CHAPTER Exponents and Polynomials 6 Solutions Keymoodle.bisd303.org/.../Ch06solutions_key.pdf · Exponents and Polynomials Solutions Key ARE YOU READY? 1. F 2. B 3. C 4. D 5. E 6.

Exponents and Polynomials

Solutions KeyARE YOU READY?

1. F 2. B

3. C 4. D

5. E 6. 4 7

7. 5 2

8. (-10)4

9. x3

10. k5

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. 8a-2

b0

= 8(-2)-2

(6) 0

= 8 · 1_____(-2)

2 · 1

= 8 · 1________(-2)(-2)

= 8 · 1__4

= 2

4a. 2r0m

-3 = 2 · r0 · m

-3

= 2 · 1 · 1___m

3

= 2___m

3

b. r-3___7 = r

-3 · 1__7

= 1__r3 · 1__

7

= 1___7r

3

c. g

4____h

-6 = g

4 · 1____h

-6

= g4 · h

6

= g4h

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 ___ - 5

= 4 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

6CHAPTER

6-1

195 Holt McDougal Algebra 1

6. 1 -8 = 1__

1 8 = 1____________________

1 · 1 · 1 · 1 · 1 · 1 · 1 · 1 = 1

7. - 8 -3 = - 1__

8 3 = - 1_______

8 · 8 · 8 = - 1____512

8. 10 -2 = 1___

10 2 = 1______

10 · 10 = 1____

100

9. (4.2) 0 = 1

10. (-3) -3 = 1_____

(-3) 3 = 1____________

(-3)(-3)(-3) = - 1___

27

11. 4 -2 = 1__

4 2 = 1____

4 · 4 = 1___16

12. b-2

= (-3) -2

= 1_____(-3)

2

= 1________(-3)(-3)

= 1__9

13. (2t)-4

= (2(2))-4

= 4 -4

= 1__4

4

= 1__________4 · 4 · 4 · 4

= 1____256

14. (m - 4)-5

= (6 - 4)-5

= 2 -5

= 1__2

5

= 1____________2 · 2 · 2 · 2 · 2

= 1___32

15. 2x0y

-3

= 2(7)0(-4)

-3

= 2 · 1 · 1_____(-4)

3

= 2 · 1____________(-4)(-4)(-4)

= 2 · 1____-64

= - 1___32

16. 4m0 = 4 · m

0

= 4 · 1= 4

17. 3k-4 = 3 · k

-4

= 3 · 1__k

4

= 3__k

4

18. 7___r

-7 = 7 · 1___

r-7

= 7 · r7

= 7r7

19. x10____

d-3

= x10 · 1____

d-3

= x10 · d

3

= x10

d3

20. 2x0y

-4 = 2 · x0 · y

-4

= 2 · 1 · 1__y

4

= 2__y

4

21. f-4____

g-6

= f-4 · 1____

g-6

= 1__f4 · g

6

= g

6__f4

22. c

4____d

-3 = c

4 · 1____d

-3

= c4 · d

3

= c4d

3

23. p7q

-1 = p7 · q

-1

= p7 · 1__

q

= p

7__q

PRACTICE AND PROBLEM SOLVING

24. 2 -1 = 1__

2 1 = 1__

2

2 -1

oz is equal to 1__2 oz.

25. 8 0 = 1

26. 5 -4 = 1__

5 4 = 1__________

5 · 5 · 5 · 5 = 1____625

27. 3 -4 = 1__

3 4 = 1__________

3 · 3 · 3 · 3 = 1___81

28. - 9 -2 = - 1__

9 2 = - 1____

9 · 9 = - 1___81

29. - 6 -2 = - 1__

6 2 = - 1____

6 · 6 = - 1___36

30. 7 -2 = 1__

7 2 = 1____

7 · 7 = 1___49

31. (2__5)

0 = 1

32. 13 -2 = 1___

13 2 = 1______

13 · 13 = 1____

169

33. (-3)-1 = 1_____

(-3)1 = 1____

(-3) = - 1__

3

34. (-4)2 = (-4)(-4) = 16

35. (1__2)

-2 = 1____(1_

2)2 = 1____

1_2

· 1_2

= 1__1_4

= 4

36. - 7 -1 = - 1__

7 1 = - 1__

737. x

-4

= 4 -4

= 1__4

4

= 1__________4 · 4 · 4 · 4

= 1____256

38. (2__3

v)-3

= (2__3 (9))

-3

= 6 -3

= 1__6

3

= 1_______6 · 6 · 6

= 1____216

39. (10 - d) 0

= (10 - 11) 0

= (-1) 0

= 1

40. 10m-1

n-5

= 10(10)-1

(-2)-5

= 10 · 1___10

1 · 1_____

(-2)5

= 10 · 1___10

· 1___________________(-2)(-2)(-2)(-2)(-2)

= - 1___32


41. (3ab) -2

= (3(1__2)(8))

-2

= 12 -2

= 1___12

2

= 1______12 · 12

= 1____144

42. 4wv x

v

= 4(3)0(-5)

0

= 4 · 1 · 1= 4

43. k-4 = 1__

k4

44. 2z-8 = 2 · z

-8

= 2 · 1__z

8

= 2__z

8

45. 1_____2b

-3 = 1__

2 · 1____

b-3

= 1__2 · b

3

= b

3__2

46. c-2

d = c-2 · d

= 1__c

2 · d

= d__c

2

47. -5x-3 = -5 · x

-3

= -5 · 1__x

3

= - 5__x

3

48. 4x-6

y-2 = 4 · x

-6 · y-2

= 4 · 1__x

6 · 1__

y2

= 4____x

6y

2

49. 2f0_____

7g-10

= 2__7 · f

0 · 1____g

-10

= 2__7 · 1 · g

10

= 2g

10____

7

50. r-5___

s-1

= r-5 · 1___

s-1

= 1__r5 · s

= s__r5

51. s

5____t

-12 = s

5 · 1____t

-12

= s5 · t

12

= s5t12

52. 3w

-5_____x

-6 = 3 · w

-5 · 1___x

-6

= 3 · 1___w

5 · x

6

= 3x

6___w

5

53. b0c

0 = b0 · c

0

= 1 · 1= 1

54. 2__3 m

-1n

5 = 2__3 · m

-1 · n5

= 2__3 · 1__

m· n

5

= 2n5___

3m

55. q

-2r0

_____s

0 = q

-2 · r0 · 1__

s0

= 1__q

2 · 1 · 1__

1

= 1__q

2

56. a

-7b

2_____c

3d

-4 = a

-7 · b2 · 1__

c3 · 1____

d-4

= 1__a

7 · b

2 · 1__c

3 · d

4

= b

2d

4____a

7c

3

57. h3k

-1_____6m

2 = 1__

6 · h

3 · k-1 · 1___

m2

= 1__6 · h

3 · 1__k

· 1___m

2

= h3_____

6m2k

58. z-5

= 2 -5

= 1__2

5

= 1____________2 · 2 · 2 · 2 · 2

= 1___32

59. (x + y)-4

= (3 + (-1))-4

= 2 -4

= 1__2

4

= 1__________2 · 2 · 2 · 2

= 1___16

60. (yz)0

= ((-1)(2))0

= (-2)0

= 1

61. (xyz)-1

= ((3)(-1)(2))-1

= (-6)-1

= 1____(-6)

= - 1__6

62. (xy - 3)-2

= ((3)(-1) - 3)-2

= (-6)-2

= 1_____(-6)

2

= 1________(-6)(-6)

= 1___36

63. x-y

= 3 -(-1)

= 3 1

= 3

64. (yz)-x

= ((-1)(2))-3

= (-2)-3

= 1_____(-2)

3

= 1____________(-2)(-2)(-2)

= - 1__8

65. xy-4

= (3)(-1)-4

= 3 · 1_____(-1)

4

= 3· 1_______________(-1)(-1)(-1)(-1)

= 3 · 1= 3

66. Equation A is incorrect because 5 was incorrectly moved to the denominator. The negative exponent applies only to the base x.

67. a3b

-2 = a3 · b

-2

= a3 · 1__

b2

= a

3__b

2

68. c-4

d3 = c

-4 · d3

= 1__c

4 · d

3

= d

3__c

4

69. v0w

2y

-1 = v0 · w

2 · y-1

= 1 · w2 · 1__

y

= w2___

y

70. (a2b

-7)0 = 1


71. -5y-6 = -5 · y

-6

= -5 · 1__y

6

= - 5__y

6

72. 2a

-5_____b

-6 = 2 · a

-5 · 1____b

-6

= 2 · 1__a

5 · b

6

= 2b

6___a

5

73. 2a

3____b

-1 = 2 · a

3 · 1____b

-1

= 2 · a3 · b

= 2a3b

74. m2____

n-3

= m2 · 1____

n-3

= m2 · n

3

= m2n

3

75. x-8____

3y12

= 1__3 · x

-8 · 1___y

12

= 1__3 · 1__

x8 · 1___

y12

= 1______3x

8y

12

76. - 20p-1

______5q

-3 = - 20___

5 · p

-1 · 1____q

-3

= -4 · 1__p

· q3

= - 4q3

___p

77. Red blood cell: 125,000 -1 = 1_______

125,000

White blood cell: 3(500)-2 = 3 · 500

-2

= 3 · 1____500

2

= 3 · 1________500 · 500

= 3 · 1_______250,000

= 3_______

250,000

Platelet: 3(1000)-2 = 3 · 1000

-2

= 3 · 1_____1000

2

= 3 · 1__________1000 · 1000

= 3 · 1_________1,000,000

= 3_________

1,000,000

78. always 79. never

80. sometimes 81. sometimes

82. never 83. sometimes

84. 2 3 · 2

-3

= 2 3 · 1__

2 3

= (2 · 2 · 2) · 1_______2 · 2 · 2

= 8 · 1__8

= 1

3 2 · 3

-2

= 3 2 · 1__

3 2

= (3 · 3) · 1____3 · 3

= 9 · 1__9

= 1

an · a

-n = 1

85. Possible answer: Look at the pattern below. As the exponent goes down by 1, the value is half of what it was before.

2 3 = 8, 2

2 = 4, 2 1 = 2, 2

0 = 1, 2 -1 = 1__

2 , 2

-2 = 1__4 ,

2 -3 = 1__

8 = 1__

2 3

86. 1__4 = 1____

2 · 2 = 1__2

2 = 2

-2; -2

87. 9 -2 = 1__

9 2 = 1____

9 · 9 = 1___81

; 81

88. 1___64

= 1____8 · 8 = 1__

8 2 = 8

-2; 8

89. 3 -1 = 1__

3 1 = 1__

3 ; 1

90. 7 -2 = 1__

7 2 = 1____

7 · 7 = 1___49

; 49

91. 1_____1000

= 1__________10 · 10 · 10

= 1___10

3 = 10

-3; -3

92. 3 · 4 -2 = 3 · 1__

4 2 = 3 · 1____

4 · 4 = 3 · 1___16

= 3___16

; 16

93. 2 · 1__5

= 2 · 5 -1

; -1 94a. fw = v

b. fw = v

fw__f

= v__f

w = v__f

w = v · 1__f

w = v · f-1

w = vf-1

c. 1__s = s

-1

TEST PREP

95. D; Since 0.04 = 1___25

= 1____5 · 5 = 1__

5 2 = 5

-2, A, B, and

C are all equal and do not equal -25.

96. J 6

-2 = 1__6

2 = 1____

6 · 6

97. A

a

3b

-2_____c

-1 = a

3 · b-2 · 1___

c-1

= a3 · 1__

b2 · c

= a

3c___

b2

98. 5__4 , or 1.25

2 -2 + (6 + 2)

0

= 2 -2 + 8

0

= 1__2

2 + 1

= 1____2 · 2 + 1

= 1__4 + 4__

4

= 5__4 , or 1.25

99. 1__a

n; a

-n = 1__a

n and b

0 = 1 if b ≠ 0. So you have

1__a

n· 1, or simply 1__

an.


CHALLENGE AND EXTEND

100. x -4 -3 -2 -1 0 1 2 3 4

y = 2 x 1___

161__8

1__4

1__2

1 2 4 8 16

8

4

12

16

0 4 2 -4 -2

Possible answer: y increases more rapidly as xincreases.

101. n -1 -2 -3 -4 -5

1 n

1 1 1 1 1

(-1) n -1 1 -1 1 -1

1 n = 1; (-1)

n = -1 if n is odd, and (-1)n = 1 if n is

even.

RATIONAL EXPONENTS

CHECK IT OUT!

1a. 81 1__4 =

4√ %% 81 = 3 b. 121 1__2 + 256

1__4

= √ %% 121 + 4√ %% 256

= 11 + 4 = 15

2a. 16 3__4 = 16

1__4

· 3

= (16 1__4)

3

= ( 4√ %% 16 ) 3

= 2 3

= 8

b. 1 2__5 = 1

1__5 · 2

= (1 1__5)

2

= ( 5√ % 1 ) 2

= 1

c. 27 4__3 = 27

1__3

· 4

= (27 1__3)

4

= ( 3√ %% 27 ) 4

= 3 4

= 81

3. C = 72m

3__4

= 72(81) 3__4

= 72 · ( 4√ %% 81 ) 3

= 72 · ( 4√ % 3 4 )

3

= 72 · (3) 3

= 72 · 27 = 1944 The panda needs1944 Calories per day.

