) = 4(1) - 39(1) + 93.5(1) = 4 - 39 + 93.5 = 58.5 The volume of the box when c = 1 in. is 58.5 in 3 .
b. 4 c 3 - 39 c 2 + 93.5c = 4 (1.5) 3 - 39 (1.5) 2 + 93.5(1.5) = 4(3.375) - 39(2.25) + 93.5(1.5) = 13.5 - 87.75 + 140.25 = 66 The volume of the box when c = 1.5 in. is 66 in 3 .
c. 4 c 3 - 39 c 2 + 93.5c = 4 (4.25) 3 - 39 (4.25) 2 + 93.5(4.25) = 4(76.765) - 39(18.063) + 93.5(4.25) = 307.063 - 704.438 + 397.375 = 0 The volume of the box when c = 4.25 in. is 0 in 3 .
d. Yes; the width of the cardboard is 8.5 in., so 4.25 in. cuts will meet, leaving nothing to fold up.
Polynomial x = -2 x = 0 x = 5
64. 5x - 6 -16 -6 19
65. x 5 + x 3 + 4x -48 0 3270
66. -10 x 2 -40 0 -250
67. Possible answer: x 2 + 3x - 6
68. Possible answer: 5x - 2
69. Possible answer: 5 70. Possible answer: 6 x 3
71. Possible answer: x 5 - 3
72. Possible answer: 2 x 12 - x + 15
73. Possible answer: First identify the degree of each term. From left to right, the degrees are 3, 0, 2, 4, and 1. Arrange the terms in order of decreasing degree, and move the plus or minus sign in front of each term with it: -2 x 4 + 4 x 3 + 5 x 2 - x - 3.
74a. 12x: degree 1 6: degree 0 The degree of the polynomial is 1.
74b. 8 x 2 : degree 2 12x: degree 1 The degree of the polynomial is 2.
75. A is incorrect. The student incorrectly multiplied -3 by -2 before evaluating the power.
teSt prep
76. C; A has degree 8, B has degree 1, C has degree 10, and D has degree 2. So C has the greatest degree.
0.016 m 3 - 0.390 m 2 + 4.562m + 50.310 = 0.016 (5) 3 - 0.390 (5) 2 + 4.562(5) + 50.310 = 0.016(125) - 0.390(25) + 4.562(5) + 50.310 = 2 - 9.75 + 22.81 + 50.310 ≈ 65 The average length of a two-month-old baby boy is
58 cm and the average length of a five-month-old baby boy is 65 cm.
b. 0.016 m 3 - 0.390 m 2 + 4.562m + 50.310 = 0.016 (0) 3 - 0.390 (0) 2 + 4.562(0) + 50.310 = 0.016(0) - 0.390(0) + 4.562(0) + 50.310 = 0 - 0 + 0 + 50.310 = 50.310 The average length of a newborn baby boy is
50.310 cm.
c. The first three terms of the polynomial will equal 0, so just look at the constant.
80a. 4 x 5 + x
b. yes; 0 < x < 1; raising a number between 0 and 1 to a higher power results in a lesser number. So if x is between 0 and 1, the bionomial with the least degree will have the greatest value.
adding and subtracting PoLynomiaLs
CheCk it out!
1a. 2 s 2 + 3 s 2 + s = 5 s 2 + s
b. 4 z 4 - 8 + 16 z 4 + 2 = 4 z 4 + 16 z 4 - 8 + 2 = 20 z 4 - 6
44. Yes; the simplified form of both expressions is 15 m 2 + 2m - 10. No; the simplified form of the original expression is -9 m 2 - 12m + 10 and the simplified form of the new expression is -9 m 2 + 2m - 10.
45. B is incorrect. The student incorrectly tried to combine 6 n 3 and -3 n 2 , which are not like terms, and tried to combine 4 n 2 and 9n, which are not like terms.
