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

Exponents and PolynomialsSolutions 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

exerCisesguided 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

CS10_A1_MESK710372_C06.indd 195 3/30/11 11:28:49 PM

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. 2 x 0 y -3 = 2 (7) 0 (-4) -3

= 2 · 1 · 1 _____ (-4) 3

= 2 · 1 ____________ (-4)(-4)(-4)

= 2 · 1 ____ -64

= - 1 ___ 32

16. 4 m 0 = 4 · m 0 = 4 · 1 = 4

17. 3 k -4 = 3 · k -4

= 3 · 1 __ k 4

= 3 __ k 4

18. 7 ___ r -7

= 7 · 1 ___ r -7

= 7 · r 7 = 7 r 7

19. x 10 ____ d -3

= x 10 · 1 ____ d -3

= x 10 · d 3 = x 10 d 3

20. 2 x 0 y -4 = 2 · x 0 · y -4

= 2 · 1 · 1 __ y 4

= 2 __ y 4

21. f -4 ____

g -6 = f -4 · 1 ____

g -6

= 1 __ f 4

· g 6

= g 6

__ f 4

22. c 4 ____ d -3

= c 4 · 1 ____ d -3

= c 4 · d 3 = c 4 d 3

23. p 7 q -1 = p 7 · q -1

= p 7 · 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 __ 7 37. 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. 10 m -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


CS10_A1_MESK710372_C06.indd 196 3/30/11 11:28:52 PM

41. (3ab) -2

= (3 ( 1 __ 2 ) (8))

-2

= 12 -2

= 1 ___ 12 2

= 1 ______ 12 · 12

= 1 ____ 144

42. 4 w v x v = 4 (3) 0 (-5) 0 = 4 · 1 · 1 = 4

43. k -4 = 1 __ k 4

44. 2 z -8 = 2 · z -8

= 2 · 1 __ z 8

= 2 __ z 8

45. 1 _____ 2 b -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. -5 x -3 = -5 · x -3

= -5 · 1 __ x 3

= - 5 __ x 3

48. 4 x -6 y -2 = 4 · x -6 · y -2

= 4 · 1 __ x 6

· 1 __ y 2

= 4 ____ x 6 y 2

49. 2 f 0 _____ 7 g -10

= 2 __ 7 · f 0 · 1 ____

g -10

= 2 __ 7 · 1 · g 10

= 2 g 10

____ 7

50. r -5 ___

s -1 = r -5 · 1 ___

s -1

= 1 __ r 5

· s

= s __ r 5

51. s 5 ____ t -12

= s 5 · 1 ____ t -12

= s 5 · t 12 = s 5 t 12

52. 3 w -5 _____ x -6

= 3 · w -5 · 1 ___ x -6

= 3 · 1 ___ w 5

· x 6

= 3 x 6 ___ w 5

53. b 0 c 0 = b 0 · c 0 = 1 · 1 = 1

54. 2 __ 3 m -1 n 5 = 2 __

3 · m -1 · n 5

= 2 __ 3 · 1 __ m · n 5

= 2 n 5 ___ 3m

55. q -2 r 0

_____ s 0

= q -2

· r 0 · 1 __ s 0

= 1 __ q 2

· 1 · 1 __ 1

= 1 __ q 2

56. a -7 b 2 _____ c 3 d -4

= a -7 · b 2 · 1 __ c 3

· 1 ____ d -4

= 1 __ a 7

· b 2 · 1 __ c 3

· d 4

= b 2 d 4 ____ a 7 c 3

57. h 3 k -1 _____ 6 m 2

= 1 __ 6 · h 3 · k -1 · 1 ___

m 2

= 1 __ 6 · h 3 · 1 __

k · 1 ___

m 2

= h 3 _____ 6 m 2 k

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. a 3 b -2 = a 3 · b -2

= a 3 · 1 __ b 2

= a 3 __ b 2

68. c -4 d 3 = c -4 · d 3

= 1 __ c 4

· d 3

= d 3 __ c 4

69. v 0 w 2 y -1 = v 0 · w 2 · y -1

= 1 · w 2 · 1 __ y

= w 2 ___ y

70. ( a 2 b -7 ) 0 = 1


CS10_A1_MESK710372_C06.indd 197 3/30/11 11:28:54 PM

71. -5 y -6 = -5 · y -6

= -5 · 1 __ y 6

= - 5 __ y 6

72. 2 a -5 _____ b -6

= 2 · a -5 · 1 ____ b -6

= 2 · 1 __ a 5

· b 6

= 2 b 6 ___ a 5

73. 2 a 3 ____ b -1

= 2 · a 3 · 1 ____ b -1

= 2 · a 3 · b = 2 a 3 b

74. m 2 ____ n -3

= m 2 · 1 ____ n -3

= m 2 · n 3 = m 2 n 3

75. x -8 ____ 3 y 12

= 1 __ 3 · x -8 · 1 ___

y 12

= 1 __ 3 · 1 __

x 8 · 1 ___

y 12

= 1 ______ 3 x 8 y 12

76. - 20 p -1

______ 5 q -3

= - 20 ___ 5 · p -1 · 1 ____

q -3

= -4 · 1 __ p · q 3

= - 4 q 3

___ 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 a n · 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 = v f -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 3 b -2 _____ c -1

= a 3 · b -2 · 1 ___ c -1

= a 3 · 1 __ b 2

· c

= a 3 c ___ b 2

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 __ a n

.