4a. 4√ %%% x

4y

12

= (x4y

12) 1__4

= (x4) 1__4(y12)

1__4

= (x4 · 1__4) · (y12 · 1__

4)= (x1) · (y3) = x

1y

3

b. (xy

1__2)

2

______5√ % x

5

= (xy

1__2)

2

______x

= (x2) · (y1__2 · 2) · (x-1)

= (x2) · y · (x-1)

= (x2) · (x-1) · y

= x2 + (-1) · y = xy

THINK AND DISCUSS

1. Rewrite the expression as 25 to the power 1__10

, all

raised to the power 5. Then simplify the exponent

to 1_2. Finally take the square root.

2.

_

Fractional

Exponent Definition

1

b

_ n

b

m _ n

A number raised to the power of is equal to the th root of that number.

A number raised to the power of is equal to the th root of that number raised to the th power.

Numerical

Example

= =

6" √ 36 36

1 _

2

1 _

_ = =

6 = 216" √ 36 36

3 3 2 3 ( )

EXERCISES

GUIDED PRACTICE

1. 5

2. 8 1__3 =

3√ % 8 = 2 3. 16 1__2 = √ %% 16 = 4

4. 0 1__6 =

6√ % 0 = 0 5. 27 1__3 =

3√ %% 27 = 3

6. 81 1__2 = √ %% 81 = 9 7. 216

1__3 =

3√ %% 216 = 6

8. 1 1__9 =

9√ % 1 = 1 9. 625 1__4 =

4√ %% 625 = 5

10. 36 1__2 + 1

1__3

= √ %% 36 + 3√ % 1

= 6 + 1 = 7

11. 8 1__3 + 64

1__2

= 3√ % 8 + √ %% 64

= 2 + 8 = 10

12. 81 1__4 + 8

1__3

= 4√ %% 81 +

3√ % 8 = 3 + 2 = 5

13. 25 1__2 - 1

1__4

= √ %% 25 - 4√ % 1

= 5 - 1 = 4

14. 81 3__4 = (81

1__4)

3

= ( 4√ %% 81 ) 3

= 3 3

= 27

15. 8 5__3 = (8

1__3)

5

= ( 3√ % 8 ) 5

= 2 5

= 32

16. 125 2__3 = (125

1__3)

2

= ( 3√ %% 125 ) 2

= 5 2

= 25

17. 25 3__2 = (25

1__2)

3

= (√ %% 25 ) 3

= 5 3

= 125

6-2


18. 36 3__2 = (36

1__2)

3

= (√ %% 36 ) 3

= 6 3

= 216

19. 64 4__3 = (64

1__3)

4

= ( 3√ %% 64 ) 4

= 4 4

= 256

20. 1 3__4 =

4√ % 1 3

= 4√ % 1 = 1

21. 0 2__3 =

3√ % 0 2

= 4√ % 0 = 0

22. P = 4a

1__2

= 4(64) 1__2

= 4(√ %% 64 )= 4(8) = 32

The perimeter is 32 m.

23. √ %% x4y

2

= (x4y

2) 1__2

= (x4 · 1__2) · (y2 · 1__

2)= x

2 · y1 = x

2y

24. 4√ % z

4

= (z4) 1__4

= z4 · 1__

4

= z1 = z

25. √ %% x6y

6

= (x6y

6) 1__2

= (x6 · 1__2) · (y6 · 1__

2)= x

3 · y3 = x

3y

3

26. 3√ %%% a

12b

6

= (a12b

6) 1__3

= (a12 · 1__3) · (b6 · 1__

3)= a

4 · b2 = a

4b

2

27. (a1__2)

2

√ % a2

= (a1__2 · 2) · (a2)

1__2

= (a1) · (a2 · 1__2)

= a1 · a

1

= a1 + 1 = a

2

28. (x1__3)

64√ % y

4

= (x1__3 · 6) · (y4)

1__4

= (x2) · (y4 · 1__4)

= x2 · y

1 = x2y

29. (z

1__3)

3

_____√ % z

2

= z1__3 · 3

_____

(z2) 1__2

= z1_____

z2 · 1__

2

= z1__

z1 = 1

30.

3√ %% x6y

9 ______

x2

= (x6

y9)

1__3

_______x

2

= (x6 · 1__

3) · (y9 · 1__3)_____________

x2

= x

2 · y3

______x

2 = y

3


31. 100 1__2 = √ %% 100 = 10 32. 1

1__5 =

5√ % 1 = 1

33. 512 1__3 =

3√ %% 512 = 8 34. 729 1__2 = √ %% 729 = 27

35. 32 1__5 =

5√ %% 32 = 2 36. 196 1__2 = √ %% 196 = 14

37. 256 1__8 =

8√ %% 256 = 2 38. 400 1_2 = √ %% 400 = 20

39. 125 1__3 + 81

1__2 40. 25

1__2 - 81

1__4

= 3√ %% 125 + √ %% 81 = √ %% 25 - 4√ %% 81

= 5 + 9 = 14 = 5 - 3 = 2

41. 121 1__2 - 243

1__5

= √ %% 121 - 5√ %% 243

= 11 - 3 = 8

42. 256 1__4 + 0

1__3

= 4√ %% 256 +

3√ % 0 = 4 + 0 = 4

43. 4 3__2 = (√ % 4 ) 3

= 2 3 = 8

44. 27 2__3 = ( 3√ %% 27 ) 2

= 3 1 = 9

45. 256 3__4 = ( 4√ %% 256 ) 3

= 4 3 = 64

46. 64

5__6 = ( 6√ %% 64 ) 5

= 2 5 = 32

47. 100 3__2 = (√ %% 100 ) 3

= 10 3 = 1000

48. 1

5__3 = ( 3√ % 1 ) 5

= 1 5 = 1

49. 9 5__2 = (√ % 9 ) 5

= 3 5 = 243

50. 243 2__5 = ( 5√ %% 243 ) 2

= 3 2 = 9

51. B = 1__8 m

2__3

= 1__8 (64)

2__3

= 1__8 ( 3√ %% 64 ) 2

= 1__8 (4)

2

= 1__8 (16) = 2

The mass of the mouse’s brain is 2g.

52. 3√ %% a

6c

9

= (a6c

9) 1__3

= (a6 · 1__3) · (c9 · 1__

3)= a

2 · c3 = a

2c

3

53. 3√ %% 8m

3

= (8m3)

1__3

= (8 1__3) · (m3 · 1__

3)= ( 3√ % 8 ) · m

1 = 2m

54. 4√ %%% x

16y

4

= (x16y

4) 1__4

= (x16 · 1__4) · (y4 · 1__

4)= x

4 · y1 = x

4y

55. 3√ %% 27x

6

= (27x6)

1__3

= (27 1__3) · (x6 · 1__

3)= ( 3√ %% 27 ) · x

2 = 3x2

56. (x1__2y

3) 2

√ % x2

= (x1__2 · 2) · (y3 · 2) · x

= x1 · y

6 · x

= x1 + 1 · y

6

= x2 · y

6 = x2y

6

57. (a2b

4) 1__2

3√ % b6

= (a2 · 1__2) · (b4 · 1__

2) · (b6) 1__3

= (a1) · (b2) · (b6 · 1__3)

= a1 · b

2 · b2

= a1 · b

2 + 2

= a1 · b

4 = ab4


58.

3√ %% x6y

6 ______

yx2

= (x6

y6)

1__3

_______yx

2

= (x6 · 1__3) · (y6 · 1__

3) · y-1 · x

-2

= (x2) · (y2) · (y-1) · (x-2)

= x2 - 2 · y

2 - 1

= x0 · y

1 = y

59. (a2

b

1__2)

4

_______√ % b

2

= (a2 · 4) · (b

1__2 · 4)_____________

b

= (a8) · (b2) · (b-1)

= a8 · b

2 - 1

= a8 · b

1 = a8b

60. 256 x__4 = 4

( 4√ %% 256 ) x = 4

4 x = 4x = 1

61. x1__5 = 1

(x1__5)

5

= 1 5

x = 1

62. 225 1__x = 15

(225 1__x)

x

= 15 x

225 = 15 x

15 2 = 15

x

x = 2

63. x1__6 = 0

(x1__6)

6

= 0 6

x = 0

64. 64 x__3 = 16

( 3√ %% 64 ) x = 16

4 x = 16x = 2

65. x3__4 = 125

(x3__4)

4__3 = 125

4__3

x = ( 3√ %% 125 ) 4

x = 5 4

x = 625

66. 27 4__x = 81

(27 4__x)

x__4 = 81

x__4

27 = ( 4√ %% 81 ) x

27 = 3 x

x = 3

67. 36 x__2 = 216

(√ %% 36 ) x = 216

6 x = 216x = 3

68. ( 81____169)

1__2 = √ %% 81____

169

= √ %% 81 _____

√ %% 169

= 9___13

69. ( 8___27)

1__3 =

3√ %% 8___27

= 3√ % 8 ____

3√ %% 27

= 2__3

70. (256____81 )

1__4 =

4√ %% 256____81

= 4√ %% 256 _____4√ %% 81

= 4__3

71. ( 1___16)

1__2 = √ %% 1___

16

= √ % 1 ____

√ %% 16

= 1__4

72. ( 9___16)

3__2 = (√ %% 9___

16 )

3

= ( √ % 9 ____√ %% 16 )

3

= (3__4)

3

= 27___64

73. ( 8___27)

2__3 = ( 3√ %% 8___

27 )

2

= (3√ % 8 ____

3√ %% 27 ) 2

= (2__3)

2

= 4__9

74. (16___81)

3__4 = ( 4√ %% 16___

81 )

3

= (4√ %% 16 ____4√ %% 81 )

3

= (2__3)

3

= 8___

27

75. ( 4___49)

3_2 = (√ %% 4___

49 )

3

= ( √ % 4 _√ %% 49 )

3

= (2__7)

3

= 8____

343

76. ( 4___25)

3__2 = (√ %% 4___

25 )

3

= ( √ % 4 ____√ %% 25 )

3

= (2__5)

3

= 8____

125

77. ( 1___81)

3__4 = ( 4√ %% 1___

81 )

3

= (4√ % 1 ____

4√ %% 81 ) 3

= (1__3)

3

= 1___27

78. (27___64)

2__3 =

3√ %% 27___64

= (3√ %% 27 ____3√ %% 64 )

2

= (3__4)

2

= 9___

16

79. ( 8____125)

4__3 = ( 3√ %% 8____

125 )

4

= (3√ % 8 _____

3√ %% 125 ) 4

= (2__5)

4

= 16____625

80. Lion: Wolf:

L = 12m

1__5 L = 12m

1__5

= 12(243) 1__5 = 12(32)

1__5

= 12( 5√ %% 243 ) = 12( 5√ %% 32 )= 12(3) = 36 = 12(2) = 24

The lion’s lifespan is 36 - 24 = 12 years longerthan the wolf’s.

81. r = 0.62V

1__3

= 0.62(27) 1__3

= 0.62( 3√ %% 27 )= 0.62(3) = 1.86

The radius is 1.86 in.

82. (b1_3)

3

= b1_3

· 3 = b1 = b. Also, by definition ( 3√ % b )3 = b.

Therefore b1_3 =

3√ % b .

83. n2__3 will be less than n because 2__

3 < 1. n

3__2 will be

greater than n because 3__2

> 1.

84. A is incorrect; the first line should be 64 3_2 = (√ %% 64 ) 3.


85a. d = (0.8 L__B)

1__2

= (0.8(4000_____32 ))

1__2

= (0.8(125)) 1__2

= (100) 1__2

= √ %% 100 = 10 Distance to light source is 10 in.

b. d = (0.8 L__B)

1__2

= (0.8(4000_____8 ))

1__2

= (0.8(500)) 1__2

= (400) 1__2

= √ %% 400 = 20 Distance doubles to 20 in.

86. 43__2 = 4

3 · 1__2 = (4

3) 1__2 = 64

1__2 = 8

4 3__2 = 4

1__2 · 3 = (4

1__2)

3

= 2 3 = 8

It is often easier to take the square root first so that the remaining numbers in the calculation are smaller.

87. B;

9 1__2 + 8

1__3 = √ % 9 +

3√ % 8 = 3 + 2 = 5

88. F; 4 3__2 = (√ % 4 ) 3

= 2 3 = 8

89. C; 3√ %% a

9b

3

= (a9b

3) 1__3

= (a9 · 1__3) · (b3 · 1__

3)= a

3 · b1

= a3b

90. H; 3√ %% 16

2 = ( 3√ % 2

4 )

2

= (2 4__3)

2

= 2 4__3 · 2 = 2

8__3

which is not an integer


91. (a1__3)(a

1__3)(a

1__3) = a

(1__3 + 1__

3 + 1__

3) = a

1

= a

92. (x1__2)

5

(x3__2) = (x

5__2)(x

3__2)

= x(5__2

+ 3__2)

= x8__2

= x4

93. (x1__3)

4

(x5) 1__3 = (x

4__3)(x

5__3)

= x(4__3

+ 5__3)

= x9__3

= x3

94. y5 = 32

(y 5)

1__5 = 32

1__5

y 5 · 1__

5 = 5√ %% 32

y1 = 2

y = 2

95. 27x3 = 729

27x3____

27 = 729____

27

x3 = 27

(x3)1__3 = 27

1__3

x3 · 1__

3 = 3√ %% 27

x1 = 3

x = 3

96. 1 = 1__8

x3

(8)1 = (8) 1__8 x

3

8 = x3

8 1__3 = (x3)

1__3

3√ % 8 = x

3 · 1__3

2 = x1

2 = x

97. S = (4π ) 1__3(3V)

2__3

= (4π) 1__3(3(36π))

2__3

= (4π) 1__3(108π )

2__3

= 4 1__3 · π

1__3 · 108

2__3 · π

2__3

= 4 1__3 · 108

2__3 · π

1__3 + 2__

3

= (2 2)

1__3 · 108

2__3 · π

1

= 2 2__3 · 108

2__3 · π

= (2 · 108) 2__3 · π

= 216 2__3 · π

= ( 3√ %% 216 ) 2 · π

= 6 2 · π = 36π cm

2

Both volume and surface area are described by 36π

(although the units are different).