Polynomial 1 Polynomial 2 Sum
46. x 2 - 6 3 x 2 - 10x + 2 4 x 2 - 10x - 4
47. 12x + 5 3x + 6 15x + 11
48. x 4 - 3 x 2 - 9 5 x 4 + 8 6 x 4 - 3 x 2 - 1
49. 7 x 3 - 6x - 3 6x + 14 7 x 3 + 11
50. 2 x 3 + 5 x 2 7 x 3 - 5 x 2 + 1 9 x 3 + 1
51. 2 x 2 + x - 5 x + x 2 + 6 3 x 2 + 2x + 1
52. No; polynomial addition simply involves combining like terms. No matter what order the terms are combined in, the sum will be the same. Yes; in polynomial subtraction, the subtraction sign is distributed among all terms in the second polynomial, changing all the signs to their opposites.
c. P = 4x + 2 = 4(15) + 2 = 60 + 2 = 62 He will need 62 ft of fencing.
teSt prep
54. C; Since -14 y 2 + 9 y 2 + 2 y 2 = -3 y 2 , and 3 - 2 = 1, the term must be in the form ay. So -12y + ay - 6y = -15y gives -12 + a - 6 = -15 or a = 3. So the missing term is 3y.
55. G; Since 2 t 3 - 4t - (-7t - 3t) = 2 t 3 + 6t ≠ -5 t 3 - t, G is correct.
= 3 x 3 - 6 x 2 + 15x + 2 x 2 - 4x + 10 = 3 x 3 - 4 x 2 + 11x + 10
5a. Let x represent the width of the rectangle. A = ℓw = (x - 4)(x) = x(x) - 4(x) = x 2 - 4x The area is represented by x 2 - 4x.
b. A = x 2 - 4x = (6) 2 - 4(6) = 36 - 24 = 12 The area is 12 m 2 .
think and disCuss
1. Possible answer: Both numbers and polynomials are set up in two rows and require you to multiply each item in the top row by an item in the bottom row. In the end, you add vertically to get the answer. When you are multiplying polynomials, the items are monomial terms. When your are multiplying numbers, the items are digits.
2.
x 2 + 2x + x + 2 = x 2 + 3x + 2
Vertical method: (x + 2)(x 2 + 3x + 2)
Rectangle model: (x + 2)(x 2 + 2x + 1)
2 x x
2 4x 2 + 2
+ 2x + 1
x3 + 4x 2 + 5x + 2
x 2 + 3 x + 2
−−−−−−−−−− × x + 2 2 x 2 + 6 x + 4
−−−−−−−−−−− + x 3 + 3 x 2 + 2 x x 3 + 5 x 2 + 8 x + 4
x 2
x 2 x 3
x 2
Multiplying Polynomials
Distributive Property: 5x (x + 2) = 5x 2 + 10x
FOIL method: (x + 1)(x + 2) =
exerCisesguided practice
1. (2 x 2 ) (7 x 4 )
= (2 · 7) ( x 2 · x 4 ) = 14 x 6
2. (-5m n 3 ) (4 m 2 n 2 )
= (-5 · 4) (m · m 2 ) ( n 3 · n 2 ) = -20 m 3 n 5
3. (6r s 2 ) ( s 3 t 2 ) ( 1 __ 2 r 4 t 3 )
= (6 · 1 __ 2 ) (r · r 4 ) ( s 2 · s 3 ) ( t 2 · t 3 )
= 3 r 5 s 5 t 5
4. ( 1 __ 3 a 5 ) (12a)
= ( 1 __ 3 · 12) ( a 5 · a)
= 4 a 6
5. (-3 x 4 y 2 ) (-7 x 3 y)
= (-3 · (-7)) ( x 4 · x 3 ) ( y 2 · y)
= 21 x 7 y 3
6. (-2p q 3 ) (5 p 2 q 2 ) (-3 q 4 )
= (-2 · 5 · (-3)) (p · p 2 ) ( q 3 · q 2 · q 4 )
= 30 p 3 q 9
7. 4 ( x 2 + 2x + 1)
= 4 ( x 2 ) + 4(2x) + 4(1) = 4 x 2 + 8x + 4
8. 3ab (2 a 2 + 3 b 3 )
= 3ab (2 a 2 ) + 3ab (3 b 3 )
= (3 · 2) (a · a 2 ) (b) + (3 · 3)(a) (b · b 3 ) = 6 a 3 b + 9a b 4
9. 2 a 3 b (3 a 2 b + a b 2 )
= 2 a 3 b (3 a 2 b) + 2 a 3 b (a b 2 )