CS10_A1_MESK710372_C06.indd 198 3/30/11 11:28:57 PM

challenge and extend

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

y = 2 x 1 ___ 16

1 __ 8 1 __

4 1 __

2 1 2 4 8 16

8

4

12

16

x

y

0 4 2 -4 -2

Possible answer: y increases more rapidly as x increases.

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 = 72 m 3 __ 4

= 72(81 ) 3 __ 4

= 72 · ( 4 √ 81 )

3

= 72 · ( 4

√ 3 4 ) 3

= 72 · (3 ) 3 = 72 · 27 = 1944 The panda needs 1944 Calories per day.

4a. 4 √ x 4 y 12

= ( x 4 y 12 ) 1 __ 4

= ( x 4 ) 1 __ 4 ( y 12 )

1 __ 4

= ( x 4 · 1 __

4 ) · ( y

12 · 1 __ 4 )

= ( x 1 ) · ( y 3 ) = x 1 y 3

b. (x y

1 __ 2 )

2

______

5

√ x 5

= (x y

1 __ 2 )

2

______ x

= ( x 2 ) · ( y 1 __ 2 · 2

) · ( x -1 )

= ( x 2 ) · y · ( x -1 )

= ( x 2 ) · ( x -1 ) · y

= x 2 + (-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 n th root of that number.

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

Numerical Example

= =

6 � √ 36 36 1 _ 2 1 _

n

m n

_ = =

6 = 216 � √ 36 36 3 3 2 3 ( )


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


CS10_A1_MESK710372_C06.indd 199 3/30/11 11:28:59 PM

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 = 4 a 1 __ 2

= 4(64 ) 1 __ 2

= 4 ( √ 64 ) = 4(8) = 32 The perimeter is 32 m.

23. √ x 4 y 2

= ( x 4 y 2 ) 1 __ 2

= ( x 4 · 1 __

2 ) · ( y

2 · 1 __ 2

)

= x 2 · y 1 = x 2 y

24. 4

√ z 4

= ( z 4 ) 1 __ 4

= z 4 · 1 __

4

= z 1 = z

25. √ x 6 y 6

= ( x 6 y 6 ) 1 __ 2

= ( x 6 · 1 __

2 ) · ( y

6 · 1 __ 2

)

= x 3 · y 3 = x 3 y 3

26. 3

√ a 12 b 6

= ( a 12 b 6 ) 1 __ 3

= ( a 12 · 1 __

3 ) · ( b

6 · 1 __ 3 )

= a 4 · b 2 = a 4 b 2

27. ( a 1 __ 2 )

2

√ a 2

= ( a 1 __ 2 · 2

) · ( a 2 ) 1 __ 2

= ( a 1 ) · ( a 2 · 1 __

2 )

= a 1 · a 1 = a 1 + 1 = a 2

28. ( x 1 __ 3

)

6

4 √ y 4

= ( x 1 __ 3

· 6 ) · ( y 4 )

1 __ 4

= ( x 2 ) · ( y 4 · 1 __

4 )

= x 2 · y 1 = x 2 y

29. ( z

1 __ 3 )

3

_____

√ z 2

= z 1 __ 3 · 3

_____ ( z 2 )

1 __ 2

= z 1 _____ z

2 · 1 __ 2

= z 1 __ z 1

= 1

30. 3 √ x 6 y 9

______ x 2

= ( x 6 y 9 )

1 __ 3

_______

x 2

= ( x

6 · 1 __ 3

) · ( y

9 · 1 __ 3 ) _____________

x 2

= x 2 · y 3

______ 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 6 c 9

= ( a 6 c 9 ) 1 __ 3

= ( a 6 · 1 __

3 ) · ( c

9 · 1 __ 3 )

= a 2 · c 3 = a 2 c 3

53. 3

√ 8 m 3

= (8 m 3 ) 1 __ 3

= ( 8 1 __ 3 ) · ( m

3 · 1 __ 3

)

= ( 3 √ 8 ) · m 1 = 2m

54. 4 √ x 16 y 4

= ( x 16 y 4 ) 1 __ 4

= ( x 16 · 1 __

4 ) · ( y

4 · 1 __ 4 )

= x 4 · y 1 = x 4 y

55. 3

√ 27 x 6

= (27 x 6 ) 1 __ 3

= ( 27 1 __ 3 ) · ( x

6 · 1 __ 3 )

= ( 3 √ 27 ) · x 2 = 3 x 2

56. ( x 1 __ 2 y 3 )

2

√ x 2

= ( x 1 __ 2 · 2

) · ( y 3 · 2 ) · x

= x 1 · y 6 · x

= x 1 + 1 · y 6

= x 2 · y 6 = x 2 y 6

57. ( a 2 b 4 ) 1 __ 2 3

√ b 6

= ( a 2 · 1 __

2 ) · ( b

4 · 1 __ 2 ) · ( b 6 )

1 __ 3

= ( a 1 ) · ( b 2 ) · ( b 6 · 1 __

3 )

= a 1 · b 2 · b 2

= a 1 · b 2 + 2

= a 1 · b 4 = a b 4


CS10_A1_MESK710372_C06.indd 200 3/30/11 11:29:03 PM

58. 3 √ x 6 y 6

______ y x 2

= ( x 6 y 6 )

1 __ 3

_______

y x 2

= ( x 6 · 1 __

3 ) · ( y

6 · 1 __ 3

) · y -1 · x -2

= ( x 2 ) · ( y 2 ) · ( y -1 ) · ( x -2 )

= x 2 - 2 · y 2 - 1

= x 0 · y 1 = y

59. ( a 2 b

1 __ 2

)

4

_______

√ b 2

= ( a 2 · 4 ) · ( b

1 __ 2

· 4 ) _____________

b

= ( a 8 ) · ( b 2 ) · ( b -1 )

= a 8 · b 2 - 1

= a 8 · b 1 = a 8 b

60. 256 x __ 4

= 4

( 4 √ 256 )

x = 4

4 x = 4 x = 1

61. x 1 __ 5 = 1

( x 1 __ 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. x 1 __ 6 = 0

( x 1 __ 6 )

6

= 0 6 x = 0

64. 64 x __ 3

= 16

( 3 √ 64 )

x = 16

4 x = 16 x = 2

65. x 3 __ 4 = 125

( x 3 __ 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 = 216 x = 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 = 12 m 1 __ 5 L = 12 m

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 longer than the wolf’s.