READY TO GO ON? Section A Quiz

1. t-6

= 2 -6

= 1__2

6

= 1_______________2 · 2 · 2 · 2 · 2 · 2

= 1___64

2. n-3

= (-5)-3

= 1_____(-5)

3

= 1____________(-5)(-5)(-5)

= 1_____-125

= - 1____125


3. r0s

-2

= 8 0 10

-2

= 1 · 1___10

2

= 1______10 · 10

= 1____100

4. 5k-3 = 5 · k

-3

= 5 · 1__k

3

= 5__k

3

5. x4___

y-6

= x4 · 1___

y-6

= x4 · y

6

= x4y

6

6. 8f-4

g0 = 8 · f

-4 · g0

= 8 · 1__f4 · 1

= 8__f4

7. a

-3____b

-2 = a

-3 · 1____b

-2

= 1__a

3 · b

2

= b

2__a

3

8. 10 -3 = 1___

10 3 = 1__________

10 · 10 · 10 = 1_____

1000 = 0.001

10 -2 = 1___

10 2 = 1______

10 · 10 = 0.01

10 -1 = 1___

10 1 = 1___

10 = 0.1

10 1 = 10

10 2 = 10 · 10 = 100

10 3 = 10 · 10 · 10 = 1000

9. 81 1__2= √ %% 81 = 9

10. 125 1__3 =

3√ %% 125 = 5

11. 4 3__2 = √ % 4

3 = √ %% 64 = 8

12. 0 2__9 = 0

13. √ %% x8y

4 = (x8

y4)

1__2

= (x8) 1__2(y4)

1__2

= (x8· 1__2)(y4· 1__

2)

= (x4)(y2) = x4y

2

14. 3√ % r

9 = (r9)

1__3

= r9· 1__

3 = r3

15. 6√ %% z

12 = (z12)

1__6

= z12· 1__

6 = z2

16. 3√ %%% p

3q

12 = (p3

q12)

1__3

= (p3) 1__3(q12)

1__3

= (p3· 1__3)(q12· 1__

3)

= (p1)(q4) = pq4

POLYNOMIALS

CHECK IT OUT!

1a. The degree is 3. b. The degree is 1.

c. The degree is 3.

2a. 5x: degree 1 -6: degree 0The degree of the polynomial is 1.

b. x3y

2: degree 5

- x4: degree 4

x2y

3: degree 5

2: degree 0

The degree of the polynomial is 5.

3a. 16 - 4x2 + x

5 + 9x3 → x

5 + 9x3 - 4x

2 + 16 The leading coefficient is 1.

b. 18y5 - 3y

8 + 14y → -3y8 + 18y

5 + 14y

The leading coefficient is -3.

4a. Degree: 3 Terms: 4

x3 + x

2 -x + 2 is a cubic polynomial.

b. Degree: 0 Terms: 16 is a constant monomial.

c. Degree: 8 Terms: 3

-3y8 + 18y

5 + 14y is an 8th-degree trinomial.

5. -16t2 + 400t + 6

= -16(5)2 + 400(5) + 6

= -16(25) + 400(5) + 6= -400 + 2000 + 6= 1606

When the firework explodes, it will be 1606 ft above the ground.

THINK AND DISCUSS

1. Possible answer: 2x2 + 3x

-3 contains an

expression with a negative exponent. 1 - a__b

contains a variable within a denominator.

2.Polynomials

2

Monomials

3 + 2

Binomials

22 + 6 - 7

Trinomials

EXERCISES

GUIDED PRACTICE

1. d 2. c

3. a 4. The degree is 0.

5. The degree is 3. 6. The degree is 8.

7. The degree is 0.

6-3


8. x2: degree 2 -2x: degree 1

1: degree 0The degree of the polynomial is 2.

9. 0.75a2b: degree 3 2a

3b

5: degree 8


10. 15y: degree 1 -84y3: degree 3

100: degree 0 -3y2: degree 2


11. r3: degree 3 r

2: degree 2

-5: degree 0 The degree of the polynomial is 3.

12. a3: degree 3 a

2: degree 2

-2a: degree 1 The degree of the polynomial is 3.

13. 3k4: degree 4 k

3: degree 3

-2k2: degree 2 k: degree 1


14. -2b + 5 + b2 → b

2 - 2b + 5 The leading coefficient is 1.

15. 9a8 - 8a

9 → -8a9 + 9a

8


16. 5s2 - 3s + 3 - s

7 → - s7 + 5s2 - 3s + 3


17. 2x + 3x2 - 1 → 3x

2 + 2x - 1The leading coefficient is 3.

18. 5g - 7 + g2 → g

2 + 5g - 7The leading coefficient is 1.

19. 3c2 + 5c

4 + 5c3 - 4 → 5c

4 + 5c3 + 3c

2 - 4 The leading coefficient is 5.

20. Degree: 2 Terms: 3

x2 + 2x + 3 is a quadratic trinomial.

21. Degree: 1 Terms: 2x - 7 is a linear binomial.


8 + k + 5k4 is a quartic trinomial.


q2 + 6 - q

3 + 3q4 is a quartic polynomial.


5k2 + 7k

3 is a cubic binomial.


2a3 + 4a

2 - a4 is a quartic trinomial.

26. 3.14r2 + 3.14rℓ

= 3.14(6)2 + 3.14(6)(10)

= 3.14(36) + 3.14(6)(10)= 113.04 + 188.4= 301.44

The surface area of the cone is approximately 301.44 cm

2.






35. a2: degree 2 a

4: degree 4

-6a: degree 1 The degree of the polynomial is 4.

36. 3 2b: degree 1 -5: degree 0


37. 3.5y2: degree 2 -4.1y: degree 1

-6: degree 0The degree of the polynomial is 2.

38. -5f4: degree 4 2f

6: degree 6

10f8: degree 8


39. 4n3: degree 3 -2n: degree 1


40. 4r3: degree 3 4r

6: degree 6


41. 2.5 + 4.9t3 - 4t

2 + t → 4.9t3 - 4t

2 + t + 2.5 The leading coefficient is 4.9.

42. 8a - 10a2 + 2 → -10a

2 + 8a + 2The leading coefficient is -10.

43. x7 - x + x

3 - x5 + x

10 → x10 + x

7 - x5 + x

3 - x

The leading coefficient is 1.

44. -m + 7 - 3m2 → -3m

2 - m + 7The leading coefficient is -3.

45. 3x2 + 5x - 4 + 5x

3 → 5x3 + 3x

2 + 5x - 4The leading coefficient is 5.

46. -2n + 1 - n2 → - n2 - 2n + 1


47. 4d + 3d2 - d

3 + 5 → - d3 + 3d2 + 4d + 5


48. 3s2 + 12s

3 + 6 → 12s3 + 3s

2 + 6The leading coefficient is 12.

49. 4x2 - x

5 - x3 + 1 → - x5 - x

3 + 4x2 + 1


50. Degree: 0 Terms: 112 is a constant monomial.

51. Degree: 1 Terms: 1 6k is a linear monomial.


3.5x3 - 4.1x - 6 is a cubic trinomial.


4g + 2g2 - 3 is a quadratic trinomial.


2x2 - 6x is a quadratic binomial.


6 - s3 - 3s

4 is a quartic trinomial.



c2 + 7 - 2c

3 is a cubic trinomial.


- y2 is a quadratic monomial.

58. 3.675v + 0.096v2

= 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. 4c3 - 39c

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. 4c3 - 39c

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. 4c3 - 39c

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. x5 + x

3 + 4x -48 0 3270

66. -10x2 -40 0 -250

67. Possible answer: x2 + 3x - 6

68. Possible answer: 5x - 2

69. Possible answer: 5 70. Possible answer: 6x3

71. Possible answer: x5 - 3

72. Possible answer: 2x12 - 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: -2x4 + 4x

3 + 5x2 - x - 3.

74a. 12x: degree 1 6: degree 0 The degree of the polynomial is 1.

74b. 8x2: degree 2 12x: degree 1


75. A is incorrect. The student incorrectly multiplied -3by -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.

77. J-3x

3 + 4x2 - 5x + 7

= -3(-1)3 + 4(-1)

2 - 5(-1) + 7= -3(-1) + 4(1) - 5(-1) + 7= 3 + 4 + 5 + 7= 19

78. Time (s) Height (ft)

1 59

2 86

3 81

4 44

The rocket will be the highest after 2 s.


79a. 0.016m3 - 0.390m

2 + 4.562m + 50.310 = 0.016(2)

3 - 0.390(2)2 + 4.562(2) + 50.310

= 0.016(8) - 0.390(4) + 4.562(2) + 50.310 = 0.128 - 1.56 + 9.124 + 50.310

≈ 58

0.016m3 - 0.390m

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. 0.016m3 - 0.390m

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. 4x5 + 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. 2s2 + 3s

2 + s

= 5s2 + s

b. 4z4 - 8 + 16z

4 + 2

= 4z4 + 16z

4 - 8 + 2

= 20z4 - 6

6-4


c. 2x8 + 7y

8 - x8 - y

8

= 2x8 - x

8 + 7y8 - y

8

= x8 + 6y

8

d. 9b3c

2 + 5b3c

2 - 13b3c

2

= b3c

2

2. (5a3 + 3a

2 - 6a + 12a2) + (7a

3 - 10a)

= (5a3 + 7a

3) + (3a2 + 12a

2) + (-6a - 10a)

= 12a3 + 15a

2 - 16a

3. (2x2 - 3x

2 + 1) - (x2 + x + 1)

= (2x2 - 3x

2 + 1) + (- x2 - x - 1)

= (2x2 - 3x

2 - x2) + (-x) + (1 - 1)

= -2x2 - x

4. (-0.03x2 + 25x - 1500)

____________________________________________+ (-0.02x2 + 21x - 1700)

-0.05x2 + 46x - 3200

THINK AND DISCUSS

1. -12x2 and -9x

2; -4.7y and y; 1__

5 x

2y and 5x

2y

2. Take the opposite of each term: -9t2 + 5t - 8.

3.

Adding: Subtracting:

Polynomials

(16 5 - 8 + 12) - (2 5 + - 1) =

(16 5 - 8 + 12) + ( - 2 5 - + 1) =

14 5 - 9 + 13

(18 2 + 9 2 + ) + (7 2 + 6 2 + 2 ) =

25 2 + 15 2 + 3

EXERCISES

GUIDED PRACTICE

1. 7a2 - 10a

2 + 9a

= -3a2 + 9a

2. 13x2 + 9y

2 - 6x2

= 13x2 - 6x

2 + 9y2

= 7x2 + 9y

2

3. 0.07r4 + 0.32r

3 + 0.19r4

= 0.07r4 + 0.19r

4 + 0.32r3

= 0.26r4 + 0.32r

3

4. 1__4 p

3 + 2__3 p

3

= 11___12

p3

5. 5b3c + b

3c - 3b

3c

= 3b3c

6. -8m + 5 - 16 + 11m

= -8m + 11m + 5 - 16= 3m - 11

7. (5n3 + 3n + 6) + (18n

3 + 9)

= (5n3 + 18n

3) + 3n + (6 + 9)

= 23n3 + 3n + 15

8. (3.7q2 - 8q + 3.7) + (4.3q

2 - 2.9q + 1.6)= (3.7q

2 + 4.3q2) + (-8q - 2.9q) + (3.7 + 1.6)

= 8q2 - 10.9q + 5.3

9. (-3x + 12) + (9x2 + 2x - 18)

= 9x2 + (-3x + 2x) + (12 - 18)

= 9x2 - x - 6

10. (9x4 + x

3) + (2x4 + 6x

3 - 8x4 + x

3)

= (9x4 + 2x

4 - 8x4) + (x3 + 6x

3 + x3)

= 3x4 + 8x

3

11. (6c4 + 8c + 6) - (2c

4)

= (6c4 + 8c + 6) + (-2c

4)

= (6c4 - 2c

4) + 8c + 6

= 4c4 + 8c + 6

12. (16y2 - 8y + 9) - (6y

2 - 2y + 7y)= (16y

2 - 8y + 9) + (-6y2 + 2y - 7y)

= (16y2 - 6y

2) + (-8y + 2y - 7y) + 9

= 10y2 - 13y + 9

13. (2r + 5) - (5r - 6)= (2r + 5) + (-5r + 6)= (2r - 5r) + (5 + 6)= -3r + 11

14. (-7k2 + 3) - (2k

2 + 5k - 1)

= (-7k2 + 3) + (-2k

2 - 5k + 1)

= (-7k2 - 2k

2) + (-5k) + (3 + 1)

= -9k2 - 5k + 4

15. m∠ABD = (8a2 - 2a + 5) + (7a + 4)

= 8a2 + (-2a + 7a) + (5 + 4)

= 8a2 + 5a + 9


16. 4k3 + 6k

2 + 9k3

= 4k3 + 9k

3 + 6k2

= 13k3 + 6k

2

17. 5m + 12n2 + 6n - 8m

= 5m - 8m + 12n2 + 6n

= 12n2 + 6n - 3m

18. 2.5a4 - 8.1b

4 - 3.6b4

= 2.5a4 - 11.7b

4

19. 2d5 + 1 - d

5

= 2d5 - d

5 + 1

= d5 + 1

20. 7xy - 4x2y - 2xy

= 7xy - 2xy - 4x2y

= -4x2y + 5xy

21. -6x3 + 5x + 2x

3 + 4x3

= -6x3 + 2x

3 + 4x3 + 5x

= 5x

22. x2 + x + 3x + 2x

2

= x2 + 2x

2 + x + 3x

= 3x2 + 4x

23. 3x3 - 4 - x

3 - 1

= 3x3 - x

3 - 4 - 1

= 2x3 - 5


24. 3b3 - 2b - 1 - b

3 - b

= 3b3 - b

3 - 2b - b - 1

= 2b3 - 3b - 1

25. (2t2 - 8t) + (8t

2 + 9t)