= (2 · 3) ( a 3 · a 2 ) (b · b) + (2) ( a 3 · a) (b · b 2 ) = 6 a 5 b 2 + 2 a 4 b 3
10. -3x ( x 2 - 4x + 6)
= -3x ( x 2 ) - 3x(-4x) - 3x(6) = -3 x 3 + 12 x 2 - 18x
11. 5 x 2 y (2x y 3 - y)
= 5 x 2 y (2x y 3 ) + 5 x 2 y(-y)
= (5 · 2) ( x 2 · x) (y · y 3 ) + (5 · (-1)) ( x 2 ) (y · y)
= 10 x 3 y 4 - 5 x 2 y 2
12. 5 m 2 n 3 · m n 2 (4m - n)
= (5) ( m 2 · m) ( n 3 · n 2 ) (4m - n) = 5 m 3 n 5 (4m - n) = 5 m 3 n 5 (4m) + 5 m 3 n 5 (-n)
= (5 · 4) ( m 3 · m) ( n 5 ) + (5 · (-1)) ( m 3 ) ( n 5 · n)
= 20 m 4 n 5 - 5 m 3 n 6
13. (x + 1)(x - 2) = x(x) + x(-2) + 1(x) + 1(-2) = x 2 -2x + x - 2 = x 2 - x - 2
14. (x + 1) 2 = (x + 1)(x + 1) = x(x) + x(1) + 1(x) + 1(1) = x 2 + x + x + 1 = x 2 + 2x + 1
= 2p (4 p 2 ) + 2p(-12pq) + 2p (9 q 2 ) - 3q (4 p 2 )
- 3q(-12pq) - 3q (9 q 2 )
= 8 p 3 - 24 p 2 q + 18p q 2 - 12 p 2 q + 36p q 2 - 27 q 3 = 8 p 3 - 36 p 2 q + 54p q 2 - 27 q 3
82a. x
x
10
25
b. The length is 25 + x + x = 2x + 25. The width is 10 + x + x = 2x + 10.
c. A = ℓw = (2x + 25)(2x + 10) = 2x(2x) + 2x(10) + 25(2x) + 25(10) = 4 x 2 + 20x + 50x + 250 = 4 x 2 + 70x + 250
83. Possible answer: Each letter in FOIL represents a pair of terms in a certain position within the factors. The letters must account for every pairing of terms while describing first, outside, inside, and last positions. This is only possible with two binomials.
= ( x 2 + 5x) (x + 2) = x 2 (x) + x 2 (2) + 5x(x) + 5x(2) = x 3 + 2 x 2 + 5 x 2 + 10x = x 3 + 7 x 2 + 10x The area is represented by x 3 + 7 x 2 + 10x.
85. Yes; x = 0
86. Let x represent the width of the rectangle. A = ℓw = (x + 1)(x) = x(x) + 1(x) = x 2 + x Since (4.5) 2 + 4.5 = 20.25 + 4.5 ≈ 25, the width of
the rectangle is about 4.5 ft.
teSt prep
87. C (a + 1)(a - 6) = a(a) + a(-6) + 1(a) + 1(-6) = a 2 - 6a + a - 6 = a 2 - 5a - 6
A = P 4 x 2 + 32x + 64 = 4 x 2 + 192 _______________ -4 x 2 ___________ -4 x 2 32x + 64 = 192 ________ - 64 ____ -64 32x = 128
32x ____ 32
= 128 ____ 32
x = 4
95. x(x + 1)(x + 2) = (x(x) + x(1)) (x + 2)
= ( x 2 + x) (x + 2) = x 2 (x) + x 2 (2) + x(x) + x(2) = x 3 + 2 x 2 + x 2 + 2x = x 3 + 3 x 2 + 2x
96. x m ( x n + x n - 2 ) = x 5 + x 3
x m ( x n ) + x m ( x n - 2 ) = x 5 + x 3 x m + n + x m + n - 2 = x 5 + x 3 Therefore, it must be true that: m + n = 5 → m + n = 5 m + n - 2 = 3 → m + n = 5 Therefore, the system is consistent and dependent,
so there is an infinite number of solutions. One is m = 2; n = 3.
97. 2 x a (5 x 2a - 3 + 2 x 2a + 2 ) = 10 x 3 + 4 x 8
2 x a (5 x 2a - 3 ) + 2 x a (2 x 2a + 2 ) = 10 x 3 + 4 x 8 10 x 3a - 3 + 4 x 3a + 2 = 10 x 3 + 4 x 8 Therefore, it must be true that: 3a - 3 = 3 and 3a + 2 = 8 ______ + 3 ___ +3 ______ - 2 ___ -2 3a = 6 and 3a = 6 3a = 6
3a ___ 3 = 6 __
3
a = 2
sPEciaL Products of binomiaLs
CheCk it out!