81. r = 0.62 V 1 __ 3

= 0.62(27 ) 1 __ 3

= 0.62 ( 3 √ 27 )

= 0.62(3) = 1.86 The radius is 1.86 in.

82. ( b 1 _

3 )

3 = b

1 _

3 · 3 = b 1 = b. Also, by definition (

3 √ b )

3 = b.

Therefore b 1 _

3 =

3 √ b .

83. n 2 __ 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 .


CS10_A1_MESK710372_C06.indd 201 3/30/11 11:29:06 PM

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. 4 3 __ 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 = 8It 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 9 b 3

= ( a 9 b 3 ) 1 __ 3

= ( a 9 · 1 __

3 ) · ( b

3 · 1 __ 3

)

= a 3 · b 1

= a 3 b

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. ( a 1 __ 3 ) ( a

1 __ 3 ) ( a

1 __ 3

) = a

( 1 __ 3

+ 1 __ 3

+ 1 __ 3

)

= a 1 = a

92. ( x 1 __ 2 )

5

( x 3 __ 2 ) = ( x

5 __ 2

) ( x

3 __ 2

)

= x ( 5 __ 2

+ 3 __ 2

)

= x 8 __ 2

= x 4

93. ( x 1 __ 3

)

4

( x 5 ) 1 __ 3

= ( x

4 __ 3 ) ( x

5 __ 3 )

= x ( 4 __ 3 + 5 __

3 )

= x 9 __ 3

= x 3

94. y 5 = 32

( y 5 ) 1 __ 5 = 32

1 __ 5

y 5 · 1 __

5 =

5 √ 32

y 1 = 2 y = 2

95. 27 x 3 = 729

27 x 3 ____ 27

= 729 ____ 27

x 3 = 27

( x 3 ) 1 __ 3 = 27

1 __ 3

x 3 · 1 __

3 =

3 √ 27

x 1 = 3 x = 3

96. 1 = 1 __ 8 x 3

(8)1 = (8) 1 __ 8 x 3

8 = x 3

8 1 __ 3 = ( x 3 )

1 __ 3

3 √ 8 = x

3 · 1 __ 3

2 = x 1 2 = x

97. S = (4π ) 1 __ 3 (3V )

2 __ 3

= (4 π) 1 __ 3 (3(36π) )

2 __ 3

= (4 π) 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


CS10_A1_MESK710372_C06.indd 202 3/30/11 11:29:08 PM

3. r 0 s -2 = 8 0 10 -2

= 1 · 1 ___ 10 2

= 1 ______ 10 · 10

= 1 ____ 100

4. 5 k -3 = 5 · k -3

= 5 · 1 __ k 3

= 5 __ k 3

5. x 4 ___ y -6

= x 4 · 1 ___ y -6

= x 4 · y 6 = x 4 y 6

6. 8 f -4 g 0 = 8 · f -4 · g 0

= 8 · 1 __ f 4

· 1

= 8 __ f 4

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. √ x 8 y 4 = ( x 8 y 4 ) 1 __ 2

= ( x 8 ) 1 __ 2

( y 4 )

1 __ 2

= ( x 8· 1 __

2 ) ( y

4· 1 __ 2

)

= ( x 4 ) ( y 2 ) = x 4 y 2

14. 3

√ r 9 = ( r 9 ) 1 __ 3

= r 9· 1 __

3 = r 3

15. 6

√ z 12 = ( z 12 ) 1 __ 6

= z 12· 1 __

6 = z 2

16. 3 √ p 3 q 12 = ( p 3 q 12 )

1 __ 3

= ( p 3 ) 1 __ 3 ( q 12 )

1 __ 3

= ( p 3· 1 __

3 ) ( q

12· 1 __ 3 )

= ( p 1 ) ( q 4 ) = p q 4

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 0 The degree of the polynomial is 1.

b. x 3 y 2 : degree 5 - x 4 : degree 4

x 2 y 3 : degree 5 2: degree 0

The degree of the polynomial is 5.

3a. 16 - 4 x 2 + x 5 + 9 x 3 → x 5 + 9 x 3 - 4 x 2 + 16 The leading coefficient is 1.

b. 18 y 5 - 3 y 8 + 14y → -3 y 8 + 18 y 5 + 14y The leading coefficient is -3.

4a. Degree: 3 Terms: 4 x 3 + x 2 -x + 2 is a cubic polynomial.

b. Degree: 0 Terms: 1 6 is a constant monomial.

c. Degree: 8 Terms: 3 -3 y 8 + 18 y 5 + 14y is an 8th-degree trinomial.

5. -16 t 2 + 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: 2 x 2 + 3 x -3 contains an expression with a negative exponent. 1 - a __

b

contains a variable within a denominator.