= (2t2 + 8t

2) + (-8t + 9t)

= 10t2 + t‹

26. (-7x2 - 2x + 3) + (4x

2 - 9x)

= (-7x2 + 4x

2) + (-2x - 9x) + 3

= -3x2 - 11x + 3

27. (x5 - x) + (x4 + x)

= (x5 + x4) + (-x + x)

= x5 + x

4

28. (-2z3 + z + 2z

3 + z) + (3z3 - 5z

2)

= (-2z3 + 2z

3 + 3z3) + (-5z

2) + (z + z)

= 3z3 - 5z

2 + 2z

29. (t3 + 8t2) - (3t

3)

= (t3 + 8t2) + (-3t

3)

= (t3 - 3t3) + 8t

2

= -2t3 + 8t

2

30. (3x2 - x) - (x2 + 3x - x)

= (3x2 - x) + (- x2 - 3x + x)

= (3x2 - x

2) + (-x - 3x + x)

= 2x2 - 3x

31. (5m + 3) - (6m3 - 2m

2)

= (5m + 3) + (-6m3 + 2m

2)= -6m

3 + 2m2 + 5m + 3

32. (3s2 + 4s) - (-10s

2 + 6s)

= (3s2 + 4s) + (10s

2 - 6s)

= (3s2 + 10s

2) + (4s - 6s)

= 13s2 - 2s

33. width = (6w2 + 8) - 2(w2 - 3w + 2)

= (6w2 + 8) + (-2(w2) - 2(-3w) - 2(2))

= (6w2 + 8) + (-2w

2 + 6w - 4)

= (6w2 - 2w

2) + 6w + (8 - 4)

= 4w2 + 6w + 4

34. P = 2ℓ + 2w

= 2(4a + 3b) + 2(7a - 2b)= 2(4a) + 2(3b) + 2(7a) + 2(-2b)= 8a + 6b + 14a - 4b

= 8a + 14a + 6b - 4b

= 22a + 2b

35. (2t - 7) + (-t + 2)= (2t - t) + (-7 + 2)= t - 5

36. (4m2 + 3m) + (-2m

2)

= (4m2 - 2m

2) + 3m

= 2m2 + 3m

37. (4n - 2) - 2n

= (4n - 2) + (-2n)= (4n - 2n) + (-2)= 2n - 2

38. (-v - 7) - (-2v)= (-v - 7) + (2v)= (-v + 2v) + (-7)= v - 7

39. (4x2 + 3x - 6) + (2x

2 - 4x + 5)

= (4x2 + 2x

2) + (3x - 4x) + (-6 + 5)

= 6x2 - x - 1

40. (2z2 - 3z - 3) + (2z

2 - 7z - 1)

= (2z2 + 2z

2) + (-3z - 7z) + (-3 - 1)

= 4z2 - 10z - 4

41. (5u2 + 3u + 7) - (u3 + 2u

2 + 1)

= (5u2 + 3u + 7) + (- u3 - 2u

2 - 1)

= (- u3) + (5u2 - 2u

2) + 3u + (7 - 1)

= - u3 + 3u2 + 3u + 6

42. (-7h2 - 4h + 7) - (7h

2 - 4h + 11)

= (-7h2 - 4h + 7) + (-7h

2 + 4h - 11)

= (-7h2 - 7h

2) + (-4h + 4h) + (7 - 11)

= -14h2 - 4

43. P = 2ℓ + 2w

35 = 2(2x + 3) + 2(3x + 7) 35 = 2(2x) + 2(3) + 2(3x) + 2(7) 35 = 4x + 6 + 6x + 14 35 = 4x + 6x + 6 + 14 35 = 10x + 20

____-20 _______- 20 15 = 10x

15___10

= 10x____10

3__2 = x, or x = 1.5

44. Yes; the simplified form of both expressions is

15m2 + 2m - 10. No; the simplified form of the

original expression is -9m2 - 12m + 10 and the

simplified form of the new expression is

-9m2 + 2m - 10.

45. B is incorrect. The student incorrectly tried to combine 6n

3 and -3n

2, which are not like terms,

and tried to combine 4n2 and 9n, which are not

like terms.

Polynomial 1 Polynomial 2 Sum

46. x2 - 6 3x

2 - 10x + 2 4x2 - 10x - 4

47. 12x + 5 3x + 6 15x + 11

48. x4 - 3x

2 - 9 5x4 + 8 6x

4 - 3x2 - 1

49. 7x3 - 6x - 3 6x + 14 7x

3 + 11

50. 2x3 + 5x

27x

3 - 5x2 + 1 9x

3 + 1

51. 2x2 + x - 5 x + x

2 + 6 3x2 + 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.


53a. + 4

- 3

b. P = 2ℓ + 2w

= 2(x + 4) + 2(x - 3)= 2(x) + 2(4) + 2(x) + 2(-3)= 2x + 8 + 2x - 6= 2x + 2x + 8 - 6= 4x + 2

c. P = 4x + 2= 4(15) + 2= 60 + 2= 62

He will need 62 ft of fencing.

TEST PREP

54. C; Since -14y2 + 9y

2 + 2y2 = -3y

2, and

3 - 2 = 1, the term must be in the form ay. So -12y + ay - 6y = -15y gives -12 + a - 6 = -15or a = 3. So the missing term is 3y.

55. G; Since 2t3 - 4t - (-7t - 3t) = 2t

3 + 6t ≠ -5t3 - t,

G is correct.

56a. P = 2ℓ + 2w - 3= 2(2x - 1) + 2(x + 4) - 3= 2(2x) + 2(-1) + 2(x) + 2(4) - 3= 4x - 2 + 2x + 8 - 3= 4x + 2x - 2 + 8 - 3= 6x + 3

b. 6x + 3 = 50 _____- 3 ___-36x = 47

6x___6

= 47___6

x ≈ 7.837; If x = 7, Tammy will need 6(7) + 3 = 45 feet of wallpaper border. However, if x = 8, Tammy will need 6(8) + 3 = 51 feet of wallpaper border, which is more than the store has.

c. (2x - 1) ft × (x + 4) ft

= (2(7) - 1) ft × (7 + 4) ft

= 13 ft × 11 ft


57. P = b + 2s

____- 2s ______- 2s

P - 2s = b

b = (2x3 + 3x

2 + 8) - 2(x3 + 5)

= (2x3 + 3x

2 + 8) + (-2x3 - 2(5))

= (2x3 + 3x

2 + 8) + (-2x3 - 10)

= (2x3 - 2x

3) + 3x2 + (8 - 10)

= 3x2 - 2

58. Possible answer: 2m3 + 2m, 2m

3 + m

59. Possible answer: 5m3 + 2m, m

3 - m

60. Possible answer: 2m3 + m, m

3 + m

+ m3 + m

61. Possible answer: 4m3 + 3m

62. Possible answer: 2m3 + m

2 + m, m3 + m

2 + m,m

3 - 2m2 + m

MULTIPLYING POLYNOMIALS

CHECK IT OUT!

1a. (3x3)(6x

2)

= (3 · 6)(x3 · x2)

= 18x5

b. (2r2t)(5t

3)

= (2 · 5)(r2)(t · t3)

= 10r2t4

c. (1__3 x

2y)(12x

3z

2)(y4z

5)

= (1__3 · 12)(x2 · x

3)(y · y4)(z2 · z

5)

= 4x5y

5z

7

2a. 2(4x2 + x + 3)

= 2(4x2) + 2(x) + 2(3)

= 8x2 + 2x + 6

b. 3ab(5a2 + b)

= 3ab(5a2) + 3ab(b)

= (3 · 5)(a · a2)(b) + (3)(a)(b · b)

= 15a3b + 3ab

2

c. 5r2s

2(r - 3s)

= 5r2s

2(r) + 5r

2s

2(-3s)

= (5)(r2 · r)(s2) + (5 · (-3))(r2)(s2 · s)

= 5r3s

2 - 15r2s

3

3a. (a + 3)(a - 4)= a(a) + a(-4) + 3(a) + 3(-4)

= a2 - 4a + 3a - 12

= a2 - a - 12

b. (x - 3)2

= (x - 3)(x - 3)= x(x) + x(-3) - 3(x) - 3(-3)

= x2 - 3x - 3x + 9

= x2 - 6x + 9

c. (2a - b2)(a + 4b

2)

= 2a(a) + 2a(4b2) - b

2(a) - b

2(4b2)

= 2a2 + 8ab

2 - ab2 - 4b

4

= 2a2 + 7ab

2 - 4b4

4a. (x + 3)(x2 - 4x + 6)

= x(x2 - 4x + 6) + 3(x2 - 4x + 6)

= x(x2) + x(-4x) + x(6) + 3(x2) + 3(-4x) + 3(6)

= x3 - 4x

2 + 6x + 3x2 - 12x + 18

= x3 - x

2 - 6x + 18

6-5


b. (3x + 2)(x2 - 2x + 5)

= 3x(x2 - 2x + 5) + 2(x2 - 2x + 5)

= 3x(x2) + 3x(-2x) + 3x(5) + 2(x2) + 2(-2x)+ 2(5)

= 3x3 - 6x

2 + 15x + 2x2 - 4x + 10

= 3x3 - 4x

2 + 11x + 10

5a. Let x represent the width of the rectangle.A = ℓw

= (x - 4)(x)= x(x) - 4(x)

= x2 - 4x

The area is represented by x2 - 4x.

b. A = x2 - 4x

= (6)2 - 4(6)

= 36 - 24 = 12The 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.

2 + 2 + + 2 = 2 + 3 + 2

Vertical method:

( + 2)( 2 + 3 + 2)

Rectangle model:

( + 2)( 2 + 2 + 1)

2

2 4 2 + 2

+ 2 + 1

3 + 4 2 + 5 + 2

2 + 3 + 2

−−−−−−−−−− × + 2

2 2 + 6 + 4

−−−−−−−−−−− + 3 + 3 2 + 2

3 + 5 2 + 8 + 4

2

2 3

2

Multiplying Polynomials

Distributive Property:

5 ( + 2) =

5 2 + 10

FOIL method: ( + 1)( + 2) =

EXERCISES

GUIDED PRACTICE

1. (2x2)(7x

4)

= (2 · 7)(x2 · x4)

= 14x6

2. (-5mn3)(4m

2n

2)

= (-5 · 4)(m · m2)(n3 · n

2)= -20m

3n

5

3. (6rs2)(s3

t2)(1__

2 r

4t3)

= (6 · 1__2)(r · r

4)(s2 · s3)(t2 · t

3)

= 3r5s

5t5

4. (1__3 a

5)(12a)

= (1__3 · 12)(a5 · a)

= 4a6

5. (-3x4y

2)(-7x3y)

= (-3 · (-7))(x4 · x3)(y2 · y)

= 21x7y

3

6. (-2pq3)(5p

2q

2)(-3q4)

= (-2 · 5 · (-3))(p · p2)(q3 · q

2 · q4)

= 30p3q

9

7. 4(x2 + 2x + 1)

= 4(x2) + 4(2x) + 4(1)

= 4x2 + 8x + 4

8. 3ab(2a2 + 3b

3)

= 3ab(2a2) + 3ab(3b

3)

= (3 · 2)(a · a2)(b) + (3 · 3)(a)(b · b

3)= 6a

3b + 9ab

4

9. 2a3b(3a

2b + ab

2)

= 2a3b(3a

2b) + 2a

3b(ab

2)

= (2 · 3)(a3 · a2)(b · b) + (2)(a3 · a)(b · b

2)= 6a

5b

2 + 2a4b

3

10. -3x(x2 - 4x + 6)

= -3x(x2) - 3x(-4x) - 3x(6)

= -3x3 + 12x

2 - 18x

11. 5x2y(2xy

3 - y)= 5x

2y(2xy

3) + 5x2y(-y)

= (5 · 2)(x2 · x)(y · y3) + (5 · (-1))(x2)(y · y)

= 10x3y

4 - 5x2y

2

12. 5m2n

3 · mn2(4m - n)

= (5)(m2 · m)(n3 · n2)(4m - n)

= 5m3n

5(4m - n)

= 5m3n

5(4m) + 5m

3n

5(-n)

= (5 · 4)(m3 · m)(n5) + (5 · (-1))(m3)(n5 · n)

= 20m4n

5 - 5m3n

6

13. (x + 1)(x - 2)= x(x) + x(-2) + 1(x) + 1(-2)

= x2 -2x + x - 2

= x2 - x - 2

14. (x + 1)2

= (x + 1)(x + 1)= x(x) + x(1) + 1(x) + 1(1)

= x2 + x + x + 1

= x2 + 2x + 1


15. (x - 2)2

= (x - 2)(x - 2)= x(x) + x(-2) - 2(x) - 2(-2)

= x2 - 2x - 2x + 4

= x2 -4x + 4

16. (y - 3)(y - 5)= y(y) + y(-5) - 3(y) - 3(-5)

= y2 - 5y - 3y + 15

= y2 - 8y + 15

17. (4a3 - 2b)(a - 3b

2)