1a. (a + b) 2 = a 2 + 2ab + b 2 (x + 6) 2 = (x) 2 + 2(x)(6) + (6) 2 = x 2 + 12x + 36
b. (a + b) 2 = a 2 + 2ab + b 2 (5a + b) 2 = (5a) 2 + 2(5a)(b) + (b) 2 = 25 a 2 + 10ab + b 2
c. (a + b) 2 = a 2 + 2ab + b 2
(1 + c 3 ) 2 = (1) 2 + 2(1) ( c 3 ) + ( c 3 )
2
= 1 + 2 c 3 + c 6
2a. (a - b) 2 = a 2 - 2ab + b 2 (x - 7) 2 = (x) 2 - 2(x)(7) + (7) 2 = x 2 - 14x + 49
b. (a - b) 2 = a 2 - 2ab + b 2 (3b - 2c) 2 = (3b) 2 - 2(3b)(2c) + (2c) 2 = 9 b 2 - 12bc + 4 c 2
c. (a - b) 2 = a 2 - 2ab + b 2
( a 2 - 4) 2 = ( a 2 )
2 - 2 ( a 2 ) (4) + (4) 2
= a 4 - 8 a 2 + 16
3a. (a + b)(a - b) = a 2 - b 2 (x + 8)(x - 8) = (x) 2 - (8) 2 = x 2 - 64
c. (a + b)(a - b) = a 2 - b 2 (9 + r)(9 - r) = (9) 2 - (r) 2 = 81 - r 2
4. Area of : (5 + x)(5 - x) = (5) 2 - (x) 2 = 25 - x 2 Area of □: x 2 Total area = area of + area of □
= (25 - x 2 ) + x 2
= 25 + (- x 2 + x 2 ) = 25 The area of the pool is 25.
think and disCuss
1. (a + b)(a - b) = a 2 - ab + ab - b 2 = a 2 - b 2
2. product
3. Special Products of Binomials
Perfect-Square Trinomials Difference of Two Squares
( a + b ) 2 = a 2 + 2 ab + b 2
( x + 4) 2 = x 2 + 8 x + 16 ( a - b ) 2 = a 2 - 2 ab + b 2 ( x - 4) 2 = x 2 - 8 x + 16
( a + b)( a - b) = a 2 b 2 - ( x + 4)( x - 4) = x 2 - 16
exerCisesguided practice
1. Possible answer: a trinomial that is the result of squaring a binomial.
2. (a + b) 2 = a 2 + 2ab + b 2 (x + 7) 2 = (x) 2 + 2(x)(7) + (7) 2 = x 2 + 14x + 49
3. (a + b) 2 = a 2 + 2ab + b 2 (2 + x) 2 = (2) 2 + 2(2)(x) + (x) 2 = 4 + 4x + x 2
4. (a + b) 2 = a 2 + 2ab + b 2 (x + 1) 2 = (x) 2 + 2(x)(1) + (1) 2 = x 2 + 2x + 1
5. (a + b) 2 = a 2 + 2ab + b 2 (2x + 6) 2 = (2x) 2 + 2(2x)(6) + (6) 2 = 4 x 2 + 24x + 36
6. (a + b) 2 = a 2 + 2ab + b 2 (5x + 9) 2 = (5x) 2 + 2(5x)(9) + (9) 2 = 25 x 2 + 90x + 81
7. (a + b) 2 = a 2 + 2ab + b 2
(2a + 7b) 2 = (2a) 2 + 2(2a)(7b) + (7b) 2
= 4 a 2 + 28ab + 49 b 2
8. (a - b) 2 = a 2 - 2ab + b 2 (x - 6) 2 = (x) 2 - 2(x)(6) + (6) 2 = x 2 - 12x + 36
9. (a - b) 2 = a 2 - 2ab + b 2 (x - 2) 2 = (x) 2 - 2(x)(2) + (2) 2 = x 2 - 4x + 4
10. (a - b) 2 = a 2 - 2ab + b 2 (2x - 1) 2 = (2x) 2 - 2(2x)(1) + (1) 2 = 4 x 2 - 4x + 1
11. (a - b) 2 = a 2 - 2ab + b 2 (8 - x) 2 = (8) 2 - 2(8)(x) + (x) 2 = 64 - 16x + x 2
12. (a - b) 2 = a 2 - 2ab + b 2 (6p - q) 2 = (6p) 2 - 2(6p)(q) + (q) 2 = 36 p 2 - 12pq + q 2
13. (a - b) 2 = a 2 - 2ab + b 2 (7a - 2b) 2 = (7a) 2 - 2(7a)(2b) + (2b) 2 = 49 a 2 - 28ab + 4 b 2
14. (a + b)(a - b) = a 2 - b 2 (x + 5)(x - 5) = (x) 2 - (5) 2 = x 2 - 25
15. (a + b)(a - b) = a 2 - b 2 (x + 6)(x - 6) = (x) 2 - (6) 2 = x 2 - 36