2. Polynomials

x2 Monomials

3x + 2 Binomials

2x2 + 6x - 7 Trinomials


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


CS10_A1_MESK710372_C06.indd 203 3/30/11 11:29:10 PM

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

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

9. 0.75 a 2 b: degree 3 2 a 3 b 5 : degree 8 The degree of the polynomial is 8.

10. 15y: degree 1 -84 y 3 : degree 3 100: degree 0 -3 y 2 : degree 2 The degree of the polynomial is 3.

11. r 3 : degree 3 r 2 : degree 2 -5: degree 0 The degree of the polynomial is 3.

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

13. 3 k 4 : degree 4 k 3 : degree 3 -2 k 2 : degree 2 k: degree 1 The degree of the polynomial is 4.

14. -2b + 5 + b 2 → b 2 - 2b + 5 The leading coefficient is 1.

15. 9 a 8 - 8 a 9 → -8 a 9 + 9 a 8 The leading coefficient is -8.

16. 5 s 2 - 3s + 3 - s 7 → - s 7 + 5 s 2 - 3s + 3 The leading coefficient is -1.

17. 2x + 3 x 2 - 1 → 3 x 2 + 2x - 1 The leading coefficient is 3.

18. 5g - 7 + g 2 → g 2 + 5g - 7 The leading coefficient is 1.

19. 3 c 2 + 5 c 4 + 5 c 3 - 4 → 5 c 4 + 5 c 3 + 3 c 2 - 4 The leading coefficient is 5.

20. Degree: 2 Terms: 3 x 2 + 2x + 3 is a quadratic trinomial.

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

22. Degree: 4 Terms: 3 8 + k + 5 k 4 is a quartic trinomial.

23. Degree: 4 Terms: 4 q 2 + 6 - q 3 + 3 q 4 is a quartic polynomial.

24. Degree: 3 Terms: 2 5 k 2 + 7 k 3 is a cubic binomial.

25. Degree: 4 Terms: 3 2 a 3 + 4 a 2 - a 4 is a quartic trinomial.

26. 3.14 r 2 + 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. a 2 : degree 2 a 4 : degree 4 -6a: degree 1 The degree of the polynomial is 4.

36. 3 2 b: degree 1 -5: degree 0 The degree of the polynomial is 1.

37. 3.5 y 2 : degree 2 -4.1y: degree 1 -6: degree 0 The degree of the polynomial is 2.

38. -5 f 4 : degree 4 2 f 6 : degree 6 10 f 8 : degree 8 The degree of the polynomial is 8.

39. 4 n 3 : degree 3 -2n: degree 1 The degree of the polynomial is 3.

40. 4 r 3 : degree 3 4 r 6 : degree 6 The degree of the polynomial is 6.

41. 2.5 + 4.9 t 3 - 4 t 2 + t → 4.9 t 3 - 4 t 2 + t + 2.5 The leading coefficient is 4.9.

42. 8a - 10 a 2 + 2 → -10 a 2 + 8a + 2 The leading coefficient is -10.

43. x 7 - x + x 3 - x 5 + x 10 → x 10 + x 7 - x 5 + x 3 - x The leading coefficient is 1.

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

45. 3 x 2 + 5x - 4 + 5 x 3 → 5 x 3 + 3 x 2 + 5x - 4 The leading coefficient is 5.

46. -2n + 1 - n 2 → - n 2 - 2n + 1 The leading coefficient is -1.

47. 4d + 3 d 2 - d 3 + 5 → - d 3 + 3 d 2 + 4d + 5 The leading coefficient is -1.

48. 3 s 2 + 12 s 3 + 6 → 12 s 3 + 3 s 2 + 6 The leading coefficient is 12.

49. 4 x 2 - x 5 - x 3 + 1 → - x 5 - x 3 + 4 x 2 + 1 The leading coefficient is -1.

50. Degree: 0 Terms: 1 12 is a constant monomial.

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

52. Degree: 3 Terms: 3 3.5 x 3 - 4.1x - 6 is a cubic trinomial.

53. Degree: 2 Terms: 3 4g + 2 g 2 - 3 is a quadratic trinomial.

54. Degree: 2 Terms: 2 2 x 2 - 6x is a quadratic binomial.

55. Degree: 4 Terms: 3 6 - s 3 - 3 s 4 is a quartic trinomial.


CS10_A1_MESK710372_C06.indd 204 3/30/11 11:29:12 PM

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.

77. J -3 x 3 + 4 x 2 - 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. 0.016 m 3 - 0.390 m 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

0.016 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

6-4


CS10_A1_MESK710372_C06.indd 205 3/30/11 11:29:14 PM

c. 2 x 8 + 7 y 8 - x 8 - y 8 = 2 x 8 - x 8 + 7 y 8 - y 8 = x 8 + 6 y 8

d. 9 b 3 c 2 + 5 b 3 c 2 - 13 b 3 c 2 = b 3 c 2

2. (5 a 3 + 3 a 2 - 6a + 12 a 2 ) + (7 a 3 - 10a)

= (5 a 3 + 7 a 3 ) + (3 a 2 + 12 a 2 ) + (-6a - 10a) = 12 a 3 + 15 a 2 - 16a

3. (2 x 2 - 3 x 2 + 1) - ( x 2 + x + 1)

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

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

4. (-0.03 x 2 + 25x - 1500)

______________________ ______________________ + (-0.02 x 2 + 21x - 1700) -0.05 x 2 + 46x - 3200

think and disCuss

1. -12 x 2 and -9 x 2 ; -4.7y and y; 1 __ 5 x 2 y and 5 x 2 y

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

3.