= 4a3(a) + 4a

3(-3b2) - 2b(a) - 2b(-3b

2)= 4a

4 - 2ab - 12a3b

2 + 6b3

18. (m2 - 2mn)(3mn + n2)

= m2(3mn) + m

2(n2) - 2mn(3mn) - 2mn(n2)= 3m

3n + m

2n

2 - 6m2n

2 - 2mn3

= 3m3n - 5m

2n

2 - 2mn3

19. (x + 5)(x2 - 2x + 3)

= x(x2 - 2x + 3) + 5(x2 - 2x + 3)

= x(x2) + x(-2x) + x(3) + 5(x2) + 5(-2x) + 5(3)

= x3 - 2x

2 + 3x + 5x2 - 10x + 15

= x3 + 3x

2 - 7x + 15

20. (3x + 4)(x2 - 5x + 2)

= 3x(x2 - 5x + 2) + 4(x2 - 5x + 2)

= 3x(x2) + 3x(-5x) + 3x(2) + 4(x2) + 4(-5x)+ 4(2)

= 3x3 - 15x

2 + 6x + 4x2 - 20x + 8

= 3x3 - 11x

2 - 14x + 8

21. (2x - 4)(-3x3 + 2x - 5)

= 2x(-3x3 + 2x - 5) - 4(-3x

3 + 2x - 5)

= 2x(-3x3) + 2x(2x) + 2x(-5) - 4(-3x

3) - 4(2x)- 4(-5)

= -6x4 + 4x

2 - 10x + 12x3 - 8x + 20

= -6x4 + 12x

3 + 4x2 - 18x + 20

22. (-4x + 6)(2x3 - x

2 + 1)

= -4x(2x3 - x

2 + 1) + 6(2x3 - x

2 + 1)

= -4x(2x3) -4x(- x2) -4x(1) + 6(2x

3) + 6(- x2)+ 6(1)

= -8x4 + 4x

3 - 4x + 12x3 - 6x

2 + 6

= -8x4 + 16x

3 - 6x2 - 4x + 6

23. (x - 5)(x2 + x + 1)

= x(x2 + x + 1) - 5(x2 + x + 1)

= x(x2) + x(x) + x(1) -5(x2) - 5(x) - 5(1)

= x3 + x

2 + x - 5x2 - 5x - 5

= x3 - 4x

2 - 4x - 5

24. (a + b)(a - b)(b - a)

= (a(a) + a(-b) + b(a) + b(-b))(b- a)

= (a2 - ab + ab - b2)(b - a)

= (a2 - b2)(b - a)

= a2(b) + a

2(-a) - b

2(b) - b

2(-a)

= a2b - a

3 - b3 + ab

2

= - a3 + a2b + ab

2 - b3

25a. A = ℓw

= (2x - 3)(x)= 2x(x) - 3(x)

= 2x2 - 3x

The area is represented by 2x2 - 3x.

b. A = 2x2 - 3x

= 2(4)2 - 3(4)

= 2(16) - 3(4)= 32 - 12 = 20

The area is 20 in 2.


26. (3x2)(8x

5)

= (3 · 8)(x2 · x5)

= 24x7

27. (-2r3s

4)(6r2s)

= (-2 · 6)(r3 · r2)(s4 · s)

= -12r5s

5

28. (15xy2)(1__

3 x

2z

3)(y3z

4)

= (15 · 1__3)(x · x

2)(y2 · y3)(z3 · z

4)

= 5x3y

5z

7

29. (-2a3)(-5a)

= (-2 · (-5))(a3 · a)

= 10a4

30. (6x3y

2)(-2x2y)

= (6 · (-2))(x3 · x2)(y2 · y)

= -12x5y

3

31. (-3a2b)(-2b

3)(- a3b

2)

= (-3 · (-2) · (-1))(a2 · a3)(b · b

3 · b2)

= -6a5b

6

32. (7x2)(xy

5)(2x3y

2)= (7 · 2)(x2 · x · x

3)(y5 · y2)

= 14x6y

7

33. (-4a3bc

2)(a3b

2c)(3ab

4c

5)

= (-4 · 3)(a3 · a3 · a)(b · b

2 · b4)(c2 · c · c

5)= -12a

7b

7c

8

34. (12mn2)(2m

2n)(mn)

= (12 · 2)(m · m2 · m)(n2 · n · n)

= 24m4n

4


35. 9s(s + 6)= 9s(s) + 9s(6)

= 9s2 + 54s

36. 9(2x2 - 5x)

= 9(2x2) + 9(-5x)

= 18x2 - 45x

37. 3x(9x2 - 4x)

= 3x(9x2) + 3x(-4x)

= 27x3 - 12x

2

38. 3(2x2 + 5x + 4)

= 3(2x2) + 3(5x) + 3(4)

= 6x2 + 15x + 12

39. 5s2t3(2s - 3t

2)

= 5s2t3(2s) + 5s

2t3(-3t

2)

= (5 · 2)(s2 · s)(t3) + (5 · (-3))(s2)(t3 · t2)

= 10s3t3 - 15s

2t5

40. x2y

3 · 5x2y(6x + y

2)= (5)(x2 · x

2)(y3 · y)(6x + y2)

= 5x4y

4(6x + y2)

= 5x4y

4(6x) + 5x

4y

4(y2)= (5 · 6)(x4 · x)(y4) + (5)(x4)(y4 · y

2)= 30x

5y

4 + 5x4y

6

41. -5x(2x2 - 3x - 1)

= -5x(2x2) - 5x(-3x) - 5x(-1)

= -10x3 + 15x

2 + 5x

42. -2a2b

3(3ab2 - a

2b)

= -2a2b

3(3ab2) - 2a

2b

3(- a2b)

= (-2 · 3)(a2 · a)(b3 · b2) - (2 · -1)(a2 · a

2)(b3 · b)= -6a

3b

5 + 2a4b

4

43. -7x3y · x

2y

2(2x - y)

= (-7)(x3 · x2)(y · y

2)(2x - y)

= -7x5y

3(2x - y)

= -7x5y

3(2x) - 7x

5y

3(-y)

= (-7 · 2)(x5 · x)(y3) + (-7 · (-1))(x5)(y3 · y)= -14x

6y

3 + 7x5y

4

44. (x + 5)(x - 3)= x(x) + x(-3) + 5(x) + 5(-3)

= x2 - 3x + 5x - 15

= x2 + 2x - 15

45. (x + 4)2

= (x + 4)(x + 4)= x(x) + x(4) + 4(x) + 4(4)

= x2 + 4x + 4x + 16

= x2 + 8x + 16

46. (m - 5)2

= (m - 5)(m - 5)= m(m) + m(-5) - 5(m) - 5(-5)

= m2 - 5m - 5m + 25

= m2 - 10m + 25

47. (5x - 2)(x + 3)= 5x(x) + 5x(3) - 2(x) - 2(3)

= 5x2 + 15x - 2x - 6

= 5x2 + 13x - 6

48. (3x - 4)2

= (3x - 4)(3x - 4)= 3x(3x) + 3x(-4) - 4(3x) - 4(-4)

= 9x2 - 12x -12x + 16

= 9x2 - 24x + 16

49. (5x + 2)(2x - 1)

= 5x(2x) + 5x(-1) + 2(2x) + 2(-1)

= 10x2 - 5x + 4x - 2

= 10x2 - x - 2

50. (x - 1)(x - 2)= x(x) + x(-2) - 1(x) - 1(-2)

= x2 - 2x - x + 2

= x2 - 3x + 2

51. (x - 8)(7x + 4)= x(7x) + x(4) - 8(7x) - 8(4)

= 7x2 + 4x - 56x - 32

= 7x2 - 52x - 32

52. (2x + 7)(3x + 7)= 2x(3x) + 2x(7) + 7(3x) + 7(7)

= 6x2 + 14x + 21x + 49

= 6x2 + 35x + 49

53. (x + 2)(x2 - 3x + 5)

= x(x2 - 3x + 5) + 2(x2 - 3x + 5)

= x(x2) + x(-3x) + x(5) + 2(x2) + 2(-3x) + 2(5)

= x3 - 3x

2 + 5x + 2x2 - 6x + 10

= x3 - x

2 - x + 10

54. (2x + 5)(x2 - 4x + 3)

= 2x(x2 - 4x + 3) + 5(x2 - 4x + 3)

= 2x(x2) + 2x(-4x) + 2x(3) + 5(x2) + 5(-4x)+ 5(3)

= 2x3 - 8x

2 + 6x + 5x2 - 20x + 15

= 2x3 - 3x

2 - 14x + 15

55. (5x - 1)(-2x3 + 4x - 3)

= 5x(-2x3 + 4x - 3) - 1(-2x

3 + 4x - 3)

= 5x(-2x3) + 5x(4x) + 5x(-3) - 1(-2x

3) - 1(4x)- 1(-3)

= -10x4 + 20x

2 - 15x + 2x3 - 4x + 3

= -10x4 + 2x

3 + 20x2 - 19x + 3

56. (x - 3)(x2 - 5x + 6)

= x(x2 - 5x + 6) - 3(x2 - 5x + 6)

= x(x2) + x(-5x) + x(6) - 3(x2) - 3(-5x) - 3(6)

= x3 - 5x

2 + 6x - 3x2 + 15x - 18

= x3 - 8x

2 + 21x - 18

57. (2x2 - 3)(4x

3 - x2 + 7)

= 2x2(4x

3 - x2 + 7) - 3(4x

3 - x2 + 7)

= 2x2(4x

3) + 2x2(- x2) + 2x

2(7) - 3(4x

3) - 3(- x2)- 3(7)

= 8x5 - 2x

4 + 14x2 - 12x

3 + 3x2 - 21

= 8x5 - 2x

4 - 12x3 + 17x

2 - 21


58. (x - 4)3

= (x - 4)(x - 4)(x - 4)

= (x(x) + x(-4) - 4(x) - 4(-4))(x - 4)

= (x2 - 4x - 4x + 16)(x - 4)

= (x2 - 8x + 16)(x - 4)

= (x - 4)(x2 - 8x + 16)

= x(x2 - 8x + 16) - 4(x2 - 8x + 16)

= x(x2) + x(-8x) + x(16) - 4(x2) - 4(-8x) - 4(16)

= x3 - 8x

2 + 16x - 4x2 + 32x - 64

= x3 - 12x

2 + 48x - 64

59. (x - 2)(x2 + 2x + 1)

= x(x2 + 2x + 1) - 2(x2 + 2x + 1)

= x(x2) + x(2x) + x(1) - 2(x2) - 2(2x) - 2(1)

= x3 + 2x

2 + x - 2x2 - 4x - 2

= x3 - 3x - 2

60. (2x + 10)(4 - x + 6x3)

= 2x(4 - x + 6x3) + 10(4 - x + 6x

3)

= 2x(4) + 2x(-x) + 2x(6x3) + 10(4) + 10(-x)

+ 10(6x3)

= 8x - 2x2 + 12x

4 + 40 - 10x + 60x3

= 12x4 + 60x

3 - 2x2 - 2x + 40

61. (1 - x)3

= (1 - x)(1 - x)(1 - x)

= (1(1) + 1(-x) - x(1) - x(-x))(1 - x)

= (1 - x - x + x2)(1 - x)

= (1 - 2x + x2)(1 - x)

= (1 - x)(1 - 2x + x2)

= 1(1 - 2x + x2) - x(1 - 2x + x

2)

= 1 - 2x + x2 -x(1) - x(-2x) -x(x2)

= 1 - 2x + x2 - x + 2x

2 - x3

= - x3 + 3x2 - 3x + 1

62a. A = ℓw

= (x + 3)(x)= x(x) + 3(x)

= x2 + 3x

The area is represented by x2 + 3x.

b. A = x2 + 3x

= (5)2 + 3(5)

= 25 + 15 = 40The area is 40 ft

2.

63. A = s2

= (4x - 6)2

= (4x - 6)(4x - 6)= 4x(4x) + 4x(-6) - 6(4x) - 6(-6)

= 16x2 - 24x - 24x + 36

= 16x2 - 48x + 36

The area is represented by 16x2 - 48x + 36.

64a.

+ 4

+ 1

b. A = ℓw

= (x + 4)(x + 1)= x(x) + x(1) + 4(x) + 4(1)= x

2 + x + 4x + 4= x

2 + 5x + 4The area is represented by x

2 + 5x + 4.

c. A = x2 + 5x + 4

= (4)2 + 5(4) + 4

= 16 + 20 + 4 = 40

The area is 40 ft 2.

ADegree

of AB

Degree

of BA · B

Degree

of A · B

2x2

2 3x5

5 6x7

765a. 5x

33 2x

2 + 1 2 10x5 +

5x3

5

b. x2 + 2 2 x

2 - x 2 x4 - x

3 +2x

2 - 2x

4

c. x - 3 1 x3 - 2x

2

+ 13 x

4 - 5x3

+ 6x2 +

x - 3

4

d. m + n

66. A = ℓw

= (2x + 3)(4x)= 2x(4x) + 3(4x)

= 8x2 + 12x

The area is represented by 8x2 + 12x.

67. A = ℓw

= 3(2x + 1)(2x + 1)= [3(2x) + 3(1)](2x + 1)= (6x + 3)(2x + 1)= 6x(2x) + 6x(1) + 3(2x) + 3(1)

= 12x2 + 6x + 6x + 3

= 12x2 + 12x + 3

The area is represented by 12x2 + 12x + 3.

68. A = ℓw

= (x - 5)(x - 5)= x(x) + x(-5) - 5(x) - 5(-5)

= x2 - 5x - 5x + 25

= x2 - 10x + 25

The area is represented by x2 - 10x + 25.