16. (a + b)(a - b) = a 2 - b 2 (5x + 1)(5x - 1) = (5x) 2 - (1) 2 = 25 x 2 - 1
17. (a + b)(a - b) = a 2 - b 2
(2 x 2 + 3) (2 x 2 - 3) = (2 x 2 ) 2 - (3) 2
= 4 x 4 - 9
18. (a - b)(a + b) = a 2 - b 2
(9 - x 3 ) (9 + x 3 ) = (9) 2 - ( x 3 ) 2
= 81 - x 6
19. (a - b)(a + b) = a 2 - b 2 (2x - 5y)(2x + 5y) = (2x) 2 - (5y) 2 = 4 x 2 - 25 y 2
20. Area of big □: (x + 3) 2 = (x) 2 + 2(x)(3) + (3) 2 = x 2 + 6x + 9 Area of small □: (x + 1) 2 = (x) 2 + 2(x)(1) + (1) 2 = x 2 + 2x + 1 Total area = area of big □ + area of small □
= ( x 2 + 6x + 9) + ( x 2 + 2x + 1)
= ( x 2 + x 2 ) + (6x + 2x) + (9 + 1) = 2 x 2 + 8x + 10 The area of the figure is 2 x 2 + 8x + 10.
practice and problem Solving
21. (a + b) 2 = a 2 + 2ab + b 2 (x + 3) 2 = (x) 2 + 2(x)(3) + (3) 2 = x 2 + 6x + 9
22. (a + b) 2 = a 2 + 2ab + b 2 (4 + z) 2 = (4) 2 + 2(4)(z) + (z) 2 = 16 + 8z + z 2
23. (a + b) 2 = a 2 + 2ab + b 2
( x 2 + y 2 ) 2 = ( x 2 )
2 + 2 ( x 2 ) ( y 2 ) + ( y 2 )
2
= x 4 + 2 x 2 y 2 + y 4
24. (a + b) 2 = a 2 + 2ab + b 2
(p + 2 q 3 ) 2 = (p) 2 + 2(p) (2 q 3 ) + (2 q 3 )
2
= p 2 + 4p q 3 + 4 q 6
25. (a + b) 2 = a 2 + 2ab + b 2 (2 + 3x) 2 = (2) 2 + 2(2)(3x) + (3x) 2 = 4 + 12x + 9 x 2
26. (a + b) 2 = a 2 + 2ab + b 2
( r 2 + 5t) 2 = ( r 2 )
2 + 2 ( r 2 ) (5t) + (5t) 2
= r 4 + 10 r 2 t + 25 t 2
27. (a - b) 2 = a 2 - 2ab + b 2
( s 2 - 7) 2 = ( s 2 )
2 - 2 ( s 2 ) (7) + (7) 2
= s 4 - 14 s 2 + 49
28. (a - b) 2 = a 2 - 2ab + b 2
(2c - d 3 ) 2 = (2c) 2 - 2(2c) ( d 3 ) + ( d 3 )
2
= 4 c 2 - 4c d 3 + d 6
29. (a - b) 2 = a 2 - 2ab + b 2 (a - 8) 2 = (a) 2 - 2(a)(8) + (8) 2 = a 2 - 16a + 64
30. (a - b) 2 = a 2 - 2ab + b 2 (5 - w) 2 = (5) 2 - 2(5)(w) + (w) 2 = 25 - 10w + w 2
31. (a - b) 2 = a 2 - 2ab + b 2 (3x - 4) 2 = (3x) 2 - 2(3x)(4) + (4) 2 = 9 x 2 - 24x + 16
32. (a - b) 2 = a 2 - 2ab + b 2
(1 - x 2 ) 2 = (1) 2 - 2(1) ( x 2 ) + ( x 2 )
2
= 1 - 2 x 2 + x 4
33. (a - b)(a + b) = a 2 - b 2 (a - 10)(a + 10) = (a) 2 - (10) 2 = a 2 - 100
34. (a + b)(a - b) = a 2 - b 2 (y + 4)(y - 4) = (y) 2 - (4) 2 = y 2 - 16
35. (a + b)(a - b) = a 2 - b 2 (7x + 3)(7x - 3) = (7x) 2 - (3) 2 = 49 x 2 - 9
36. (a - b)(a + b) = a 2 - b 2
( x 2 -2) ( x 2 + 2) = ( x 2 ) 2 - (2) 2
= x 4 - 4
37. (a + b)(a - b) = a 2 - b 2
(5 a 2 + 9) (5 a 2 - 9) = (5 a 2 ) 2 - (9) 2
= 25 a 4 - 81
38. (a + b)(a - b) = a 2 - b 2
( x 3 + y 2 ) ( x 2 - y 2 ) = ( x 3 ) 2 - ( y 2 )
2
= x 6 - y 4
39. A = π r 2 = π (x + 4) 2
= π ( (x) 2 + 2(x)(4) + (4) 2 )
= π ( x 2 + 8x + 16)
= π ( x 2 ) + π(8x) + π(16) = π x 2 + 8πx + 16π The area of the puzzle is π x 2 + 8πx + 16π.