Adding: Subtracting:

Polynomials

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

n - 8 m + 12) + ( - 2 m 5 n - m + 1) =

14 m 5 n - 9 m + 13

(18 a 2 b + 9 a 2 + b ) + (7 a 2 b + 6 a 2 + 2 b ) = 25 a 2 b + 15 a 2 + 3 b


1. 7 a 2 - 10 a 2 + 9a = -3 a 2 + 9a

2. 13 x 2 + 9 y 2 - 6 x 2 = 13 x 2 - 6 x 2 + 9 y 2 = 7 x 2 + 9 y 2

3. 0.07 r 4 + 0.32 r 3 + 0.19 r 4 = 0.07 r 4 + 0.19 r 4 + 0.32 r 3 = 0.26 r 4 + 0.32 r 3

4. 1 __ 4 p 3 + 2 __

3 p 3

= 11 ___ 12

p 3

5. 5 b 3 c + b 3 c - 3 b 3 c = 3 b 3 c

6. -8m + 5 - 16 + 11m = -8m + 11m + 5 - 16 = 3m - 11

7. (5 n 3 + 3n + 6) + (18 n 3 + 9)

= (5 n 3 + 18 n 3 ) + 3n + (6 + 9) = 23 n 3 + 3n + 15

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

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

= 8 q 2 - 10.9q + 5.3

9. (-3x + 12) + (9 x 2 + 2x - 18) = 9 x 2 + (-3x + 2x) + (12 - 18) = 9 x 2 - x - 6

10. (9 x 4 + x 3 ) + (2 x 4 + 6 x 3 - 8 x 4 + x 3 )

= (9 x 4 + 2 x 4 - 8 x 4 ) + ( x 3 + 6 x 3 + x 3 ) = 3 x 4 + 8 x 3

11. (6 c 4 + 8c + 6) - (2 c 4 )

= (6 c 4 + 8c + 6) + (-2 c 4 )

= (6 c 4 - 2 c 4 ) + 8c + 6 = 4 c 4 + 8c + 6

12. (16 y 2 - 8y + 9) - (6 y 2 - 2y + 7y)

= (16 y 2 - 8y + 9) + (-6 y 2 + 2y - 7y)

= (16 y 2 - 6 y 2 ) + (-8y + 2y - 7y) + 9

= 10 y 2 - 13y + 9

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

14. (-7 k 2 + 3) - (2 k 2 + 5k - 1)

= (-7 k 2 + 3) + (-2 k 2 - 5k + 1)

= (-7 k 2 - 2 k 2 ) + (-5k) + (3 + 1) = -9 k 2 - 5k + 4

15. m∠ABD = (8 a 2 - 2a + 5) + (7a + 4) = 8 a 2 + (-2a + 7a) + (5 + 4) = 8 a 2 + 5a + 9


16. 4 k 3 + 6 k 2 + 9 k 3 = 4 k 3 + 9 k 3 + 6 k 2 = 13 k 3 + 6 k 2

17. 5m + 12 n 2 + 6n - 8m = 5m - 8m + 12 n 2 + 6n = 12 n 2 + 6n - 3m

18. 2.5 a 4 - 8.1 b 4 - 3.6 b 4 = 2.5 a 4 - 11.7 b 4

19. 2 d 5 + 1 - d 5 = 2 d 5 - d 5 + 1 = d 5 + 1

20. 7xy - 4 x 2 y - 2xy = 7xy - 2xy - 4 x 2 y = -4 x 2 y + 5xy

21. -6 x 3 + 5x + 2 x 3 + 4 x 3 = -6 x 3 + 2 x 3 + 4 x 3 + 5x = 5x

22. x 2 + x + 3x + 2 x 2 = x 2 + 2 x 2 + x + 3x = 3 x 2 + 4x

23. 3 x 3 - 4 - x 3 - 1 = 3 x 3 - x 3 - 4 - 1 = 2 x 3 - 5


CS10_A1_MESK710372_C06.indd 206 3/30/11 11:29:16 PM

24. 3 b 3 - 2b - 1 - b 3 - b = 3 b 3 - b 3 - 2b - b - 1 = 2 b 3 - 3b - 1

25. (2 t 2 - 8t) + (8 t 2 + 9t)

= (2 t 2 + 8 t 2 ) + (-8t + 9t) = 10 t 2 + t‹

26. (-7 x 2 - 2x + 3) + (4 x 2 - 9x)

= (-7 x 2 + 4 x 2 ) + (-2x - 9x) + 3 = -3 x 2 - 11x + 3

27. ( x 5 - x) + ( x 4 + x)

= ( x 5 + x 4 ) + (-x + x) = x 5 + x 4

28. (-2 z 3 + z + 2 z 3 + z) + (3 z 3 - 5 z 2 )

= (-2 z 3 + 2 z 3 + 3 z 3 ) + (-5 z 2 ) + (z + z) = 3 z 3 - 5 z 2 + 2z

29. ( t 3 + 8 t 2 ) - (3 t 3 )

= ( t 3 + 8 t 2 ) + (-3 t 3 )

= ( t 3 - 3 t 3 ) + 8 t 2 = -2 t 3 + 8 t 2

30. (3 x 2 - x) - ( x 2 + 3x - x)

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

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

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

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

32. (3 s 2 + 4s) - (-10 s 2 + 6s)

= (3 s 2 + 4s) + (10 s 2 - 6s)

= (3 s 2 + 10 s 2 ) + (4s - 6s) = 13 s 2 - 2s

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

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

= (6 w 2 + 8) + (-2 w 2 + 6w - 4)