69a. A = ℓw

= (2x)(x)

= 2x2

The area is represented by 2x2.

b. A = 2x2

= 2(20)2

= 2(400) = 800The area is 800 m

2.


70. (1.5a3)(4a

6)

= (1.5 · 4)(a3 · a6)

= 6a9

71. (2x + 5)(x - 6)= 2x(x) + 2x(-6) + 5(x) + 5(-6)

= 2x2 - 12x + 5x - 30

= 2x2 - 7x - 30

72. (3g - 1)(g + 5)= 3g(g) + 3g(5) - 1(g) - 1(5)

= 3g2 + 15g - g - 5

= 3g2 + 14g - 5

73. (4x - 2y)(2x - 3y)= 4x(2x) + 4x(-3y) - 2y(2x) - 2y(-3y)

= 8x2 - 12xy - 4xy + 6y

2

= 8x2 - 16xy + 6y

2

74. (x + 3)(x - 3)= x(x) + x(-3) + 3(x) + 3(-3)

= x2 - 3x + 3x - 9

= x2 - 9

75. (1.5x - 3)(4x + 2)= 1.5x(4x) + 1.5x(2) - 3(4x) - 3(2)

= 6x2 + 3x - 12x - 6

= 6x2 - 9x - 6

76. (x - 10)(x + 4)= x(x) + x(4) - 10(x) - 10(4)

= x2 + 4x - 10x - 40

= x2 - 6x - 40

77. x2(x + 3)

= x2(x) + x

2(3)

= x3 + 3x

2

78. (x + 1)(x2 + 2x)

= x(x2) + x(2x) + 1(x2) + 1(2x)

= x3 + 2x

2 + x2 + 2x

= x3 + 3x

2 + 2x

79. (x - 4)(2x2 + x - 6)

= x(2x2 + x - 6) - 4(2x

2 + x - 6)

= x(2x2) + x(x) + x(-6) - 4(2x

2) - 4(x) - 4(-6)

= 2x3 + x

2 - 6x - 8x2 - 4x + 24

= 2x3 - 7x

2 - 10x + 24

80. (a + b)(a - b)2

= (a + b)(a - b)(a - b)

= (a(a) + a(-b) + b(a) + b(-b))(a - b)

= (a2 - ab + ab - b2)(a - b)

= (a2 - b2)(a - b)

= a2(a) + a

2(-b) - b

2(a) - b

2(-b)

= a3 - a

2b - ab

2 + b3

81. (2p - 3q)3

= (2p - 3q)(2p - 3q)(2p - 3q)

= (2p(2p) + 2p(-3q) - 3q(2p) - 3q(-3q))(2p - 3q)

= (4p2 - 6pq - 6pq + 9q

2)(2p - 3q)

= (4p2 - 12pq + 9q

2)(2p - 3q)

= (2p - 3q)(4p2 - 12pq + 9q

2)= 2p(4p

2 - 12pq + 9q2) - 3q(4p

2 - 12pq + 9q2)

= 2p(4p2) + 2p(-12pq) + 2p(9q

2) - 3q(4p2)

- 3q(-12pq) - 3q(9q2)

= 8p3 - 24p

2q + 18pq

2 - 12p2q + 36pq

2 - 27q3

= 8p3 - 36p

2q + 54pq

2 - 27q3

82a.

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)

= 4x2 + 20x + 50x + 250

= 4x2 + 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.

84. A = ℓwh

= (x + 5)(x)(x + 2)= (x(x) + 5(x))(x + 2)

= (x2 + 5x)(x + 2)

= x2(x) + x

2(2) + 5x(x) + 5x(2)

= x3 + 2x

2 + 5x2 + 10x

= x3 + 7x

2 + 10x

The area is represented by x3 + 7x

2 + 10x.

85. Yes; x = 0

86. Let x represent the width of the rectangle.A = ℓw

= (x + 1)(x)= x(x) + 1(x)

= x2 + 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)

= a2 - 6a + a - 6

= a2 - 5a - 6


88. H

2a(a2 - 1)

= 2a(a2) + 2a(-1)

= 2a3 - 2a

89. D

3x3y

2z · x

2yz

= (3)(x3 · x2)(y2 · y)(z · z)

= 3x5y

3z

2

This has degree 5 + 3 + 2 = 10.


90. 6x2 - 2(3x

2 - 2x + 4)

= 6x2 - 2(3x

2) - 2(-2x) - 2(4)

= 6x2 - 6x

2 + 4x - 8= 4x - 8

91. x2 - 2x(x + 3)

= x2 - 2x(x) - 2x(3)

= x2 - 2x

2 - 6x

= - x2 - 6x

92. x(4x - 2) + 3x(x + 1)= x(4x) + x(-2) + 3x(x) + 3x(1)

= 4x2 - 2x + 3x

2 + 3x

= 7x2 + x

93a. A = ℓw

= (x + 1)(x - 1)= x(x) + x(-1) + 1(x) + 1(-1)

= x2 - x + x - 1

= x2 - 1

The area is represented by x2 - 1.

b. A = ℓw

= (x + 5)(x + 3) - (x + 1)(x - 1)

= x(x) + x(3) + 5(x) + 5(3) - ( x2 - 1)= x

2 + 3x + 5x + 15 - x2 + 1

= 8x + 16

94. A = s2

= (8 + 2x)2

= (8 + 2x)(8 + 2x)= 8(8) + 8(2x) + 2x(8) + 2x(2x)

= 64 + 16x + 16x + 4x2

= 4x2 + 32x + 64

P = 4s

= 4(x2 + 48)

= 4(x2) + 4(48)

= 4x2 + 192

A = P

4x2 + 32x + 64 = 4x

2 + 192

_______________-4x2

___________-4x2

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)

= (x2 + x)(x + 2)

= x2(x) + x

2(2) + x(x) + x(2)

= x3 + 2x

2 + x2 + 2x

= x3 + 3x

2 + 2x

96. xm(xn + x

n - 2) = x5 + x

3

xm(xn) + x

m(xn - 2) = x5 + x

3

xm + n + x

m + n - 2 = x5 + 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 ism = 2; n = 3.

97. 2xa(5x

2a - 3 + 2x2a + 2) = 10x

3 + 4x8

2xa(5x

2a - 3) + 2xa(2x

2a + 2) = 10x3 + 4x

8

10x3a - 3 + 4x

3a + 2 = 10x3 + 4x

8

Therefore, it must be true that:3a - 3 = 3 and 3a + 2 = 8

______+ 3 ___+3 ______- 2 ___-23a = 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 + b2

(x + 6)2 = (x)

2 + 2(x)(6) + (6)2

= x2 + 12x + 36

b. (a + b)2 = a

2 + 2ab + b2

(5a + b)2 = (5a)

2 + 2(5a)(b) + (b)2

= 25a2 + 10ab + b

2

c. (a + b)2 = a

2 + 2ab + b2

(1 + c3)2 = (1)

2 + 2(1)(c3) + (c3)2

= 1 + 2c3 + c

6

2a. (a - b)2 = a

2 - 2ab + b2

(x - 7)2 = (x)

2 - 2(x)(7) + (7)2

= x2 - 14x + 49

b. (a - b)2 = a

2 - 2ab + b2

(3b - 2c)2 = (3b)

2 - 2(3b)(2c) + (2c)2

= 9b2 - 12bc + 4c

2

c. (a - b)2 = a

2 - 2ab + b2

(a2 - 4)2 = (a2)2 - 2(a2)(4) + (4)

2

= a4 - 8a

2 + 16

3a. (a + b)(a - b) = a2 - b

2

(x + 8)(x - 8) = (x)2 - (8)

2

= x2 - 64

6-6


b. (a + b)(a - b) = a2 - b

2

(3 + 2y2)(3 - 2y

2) = (3)2 - (2y

2)2

= 9 - 4y4

c. (a + b)(a - b) = a2 - b

2

(9 + r)(9 - r) = (9)2 - (r)

2

= 81 - r2

4. Area of 0: (5 + x)(5 - x) = (5)2 - (x)

2

= 25 - x2

Area of □: x2

Total area = area of 0 + area of □= (25 - x

2) + x2

= 25 + (- x2 + x2)

= 25The area of the pool is 25.

THINK AND DISCUSS

1. (a + b)(a - b) = a2 - ab + ab - b

2 = a2 - b

2

2. product

3.Special Products of Binomials

Perfect-Square Trinomials Difference of

Two Squares

( + ) 2 = 2 + 2 + 2

( + 4) 2 = 2 + 8 + 16

( - ) 2 = 2 - 2 + 2

( - 4) 2 = 2 - 8 + 16

( + )( - ) = 2 2 -

( + 4)( - 4) = 2 - 16

EXERCISES

GUIDED PRACTICE

1. Possible answer: a trinomial that is the result of squaring a binomial.

2. (a + b)2 = a

2 + 2ab + b2

(x + 7)2 = (x)

2 + 2(x)(7) + (7)2

= x2 + 14x + 49

3. (a + b)2 = a

2 + 2ab + b2

(2 + x)2 = (2)

2 + 2(2)(x) + (x)2

= 4 + 4x + x2

4. (a + b)2 = a

2 + 2ab + b2

(x + 1)2 = (x)

2 + 2(x)(1) + (1)2

= x2 + 2x + 1

5. (a + b)2 = a

2 + 2ab + b2

(2x + 6)2 = (2x)

2 + 2(2x)(6) + (6)2

= 4x2 + 24x + 36

6. (a + b)2 = a

2 + 2ab + b2

(5x + 9)2 = (5x)

2 + 2(5x)(9) + (9)2

= 25x2 + 90x + 81

7. (a + b)2 = a

2 + 2ab + b2

(2a + 7b) 2 = (2a)

2 + 2(2a)(7b) + (7b)2

= 4a2 + 28ab + 49b

2

8. (a - b)2 = a

2 - 2ab + b2

(x - 6)2 = (x)

2 - 2(x)(6) + (6)2

= x2 - 12x + 36

9. (a - b)2 = a

2 - 2ab + b2

(x - 2)2 = (x)

2 - 2(x)(2) + (2)2

= x2 - 4x + 4

10. (a - b)2 = a

2 - 2ab + b2

(2x - 1)2 = (2x)

2 - 2(2x)(1) + (1)2

= 4x2 - 4x + 1

11. (a - b)2 = a

2 - 2ab + b2

(8 - x)2 = (8)

2 - 2(8)(x) + (x)2

= 64 - 16x + x2

12. (a - b)2 = a

2 - 2ab + b2

(6p - q)2 = (6p)

2 - 2(6p)(q) + (q)2

= 36p2 - 12pq + q

2

13. (a - b)2 = a

2 - 2ab + b2

(7a - 2b)2 = (7a)

2 - 2(7a)(2b) + (2b)2

= 49a2 - 28ab + 4b

2

14. (a + b)(a - b) = a2 - b

2

(x + 5)(x - 5) = (x)2 - (5)

2

= x2 - 25

15. (a + b)(a - b) = a2 - b

2

(x + 6)(x - 6) = (x) 2 - (6)

2

= x2 - 36

16. (a + b)(a - b) = a2 - b

2

(5x + 1)(5x - 1) = (5x)2 - (1)

2

= 25x2 - 1

17. (a + b)(a - b) = a2 - b

2

(2x2 + 3)(2x

2 - 3) = (2x2)2 - (3)

2

= 4x4 - 9

18. (a - b)(a + b) = a2 - b

2

(9 - x3)(9 + x

3) = (9)2 - (x3)2

= 81 - x6

19. (a - b)(a + b) = a2 - b

2

(2x - 5y)(2x + 5y) = (2x)2 - (5y)

2

= 4x2 - 25y

2

20. Area of big □: (x + 3)2 = (x)

2 + 2(x)(3) + (3)2

= x2 + 6x + 9

Area of small □: (x + 1)2 = (x)

2 + 2(x)(1) + (1)2

= x2 + 2x + 1

Total area = area of big □ + area of small □= (x2 + 6x + 9) + (x2 + 2x + 1)

= (x2 + x2) + (6x + 2x) + (9 + 1)

= 2x2 + 8x + 10

The area of the figure is 2x2 + 8x + 10.