40a. x > 2; values less than or equal to 2 cause the width of the rectangle to be zero or negative, which does not make sense.
b. Area of □: (x - 1) 2 = (x) 2 - 2(x)(1) + (1) 2 = x 2 - 2x + 1 Area of : x(x - 2) = x(x) + x(-2) = x 2 - 2x Since x 2 - 2x + 1 > x 2 - 2x, the square has the greater area.
c. Difference = area of □ - area of
= ( x 2 - 2x + 1) - ( x 2 - 2x)
= ( x 2 - 2x + 1) + (- x 2 + 2x)
= ( x 2 - x 2 ) + (-2x + 2x) + 1 = 1 The difference in area is 1 square unit.
41. (a + b) 2 = a 2 + 2ab + b 2 (x + y) 2 = (x) 2 + 2(x)(y) + (y) 2 = x 2 + 2xy + y 2
42. (a - b) 2 = a 2 - 2ab + b 2 (x - y) 2 = (x) 2 - 2(x)(y) + (y) 2 = x 2 - 2xy + y 2
43. (a + b)(a - b) = a 2 - b 2
( x 2 + 4) ( x 2 - 4) = ( x 2 ) 2 - (4) 2
= x 4 - 16
44. (a + b) 2 = a 2 + 2ab + b 2
( x 2 + 4) 2 = ( x 2 )
2 + 2 ( x 2 ) (4) + (4) 2
= x 4 + 8 x 2 + 16
45. (a - b) 2 = a 2 - 2ab + b 2
( x 2 - 4) 2 = ( x 2 )
2 - 2 ( x 2 ) (4) + (4) 2
= x 4 - 8 x 2 + 16
46. (a - b) 2 = a 2 - 2ab + b 2 (1 - x) 2 = (1) 2 - 2(1)(x) + (x) 2 = 1 - 2x + x 2
A = x 2 - 9 = (12) 2 - 9 = 144 - 9 = 135 The area of the region is 135 ft 2 .
65. For a x 2 - 49 to be a perfect square, a x 2 needs to be a perfect square. Therefore, a must be a perfect square. So all the possible values of a are all the perfect squares from 1 to 100; 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
66. When one binomial is in the form a + b and the other is in the form a - b; (x + 2)(x - 2) = x 2 - 4.
teSt prep
67. B (a - b)(a - b) = a 2 - 2ab + b 2 (5x - 6y)(5x - 6y) = (5x) 2 - 2(5x)(6y) + (6y) 2 = 25 x 2 - 60xy + 36 y 2
68. J; The 25 x 2 region means ±5x is squared. The 4 region means ±2 is squared. The two 10x regions mean that the product of ±5x and ±2 is positive, so the terms have the same sign. Therefore, it must be J.
69. D; If a = 10, then b = 2 from the first equation. Notice that (10) 2 - (2) 2 = 100 - 4 = 96, so a = 10, b = 2 is a solution to both equations. Therefore, a = 10.
70. H; Notice that (r + s) 2 = r 2 + 2rs + s 2 = 64. Since rs = 15, r 2 + 2(15) + s 2 = 64, or r 2 + s 2 = 34.