= (6 w 2 - 2 w 2 ) + 6w + (8 - 4) = 4 w 2 + 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. (4 m 2 + 3m) + (-2 m 2 )

= (4 m 2 - 2 m 2 ) + 3m = 2 m 2 + 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. (4 x 2 + 3x - 6) + (2 x 2 - 4x + 5)

= (4 x 2 + 2 x 2 ) + (3x - 4x) + (-6 + 5) = 6 x 2 - x - 1

40. (2 z 2 - 3z - 3) + (2 z 2 - 7z - 1)

= (2 z 2 + 2 z 2 ) + (-3z - 7z) + (-3 - 1) = 4 z 2 - 10z - 4

41. (5 u 2 + 3u + 7) - ( u 3 + 2 u 2 + 1)

= (5 u 2 + 3u + 7) + (- u 3 - 2 u 2 - 1)

= (- u 3 ) + (5 u 2 - 2 u 2 ) + 3u + (7 - 1) = - u 3 + 3 u 2 + 3u + 6

42. (-7 h 2 - 4h + 7) - (7 h 2 - 4h + 11)

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

= (-7 h 2 - 7 h 2 ) + (-4h + 4h) + (7 - 11) = -14 h 2 - 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 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.


CS10_A1_MESK710372_C06.indd 207 3/30/11 11:29:20 PM

53a. x + 4

x - 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 -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.

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 ___ -3 6x = 47

6x ___ 6 = 47 ___

6

x ≈ 7.83 7; 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 = (2 x 3 + 3 x 2 + 8) - 2 ( x 3 + 5)

= (2 x 3 + 3 x 2 + 8) + (-2 x 3 - 2(5))

= (2 x 3 + 3 x 2 + 8) + (-2 x 3 - 10)

= (2 x 3 - 2 x 3 ) + 3 x 2 + (8 - 10) = 3 x 2 - 2

58. Possible answer: 2 m 3 + 2m, 2 m 3 + m

59. Possible answer: 5 m 3 + 2m, m 3 - m

60. Possible answer: 2 m 3 + m, m 3 + m

+ m 3 + m

61. Possible answer: 4 m 3 + 3m

62. Possible answer: 2 m 3 + m 2 + m, m 3 + m 2 + m, m 3 - 2 m 2 + m

muLtiPLying PoLynomiaLs

CheCk it out!

1a. (3 x 3 ) (6 x 2 )

= (3 · 6) ( x 3 · x 2 ) = 18 x 5

b. (2 r 2 t) (5 t 3 )

= (2 · 5) ( r 2 ) (t · t 3 ) = 10 r 2 t 4

c. ( 1 __ 3 x 2 y) (12 x 3 z 2 ) ( y 4 z 5 )

= ( 1 __ 3 · 12) ( x 2 · x 3 ) (y · y 4 ) ( z 2 · z 5 )

= 4 x 5 y 5 z 7

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

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

b. 3ab (5 a 2 + b)

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

= (3 · 5) (a · a 2 ) (b) + (3)(a)(b · b) = 15 a 3 b + 3a b 2

c. 5 r 2 s 2 (r - 3s) = 5 r 2 s 2 (r) + 5 r 2 s 2 (-3s)

= (5) ( r 2 · r) ( s 2 ) + (5 · (-3)) ( r 2 ) ( s 2 · s)

= 5 r 3 s 2 - 15 r 2 s 3

3a. (a + 3)(a - 4) = a(a) + a(-4) + 3(a) + 3(-4) = a 2 - 4a + 3a - 12 = a 2 - a - 12

b. (x - 3) 2 = (x - 3)(x - 3) = x(x) + x(-3) - 3(x) - 3(-3) = x 2 - 3x - 3x + 9 = x 2 - 6x + 9

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

= 2a(a) + 2a (4 b 2 ) - b 2 (a) - b 2 (4 b 2 ) = 2 a 2 + 8a b 2 - a b 2 - 4 b 4 = 2 a 2 + 7a b 2 - 4 b 4

4a. (x + 3) ( x 2 - 4x + 6)

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

= x ( x 2 ) + x(-4x) + x(6) + 3 ( x 2 ) + 3(-4x) + 3(6) = x 3 - 4 x 2 + 6x + 3 x 2 - 12x + 18 = x 3 - x 2 - 6x + 18

6-5


CS10_A1_MESK710372_C06.indd 208 3/30/11 11:29:22 PM

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

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

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

= 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) =


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


CS10_A1_MESK710372_C06.indd 209 3/30/11 11:29:25 PM

15. (x - 2) 2 = (x - 2)(x - 2) = x(x) + x(-2) - 2(x) - 2(-2) = x 2 - 2x - 2x + 4 = x 2 -4x + 4

16. (y - 3)(y - 5) = y(y) + y(-5) - 3(y) - 3(-5) = y 2 - 5y - 3y + 15 = y 2 - 8y + 15

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

= 4 a 3 (a) + 4 a 3 (-3 b 2 ) - 2b(a) - 2b (-3 b 2 ) = 4 a 4 - 2ab - 12 a 3 b 2 + 6 b 3

18. ( m 2 - 2mn) (3mn + n 2 )

= m 2 (3mn) + m 2 ( n 2 ) - 2mn(3mn) - 2mn ( n 2 ) = 3 m 3 n + m 2 n 2 - 6 m 2 n 2 - 2m n 3 = 3 m 3 n - 5 m 2 n 2 - 2m n 3