21. (a + b)2 = a

2 + 2ab + b2

(x + 3)2 = (x)

2 + 2(x)(3) + (3)2

= x2 + 6x + 9


22. (a + b)2 = a

2 + 2ab + b2

(4 + z)2 = (4)

2 + 2(4)(z) + (z)2

= 16 + 8z + z2

23. (a + b)2 = a

2 + 2ab + b2

(x2 + y2)2 = (x2)2 + 2(x2)(y2) + (y2)2

= x4 + 2x

2y

2 + y4

24. (a + b)2 = a

2 + 2ab + b2

(p + 2q3)2 = (p)

2 + 2(p)(2q3) + (2q

3)2

= p2 + 4pq

3 + 4q6

25. (a + b)2 = a

2 + 2ab + b2

(2 + 3x)2 = (2)

2 + 2(2)(3x) + (3x)2

= 4 + 12x + 9x2

26. (a + b)2 = a

2 + 2ab + b2

(r2 + 5t)2 = (r2)2 + 2(r2)(5t) + (5t)2

= r4 + 10r

2t + 25t

2

27. (a - b)2 = a

2 - 2ab + b2

(s2 - 7)2 = (s2)2 - 2(s2)(7) + (7)

2

= s4 - 14s

2 + 49

28. (a - b)2 = a

2 - 2ab + b2

(2c - d3)2 = (2c)

2 - 2(2c)(d3) + (d3)2

= 4c2 - 4cd

3 + d6

29. (a - b)2 = a

2 - 2ab + b2

(a - 8)2 = (a)

2 - 2(a)(8) + (8)2

= a2 - 16a + 64

30. (a - b)2 = a

2 - 2ab + b2

(5 - w)2 = (5)

2 - 2(5)(w) + (w)2

= 25 - 10w + w2

31. (a - b)2 = a

2 - 2ab + b2

(3x - 4)2 = (3x)

2 - 2(3x)(4) + (4)2

= 9x2 - 24x + 16

32. (a - b)2 = a

2 - 2ab + b2

(1 - x2)2 = (1)

2 - 2(1)(x2) + (x2)2

= 1 - 2x2 + x

4

33. (a - b)(a + b) = a2 - b

2

(a - 10)(a + 10) = (a)2 - (10)

2

= a2 - 100

34. (a + b)(a - b) = a2 - b

2

(y + 4)(y - 4) = (y)2 - (4)

2

= y2 - 16

35. (a + b)(a - b) = a2 - b

2

(7x + 3)(7x - 3) = (7x)2 - (3)

2

= 49x2 - 9

36. (a - b)(a + b) = a2 - b

2

(x2 -2)(x2 + 2) = (x2)2 - (2)2

= x4 - 4

37. (a + b)(a - b) = a2 - b

2

(5a2 + 9)(5a

2 - 9) = (5a2)2 - (9)

2

= 25a4 - 81

38. (a + b)(a - b) = a2 - b

2

(x3 + y2)(x2 - y

2) = (x3)2 - (y2)2

= x6 - y

4

39. A = π r2

= π (x + 4)2

= π ( (x)2 + 2(x)(4) + (4)

2)= π ( x2 + 8x + 16)

= π ( x2) + π(8x) + π(16)

= π x2 + 8πx + 16π

The area of the puzzle is π x2 + 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

= x2 - 2x + 1

Area of 0: x(x - 2) = x(x) + x(-2)

= x2 - 2x

Since x2 - 2x + 1 > x

2 - 2x, the square has the greater area.

c. Difference = area of □ - area of 0= (x2 - 2x + 1) - (x2 - 2x)

= (x2 - 2x + 1) + (- x2 + 2x)

= (x2 - x2) + (-2x + 2x) + 1

= 1The difference in area is 1 square unit.

41. (a + b)2 = a

2 + 2ab + b2

(x + y)2 = (x)

2 + 2(x)(y) + (y)2

= x2 + 2xy + y

2

42. (a - b)2 = a

2 - 2ab + b2

(x - y)2 = (x)

2 - 2(x)(y) + (y)2

= x2 - 2xy + y

2

43. (a + b)(a - b) = a2 - b

2

(x2 + 4)(x2 - 4) = (x2)2 - (4)2

= x4 - 16

44. (a + b)2 = a

2 + 2ab + b2

(x2 + 4)2 = (x2)2 + 2(x2)(4) + (4)2

= x4 + 8x

2 + 16

45. (a - b)2 = a

2 - 2ab + b2

(x2 - 4)2 = (x2)2 - 2(x2)(4) + (4)

2

= x4 - 8x

2 + 16

46. (a - b)2 = a

2 - 2ab + b2

(1 - x)2 = (1)

2 - 2(1)(x) + (x)2

= 1 - 2x + x2


47. (a + b)2 = a

2 + 2ab + b2

(1 + x)2 = (1)

2 + 2(1)(x) + (x)2

= 1 + 2x + x2

48. (a - b)(a + b) = a2 - b

2

(1 - x)(1 + x) = (1)2 - (x)

2

= 1 - x2

49. (a - b)(a - b) = a2 - 2ab + b

2

(x3 - a3)(x3 - a

3) = (x3)2 - 2(x3)(a3) + (a3)2

= x6 - 2x

3a

3 + a6

50. (a + b)(a + b) = a2 + 2ab + b

2

(5 + n)(5 + n) = (5)2 + 2(5)(n) + (n)

2

= 25 + 10n + n2

51. (a - b)(a + b) = a2 - b

2

(6a - 5b)(6a + 5b) = (6a)2 - (5b)

2

= 36a2 - 25b

2

52. (a - b)(a - b) = a2 - 2ab + b

2

(r - 4t4)(r - 4t

4) = (r)2 - 2(r)(4t

4) + (4t4)2

= r2 - 8rt

4 + 16t8

a b (a - b)2

a2- 2ab + b

2

1 4 (1 - 4)2 = 9 (1)

2 - 2(1)(4) + (4)2 = 9

53. 2 4 (2 - 4)2 = 4 (2)

2 - 2(2)(4) + (4)2 = 4

54. 3 2 (3 - 2)2 = 1 (3)

2 - 2(3)(2) + (2)2 = 1

a b (a + b)2

a2+ 2ab + b

2

55. 1 4 (1 + 4)2 = 25 (1)

2 + 2(1)(4) + (4)2 = 25

56. 2 5 (2 + 5)2 = 49 (2)

2 + 2(2)(5) + (5)2 = 49

57. 3 0 (3 + 0)2 = 9 (3)

2 + 2(3)(0) + (0)2 = 9

a b (a + b)(a - b) a2- b

2

58. 1 4 (1 + 4)(1 - 4) = -15 (1)2 - (4)

2 = -15

59. 2 3 (2 + 3)(2 - 3) = -5 (2)2 - (3)

2 = -5

60. 3 2 (3 + 2)(3 - 2) = 5 (3)2 - (2)

2 = 5

61. a · b = (a + b)

2 - (a - b) 2

________________4

35 · 24 = (35 + 24)

2 - (35 - 24)2

____________________4

= (59)

2 - (11)2

___________4

= 3481 - 121__________

4

= 3360_____

4= 840

62. Notice that: (a - b)

2 = a2 - 2ab - b

2 = 16x2 - 24x + c

Therefore, a2 = 16x

2 = (4x)2. So a = ±4x.

Therefore, -24x = -2ab = -2(±4x)b = ∓8xb.-24x = ∓8xb

-24x_____∓8x

= ∓8xb_____∓8x

±3 = b

So c = b2 = (±3)

2 = 9.

63. Possible answer: The square of a difference is not

the same as a difference of squares; a2 - 2ab + b

2.

64a.

+ 3

- 3

b. A = ℓw

= (x + 3)(x - 3)

= (x)2 - (3)

2

= x2 - 9

The area is represented by x2 - 9.

c. P = 2ℓ + 2w

48 = 2(x + 3) + 2(x - 3) 48 = 2(x) + 2(3) + 2(x) + 2(-3) 48 = 2x + 6 + 2x - 6 48 = 2x + 2x + 6 - 6 48 = 4x

48___4 = 4x___

4 12 = x

A = x2 - 9

= (12)2 - 9

= 144 - 9 = 135The area of the region is 135 ft

2.

65. For ax2 - 49 to be a perfect square, ax

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 + b2

(5x - 6y)(5x - 6y) = (5x)2 - 2(5x)(6y) + (6y)

2

= 25x2 - 60xy + 36y

2

68. J; The 25x2 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 + s2 = 64. Since

rs = 15, r2 + 2(15) + s

2 = 64, or r2 + s

2 = 34.



71. (x + 4)(x + 4)(x - 4)

= ((x)2 + 2(x)(4) + (4)

2)(x - 4)

= (x2 + 8x + 16)(x - 4)

= (x - 4)(x2 + 8x + 16)

= x(x2 + 8x + 16) - 4(x2 + 8x + 16)

= x(x2) + x(8x) + x(16) - 4(x2) - 4(8x) - 4(16)

= x3 + 8x

2 + 16x - 4x2 - 32x - 64

= x3 + 4x

2 - 16x - 64

72. (x + 4)(x - 4)(x - 4)

= ((x)2 - (4)

2)(x - 4)

= (x2 - 16)(x - 4)

= x2(x) + x

2(-4) - 16(x) - 16(-4)

= x3 - 4x

2 - 16x + 64

73. Let x2 + bx + c = x

2 + bx + (± √ % c ) 2 sincec = (± √ % c ) 2.x

2 + bx + (± √ % c ) 2 = (x± √ % c )(x± √ % c ) because the trinomial is a perfect square.

(x± √ % c )(x± √ % c ) = x2 ± 2 √ % c x + (± √ % c ) 2

by multiplication. Make the coefficients of x: b = ±2 √ % c .

74. Rewrite 27 as 23 + 4 and 19 as 23 - 4.27 · 19 = (23 + 4)(23 - 4)

= (23)2 - (4)

2

= 529 - 16 = 513

READY TO GO ON? Section B Quiz

1. 4r2 + 2r

6 - 3r → 2r6 + 4r

2 - 3r


2. y2 + 7 - 8y

3 + 2y → -8y3 + y

2 + 2y + 7 The leading coefficient is -8.

3. -12t3 - 4t + t

4 → t4 - 12t

3 - 4t


4. n + 3 + 3n2 → 3n

2 + n + 3 The leading coefficient is 3.

5. 2 + 3x3 → 3x

3 + 2 The leading coefficient is 3.

6. -3a2 + 16 + a

7 + a → a7 - 3a

2 + a + 16 The leading coefficient is 1.


2x3 + 5x - 4 is a cubic trinomial.


5b2 is a quadratic monomial.


6p2 + 3p - p

4 + 2p3 is a quartic polynomial.


x2 + 12 - x is a quadratic trinomial.


-2x3 - 5 + x - 2x

7 is a 7th-degree polynomial.


5 - 6b2 + b - 4b

4 is a quartic polynomial.

13. C(x) = x3 - 15x + 14

C(900) = (900)3 - 15(900) + 14

= 729,000,000 - 13,500 + 14= 728,986,514

The cost to manufacture 900 units is $728,986,514.

14. (10m3 + 4m

2) + (7m2 + 3m)

= 10m3 + (4m

2 + 7m2) + 3m

= 10m3 + 11m

2 + 3m

15. (3t2 - 2t) + (9t

2 + 4t - 6)

= (3t2 + 9t

2) + (-2t + 4t) + (-6)

= 12t2 + 2t - 6

16. (12d6 - 3d

2) + (2d4 + 1)

= 12d6 + 2d

4 - 3d2 + 1

17. (6y3 + 4y

2) - (2y2 + 3y)

= (6y3 + 4y

2) + (-2y2 - 3y)

= 6y3 + (4y

2 - 2y2) + (-3y)

= 6y3 + 2y

2 - 3y

18. (7n2 - 3n) - (5n

2 + 5n)

= (7n2 - 3n) + (-5n

2 - 5n)

= (7n2 - 5n

2) + (-3n - 5n)

= 2n2 - 8n

19. (b2 - 10) - (-5b3 + 4b)

= (b2 - 10) + (5b3 - 4b)

= 5b3 + b

2 - 4b - 10

20. P = (2s3 + 4) + (4s

2 + 1) + (5s)

= 2s3 + 4s

2 + 5s + (4 + 1)

= 2s3 + 4s

2 + 5s + 5

21. 2h3 · 5h

5

= (2 · 5)(h3 · h5)

= 10h8

22. (s8t4)(-6st

3)

= (-6)(s8 · s)(t4 · t3)

= -6s9t7

23. 2ab(5a3 + 3a

2b)

= 2ab(5a3) + 2ab(3a

2b)

= (2 · 5)(a · a3)(b) + (2 · 3)(a · a

2)(b · b)

= 10a4b + 6a

3b

2

24. (3k + 5)2

= (3k + 5)(3k + 5)= 3k(3k) + 3k(5) + 5(3k) + 5(5)

= 9k2 + 15k + 15k + 25

= 9k2 + 30k + 25

25. (2x3 + 3y)(4x

2 + y)= 2x

3(4x2) + 2x

3(y) + 3y(4x

2) + 3y(y)

= 8x5 + 2x

3y + 12x

2y + 3y

2


26. (p2 + 3p)(9p2 - 6p - 5)

= p2(9p

2 - 6p - 5) + 3p(9p2 - 6p - 5)

= p2(9p

2) + p2(-6p) + p

2(-5) + 3p(9p

2)+ 3p(-6p) + 3p(-5)

= 9p4 - 6p

3 - 5p2 + 27p

3 - 18p2 - 15p

= 9p4 + 21p

3 - 23p2 - 15p

27. A = bh

= (x + 7)(x - 3)= x(x) + x(-3) + 7(x) + 7(-3)

= x2 - 3x + 7x - 21

= x2 + 4x - 21

The area is represented by (x2 + 4x - 21) square units.

28. (a + b)2 = a

2 + 2ab + b2

(d + 9)2 = (d)

2 + 2(d)(9) + (9)2

= d2 + 18d + 81

29. (a + b)2 = a

2 + 2ab + b2

(3 + 2t)2 = (3)

2 + 2(3)(2t) + (2t)2

= 4t2 + 12t + 9

30. (a + b)2 = a

2 + 2ab + b2

(2x + 5y)2 = (2x)

2 + 2(2x)(5y) + (5y)2

= 4x2 + 20xy + 25y

2

31. (a - b)2 = a

2 - 2ab + b2

(m - 4)2 = (m)

2 - 2(m)(4) + (4)2

= m2 - 8m + 16

32. (a - b)2 = a

2 - 2ab + b2

33. (a - b)2 = a

2 - 2ab + b2

(3w - 1)2 = (3w)

2 - 2(3w)(1) + (1)2

= 9w2 - 6w + 1

34. (a + b)(a - b) = a2 - b

2

(c + 2)(c - 2) = (c)2 - (2)

2

= c2 - 4

35. (a + b)(a - b) = a2 - b

2

(5r + 6)(5r - 6) = (5r)2 - (6)

2

= 25r2 - 36

36. S = 4π r2

= 4π (x - 5)2

= 4π ( (x)2 - 2(x)(5) + (5)

2)= 4π ( x2 - 10x + 25)

= 4π ( x2) + 4π(-10x) + 4π(25)

= 4π x2 - 40πx + 100π

The area is represented by

(4π x2 - 40πx + 100π) in

2.