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

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

= x ( x 2 ) + x(-2x) + x(3) + 5 ( x 2 ) + 5(-2x) + 5(3) = x 3 - 2 x 2 + 3x + 5 x 2 - 10x + 15 = x 3 + 3 x 2 - 7x + 15

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

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

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

= 3 x 3 - 15 x 2 + 6x + 4 x 2 - 20x + 8 = 3 x 3 - 11 x 2 - 14x + 8

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

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

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

= -6 x 4 + 4 x 2 - 10x + 12 x 3 - 8x + 20 = -6 x 4 + 12 x 3 + 4 x 2 - 18x + 20

22. (-4x + 6) (2 x 3 - x 2 + 1)

= -4x (2 x 3 - x 2 + 1) + 6 (2 x 3 - x 2 + 1)

= -4x (2 x 3 ) -4x (- x 2 ) -4x(1) + 6 (2 x 3 ) + 6 (- x 2 ) + 6(1)

= -8 x 4 + 4 x 3 - 4x + 12 x 3 - 6 x 2 + 6 = -8 x 4 + 16 x 3 - 6 x 2 - 4x + 6

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

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

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

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

= ( a 2 - ab + ab - b 2 ) (b - a)

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

25a. A = ℓw = (2x - 3)(x) = 2x(x) - 3(x) = 2 x 2 - 3x The area is represented by 2 x 2 - 3x.

b. A = 2 x 2 - 3x = 2 (4) 2 - 3(4) = 2(16) - 3(4) = 32 - 12 = 20 The area is 20 in 2 .


26. (3 x 2 ) (8 x 5 )

= (3 · 8) ( x 2 · x 5 ) = 24 x 7

27. (-2 r 3 s 4 ) (6 r 2 s)

= (-2 · 6) ( r 3 · r 2 ) ( s 4 · s) = -12 r 5 s 5

28. (15x y 2 ) ( 1 __ 3 x 2 z 3 ) ( y 3 z 4 )

= (15 · 1 __ 3 ) (x · x 2 ) ( y 2 · y 3 ) ( z 3 · z 4 )

= 5 x 3 y 5 z 7

29. (-2 a 3 ) (-5a)

= (-2 · (-5)) ( a 3 · a)

= 10 a 4

30. (6 x 3 y 2 ) (-2 x 2 y)

= (6 · (-2)) ( x 3 · x 2 ) ( y 2 · y)

= -12 x 5 y 3

31. (-3 a 2 b) (-2 b 3 ) (- a 3 b 2 )

= (-3 · (-2) · (-1)) ( a 2 · a 3 ) (b · b 3 · b 2 )

= -6 a 5 b 6

32. (7 x 2 ) (x y 5 ) (2 x 3 y 2 )

= (7 · 2) ( x 2 · x · x 3 ) ( y 5 · y 2 )

= 14 x 6 y 7

33. (-4 a 3 b c 2 ) ( a 3 b 2 c) (3a b 4 c 5 )

= (-4 · 3) ( a 3 · a 3 · a) (b · b 2 · b 4 ) ( c 2 · c · c 5 ) = -12 a 7 b 7 c 8

34. (12m n 2 ) (2 m 2 n) (mn)

= (12 · 2) (m · m 2 · m) ( n 2 · n · n) = 24 m 4 n 4


CS10_A1_MESK710372_C06.indd 210 3/30/11 11:29:28 PM

35. 9s(s + 6) = 9s(s) + 9s(6) = 9 s 2 + 54s

36. 9 (2 x 2 - 5x)

= 9 (2 x 2 ) + 9(-5x) = 18 x 2 - 45x

37. 3x (9 x 2 - 4x)

= 3x (9 x 2 ) + 3x(-4x) = 27 x 3 - 12 x 2

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

= 3 (2 x 2 ) + 3(5x) + 3(4) = 6 x 2 + 15x + 12

39. 5 s 2 t 3 (2s - 3 t 2 )

= 5 s 2 t 3 (2s) + 5 s 2 t 3 (-3 t 2 )

= (5 · 2) ( s 2 · s) ( t 3 ) + (5 · (-3)) ( s 2 ) ( t 3 · t 2 )

= 10 s 3 t 3 - 15 s 2 t 5

40. x 2 y 3 · 5 x 2 y (6x + y 2 )

= (5) ( x 2 · x 2 ) ( y 3 · y) (6x + y 2 )

= 5 x 4 y 4 (6x + y 2 )

= 5 x 4 y 4 (6x) + 5 x 4 y 4 ( y 2 )

= (5 · 6) ( x 4 · x) ( y 4 ) + (5) ( x 4 ) ( y 4 · y 2 )

= 30 x 5 y 4 + 5 x 4 y 6

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

= -5x (2 x 2 ) - 5x(-3x) - 5x(-1) = -10 x 3 + 15 x 2 + 5x

42. -2 a 2 b 3 (3a b 2 - a 2 b)

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

= (-2 · 3) ( a 2 · a) ( b 3 · b 2 ) - (2 · -1) ( a 2 · a 2 ) ( b 3 · b) = -6 a 3 b 5 + 2 a 4 b 4

43. -7 x 3 y · x 2 y 2 (2x - y)

= (-7) ( x 3 · x 2 ) (y · y 2 ) (2x - y)

= -7 x 5 y 3 (2x - y) = -7 x 5 y 3 (2x) - 7 x 5 y 3 (-y)

= (-7 · 2) ( x 5 · x) ( y 3 ) + (-7 · (-1)) ( x 5 ) ( y 3 · y)