STUDY GUIDE: REVIEW

1. cubic 2. standard form of a polynomial

3. monomial 4. trinomial

INTEGER EXPONENTS

5. 2 -5 = 1__

2 5 = 1____________

2 · 2 · 2 · 2 · 2 = 1___32

2 -5

in. is equal to 1___32

in.

6. (3.6)0 = 1

7. (-1)-4 = 1_____

(-1)4 = 1_______________

(-1)(-1)(-1)(-1) = 1

8. 5 -3 = 1__

5 3 = 1_______

5 · 5 · 5 = 1____125

9. 10 -4 = 1___

10 4

= 1______________10 · 10 · 10 · 10

= 1______10,000

, or 0.0001

10. b-4

= 2 -4

= 1__2

4

= 1__________2 · 2 · 2 · 2

= 1___16

11. (2__5

b)-4

= (2__5

(10))-4

= 4 -4

= 1__4

4

= 1__________4 · 4 · 4 · 4

= 1____256

12. -2p3q

-3

= -2(3)3(-2)

-3

= -2 · 3 3 · (-2)

-3

= -2 · 27 · 1_____(-2)

3

= -54 · 1____________(-2)(-2)(-2)

= -54 · 1___-8

= 27___4

13. m-2 = 1___

m2

14. bc0 = b · c

0

= b · 1= b

15. - 1__2 x

-2y

-4 = - 1__2 · x

-2 · y-4

= - 1__2 · 1__

x2 · 1__

y4

= - 1_____2x

2y

4

16. 2b

6___c

-4 = 2 · b

6 · 1___c

-4

= 2 · b6 · c

4

= 2b6c

4

17. 3a

2c

-2______4b

0 =

3__4

· a2 · c

-2 · 1__b

0

= 3__4

· a2 · 1__

c2 · 1__

1

= 3a

2___4c

2

6-1


18. q

-1r

-2______

s-3

= q-1 · r

-2 · 1___s

-3

= 1__q

· 1__r2 · s

3

= s

3___qr

2

RATIONAL EXPONENTS

19. 81 1__2 = √ %% 81 = 9 20. 343

1__3 =

3√ %% 343 = 7

21. 64 2__3 = ( 3√ %% 64 ) 2

= 4 2 = 16

22. (2 6 )

1__2 = 2

6 · 1__2

= 2 3 = 8

23. 5√ %% z

10 = (z10)

1__5

= z10 · 1__

5 = z2

24. 3√ %%% 125x

6

= (125x6)

1__3

= (125) 1__3 · (x6)

1__3

= (5 3)

1__3 · (x6)

1__3

= (5 3 · 1__

3) · (x6 · 1__3)

= (5 1) · (x2) = 5x

2

25. √ %% x8y

6

= (x8y

6) 1__2

= (x8 · 1__2) · (y6 · 1__

2)= (x4) · (y3) = x

4y

3

26. 3√ %%% m

6n

12

= (m6n

12) 1__3

= (m6 · 1__3) · (n12 · 1__

3)= (m2) · (n4) = m

2n

4

POLYNOMIALS

27. 5 = 5x0

Degree: 028. 8st

3 = 8s1t

3

Degree: 1 + 3 = 4

29. 3z6

Degree: 6 30. 6h = 6h

1

Degree: 1

31. 2n - 4 + 3n2 → 3n

2 + 2n - 4The leading coefficient is 3.

32. 2a - a4 - a

6 + 3a3 → - a6 - a

4 + 3a3 + 2a


33. Degree: 1 Terms: 22s - 6 is a linear binomial.


-8p5 is a quintic monomial.


- m4 - m2 - 1 is a quartic trinomial.

36. Degree: 0 Terms: 12 is a constant monomial.

ADDING AND SUBTRACTINGPOLYNOMIALS

37. 3t + 5 - 7t - 2= 3t - 7t + 5 - 2= -4t + 3

38. 4x5 - 6x

6 + 2x5 - 7x

5

= -6x6 + 4x

5 + 2x5 - 7x

5

= -6x6 - x

5

39. - h3 - 2h2 + 4h

3 - h2 + 5

= - h3 + 4h3 - 2h

2 - h2 + 5

= 3h3 - 3h

2 + 5

40. (3m - 7) + (2m2 - 8m + 6)

= 2m2 + (3m - 8m) + (-7 + 6)

= 2m2 - 5m - 1

41. (12 + 6p) - (p - p2 + 4)

= (12 + 6p) + (-p + p2 - 4)

= p2 + (6p - p) + (12 - 4)

= p2 + 5p + 8

42. (3z - 9z2 + 2) + (2z

2 - 4z + 8)

= (-9z2 + 2z

2) + (3z - 4z) + (2 + 8)

= -7z2 - z + 10

43. (10g - g2 + 3) - (-4g

2 + 8g - 1)= (10g - g

2 + 3) + (4g2 - 8g + 1)

= (- g2 + 4g2) + (10g - 8g) + (3 + 1)

= 3g2 + 2g + 4

44. (-5x3 + 2x

2 - x + 5) - (-5x3 + 3x

2 - 5x - 3)

= (-5x3 + 2x

2 - x + 5) + (5x3 - 3x

2 + 5x + 3)

= (-5x3 + 5x

3) + (2x2 - 3x

2) + (-x + 5x)+ (5 + 3)

= - x2 + 4x + 8

MULTIPLYING POLYNOMIALS

45. (2r)(4r)= (2 · 4)(r · r)

= 8r2

46. (3a5)(2ab)

= (3 · 2)(a5 · a)(b)

= 6a6b

47. (-3xy)(-6x2y)

= (-3 · (-6))(x · x2)(y · y)

= 18x3y

2

48. (3s3t2)(2st

4)(1__2 s

2t8)

= (3 · 2 · 1__2)(s

3 · s · s2)(t2 · t

4 · t8)

= 3s6t14

49. 2(x2 - 4x + 6)

= 2(x2) + 2(-4x) + 2(6)

= 2x2 - 8x + 12

50. -3ab(ab - 2a2b + 5a)

= -3ab(ab) - 3ab(-2a2b) - 3ab(5a)

= (-3)(a · a)(b · b) + (-3)(- 2)(a · a2)(b · b)

+ (-3 · 5)(a · a)(b)

= -3a2b

2 + 6a3b

2 - 15a2b

6-2

6-3

6-4

6-5


51. (a + 3)(a - 6)= a(a) + a(-6) + 3(a) + 3(-6)

= a2 - 6a + 3a - 18

= a2 - 3a - 18

52. (b - 9)(b + 3)= b(b) + b(3) - 9(b) - 9(3)= b

2 + 3b - 9b - 27= b

2 - 6b - 27

53. (x - 10)(x - 2)= x(x) + x(-2) - 10(x) - 10(-2)

= x2 - 2x - 10x + 20

= x2 - 12x + 20

54. (t - 1)(t + 1)= t(t) + t(1) - 1(t) - 1(1)

= t2 + t - t - 1

= t2 - 1

55. (2q + 6)(4q + 5)= 2q(4q) + 2q(5) + 6(4q) + 6(5)

= 8q2 + 10q + 24q + 30

= 8q2 + 34q + 30

56. (5g - 8)(4g - 1)= 5g(4g) + 5g(-1) - 8(4g) - 8(-1)

= 20g2 - 5g - 32g + 8

= 20g2 - 37g + 8

SPECIAL PRODUCTS OF BINOMIALS

57. (a - b) 2 = a

2 - 2ab + b2

(p - 4) 2 = (p)

2 - 2(p)(4) + (4) 2

= p2 - 8p + 16

58. (a + b)2 = a

2 + 2ab + b2

(x + 12)2 = (x)

2 + 2(x)(12) + (12)2

= x2 + 24x + 144

59. (a + b)2 = a

2 + 2ab + b2

(m + 6)2 = (m)

2 + 2(m)(6) + (6)2

= m2 + 12m + 36

60. (a + b)2 = a

2 + 2ab + b2

(3c + 7)2 = (3c)

2 + 2(3c)(7) + (7)2

= 9c2 + 42c + 49

61. (a - b)2 = a

2 - 2ab + b2

(2r - 1)2 = (2r)

2 - 2(2r)(1) + (1)2

= 4r2 - 4r + 1

62. (a - b)2 = a

2 - 2ab + b2

(3a - b)2 = (3a)

2 - 2(3a)(b) + (b)2

= 9a2 - 6ab + b

2

63. (a - b)2 = a

2 - 2ab + b2

(2n - 5)2 = (2n)

2 - 2(2n)(5) + (5)2

= 4n2 - 20n + 25

64. (a - b)2 = a

2 - 2ab + b2

(h - 13)2 = (h)

2 - 2(h)(13) + (13)2

= h2 - 26h + 169

65. (a - b)(a + b) = a2 - b

2

(x - 1)(x + 1) = (x)2 - (1)

2

= x2 - 1

66. (a + b)(a - b) = a2 - b

2

(z + 15)(z - 15) = (z)2 - (15)

2

= z2 - 225

67. (a - b)(a + b) = a2 - b

2

(c2 - d)(c2 + d) = (c2)2 - (d)2

= c4 - d

2

68. (a + b)(a - b) = a2 - b

2

(3k2 + 7)(3k2 - 7) = (3k2)2 - (7)2

= 9k4 - 49

CHAPTER TEST

1. (1__3

b)-2

= (1__3 (12))

-2

= 4 -2

= 1__4

2

= 1____4 · 4

= 1___16

2. (14 - a0b

2)-3

= (14 - (-2)0 (4)

2)-3

= (14 - 1 · (4 · 4))-3

= (14 - 16)-3

= (-2)-3

= 1_____(-2)

3

= 1____________(-2)(-2)(-2)

= - 1__8

3. 2r-3 = 2 · r

-3

= 2 · 1__r3

= 2__r3

4. -3f0g

-1 = -3 · f0 · g

-1

= -3 · 1 · 1__g

= - 3__g

5. m2n

-3 = m2 · n

-3

= m2 · 1______n

3

= m2___

n3

6. 1__2 s

-5t3 = 1__

2 · s

-5 · t3

= 1__2 · 1__

s5 · t

3

= t3___

2s5

7. S = 3.14r2 + 3.14rℓ

= 3.14(3)2 + 3.14(3)(5)

= 3.14(9) + 3.14(3)(5)= 28.26 + 47.1= 75.36

The area of the cone is approximately 75.36 cm 2.

8. ( 27____125)

1__3 = 27

1__3_____

125 1__3

= 3√ %% 27 _____

3√ %% 125 =

3__5

9. 3√ %% 43

3 = (43

3) 1__3

= 43 3 · 1__

3

= 43 1 = 43

6-6


10. √ %% 25y8 = (25y

8) 1__2

= (25) 1__2 · (y8)

1__2

= √ %% 25 · (y8 · 1__2)

= 5 · y4 = 5y

4

11. 5√ %% 3

5t10

= (3 5t10)

1__5

= (3 5 · 1__

5) · (t10 · 1__5)

= (3 1) · (t2) = 3t

2

12. 3a - 4b + 2a

= 3a + 2a - 4b

= 5a - 4b

13. (2b2 - 4b

3) - (6b3 + 8b

2)

= (2b2 - 4b

3) + (-6b3 - 8b

2)

= (-4b3 - 6b

3) + (2b2 - 8b

2)= -10b

3 - 6b2

14. -9g2 + 3g - 4g

3 - 2g + 3g2 - 4

= -4g3 - 9g

2 + 3g2 + 3g - 2g - 4

= -4g3 - 6g

2 + g - 4

15. -5(r2s - 6)

= -5(r2s) - 5(-6)

= -5r2s + 30

16. (2t - 7)(t + 4)= 2t(t) + 2t(4) - 7(t) - 7(4)

= 2t2 + 8t - 7t - 28

= 2t2 + t - 28

17. (4g - 1)(4g2 - 5g - 3)

= 4g(4g2 - 5g - 3) - 1(4g

2 - 5g - 3)= 4g(4g

2) + 4g(-5g) + 4g(-3) - 1(4g2) - 1(-5g)

- 1(-3)

= 16g3 - 20g

2 - 12g - 4g2 + 5g + 3

= 16g3 - 24g

2 - 7g + 3

18. (a + b)2 = a

2 + 2ab + b2

(m + 6)2 = (m)

2 + 2(m)(6) + (6)2

= m2 + 12m + 36

19. (a - b)(a + b) = a2 - b

2

(3t - 7)(3t + 7) = (3t)2 - (7)

2

= 9t2 - 49

20. (a - b)2 = a

2 - 2ab + b2

(3x2 - 7)

2 = (3x2)2 - 2(3x

2)(7)+ (7)2

= 9x4 - 42x

2 + 49

21a. A = 1__2

bh

= 1__2 (2x + 6)(x - 4)

= (1__2 (2x) + 1__

2 (6))(x - 4)

= (x + 3)(x - 4)= x(x) + x(-4) + 3(x) + 3(-4)

= x2 - 4x + 3x - 12

= x2 - x - 12

The area is represented by x2 - x - 12.

b. A = x2 - x - 12

= (4.5)2 - (4.5) - 12

= 20.25 - 4.5 - 12= 3.75

The area is 3.75 in 2.


CHAPTER Exponents and Polynomials 6 Solutions Keymoodle.bisd303.org/.../Ch06solutions_key.pdf · Exponents and Polynomials Solutions Key ARE YOU READY? 1. F 2. B 3. C 4. D 5. E 6.

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