= -14 x 6 y 3 + 7 x 5 y 4

44. (x + 5)(x - 3) = x(x) + x(-3) + 5(x) + 5(-3) = x 2 - 3x + 5x - 15 = x 2 + 2x - 15

45. (x + 4) 2 = (x + 4)(x + 4) = x(x) + x(4) + 4(x) + 4(4) = x 2 + 4x + 4x + 16 = x 2 + 8x + 16

46. (m - 5) 2 = (m - 5)(m - 5) = m(m) + m(-5) - 5(m) - 5(-5) = m 2 - 5m - 5m + 25 = m 2 - 10m + 25

47. (5x - 2)(x + 3) = 5x(x) + 5x(3) - 2(x) - 2(3) = 5 x 2 + 15x - 2x - 6 = 5 x 2 + 13x - 6

48. (3x - 4) 2 = (3x - 4)(3x - 4) = 3x(3x) + 3x(-4) - 4(3x) - 4(-4) = 9 x 2 - 12x -12x + 16 = 9 x 2 - 24x + 16

49. (5x + 2) (2x - 1) = 5x (2x) + 5x (-1) + 2 (2x) + 2 (-1) = 10 x 2 - 5x + 4x - 2= 10 x 2 - x - 2

50. (x - 1)(x - 2) = x(x) + x(-2) - 1(x) - 1(-2) = x 2 - 2x - x + 2 = x 2 - 3x + 2

51. (x - 8)(7x + 4) = x(7x) + x(4) - 8(7x) - 8(4) = 7 x 2 + 4x - 56x - 32 = 7 x 2 - 52x - 32

52. (2x + 7)(3x + 7) = 2x(3x) + 2x(7) + 7(3x) + 7(7) = 6 x 2 + 14x + 21x + 49 = 6 x 2 + 35x + 49

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

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

= x ( x 2 ) + x(-3x) + x(5) + 2 ( x 2 ) + 2(-3x) + 2(5) = x 3 - 3 x 2 + 5x + 2 x 2 - 6x + 10 = x 3 - x 2 - x + 10

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

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

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

= 2 x 3 - 8 x 2 + 6x + 5 x 2 - 20x + 15 = 2 x 3 - 3 x 2 - 14x + 15

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

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

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

= -10 x 4 + 20 x 2 - 15x + 2 x 3 - 4x + 3 = -10 x 4 + 2 x 3 + 20 x 2 - 19x + 3

56. (x - 3) ( x 2 - 5x + 6)

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

= x ( x 2 ) + x(-5x) + x(6) - 3 ( x 2 ) - 3(-5x) - 3(6) = x 3 - 5 x 2 + 6x - 3 x 2 + 15x - 18 = x 3 - 8 x 2 + 21x - 18

57. (2 x 2 - 3) (4 x 3 - x 2 + 7)

= 2 x 2 (4 x 3 - x 2 + 7) - 3 (4 x 3 - x 2 + 7)

= 2 x 2 (4 x 3 ) + 2 x 2 (- x 2 ) + 2 x 2 (7) - 3 (4 x 3 ) - 3 (- x 2 ) - 3(7)

= 8 x 5 - 2 x 4 + 14 x 2 - 12 x 3 + 3 x 2 - 21 = 8 x 5 - 2 x 4 - 12 x 3 + 17 x 2 - 21


CS10_A1_MESK710372_C06.indd 211 3/30/11 11:29:31 PM

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

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

= ( x 2 - 8x + 16) (x - 4)

= (x - 4) ( x 2 - 8x + 16)

= x ( x 2 - 8x + 16) - 4 ( x 2 - 8x + 16)

= x ( x 2 ) + x(-8x) + x(16) - 4 ( x 2 ) - 4(-8x) - 4(16) = x 3 - 8 x 2 + 16x - 4 x 2 + 32x - 64 = x 3 - 12 x 2 + 48x - 64

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

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

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

60. (2x + 10) (4 - x + 6 x 3 )

= 2x (4 - x + 6 x 3 ) + 10 (4 - x + 6 x 3 )

= 2x(4) + 2x(-x) + 2x (6 x 3 ) + 10(4) + 10(-x) + 10 (6 x 3 )

= 8x - 2 x 2 + 12 x 4 + 40 - 10x + 60 x 3 = 12 x 4 + 60 x 3 - 2 x 2 - 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 + x 2 ) (1 - x)

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

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

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

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

62a. A = ℓw = (x + 3)(x) = x(x) + 3(x) = x 2 + 3x The area is represented by x 2 + 3x.

b. A = x 2 + 3x = (5) 2 + 3(5) = 25 + 15 = 40 The area is 40 ft 2 .

63. A = s 2 = (4x - 6) 2 = (4x - 6)(4x - 6) = 4x(4x) + 4x(-6) - 6(4x) - 6(-6) = 16 x 2 - 24x - 24x + 36 = 16 x 2 - 48x + 36 The area is represented by 16 x 2 - 48x + 36.

64a.

x + 4

x + 1

b. A = ℓw = (x + 4)(x + 1) = x(x) + x(1) + 4(x) + 4(1) = x 2 + x + 4x + 4 = x 2 + 5x + 4 The area is represented by x 2 + 5x + 4.

c. A = x 2 + 5x + 4 = (4) 2 + 5(4) + 4 = 16 + 20 + 4 = 40 The area is 40 ft 2 .

adegree

of aB

degree of B

a · Bdegree of a · B

2 x 2 2 3 x 5 5 6 x 7 7 65a. 5 x 3 3 2 x 2 + 1 2 10 x 5 +

5 x 3 5

b. x 2 + 2 2 x 2 - x 2 x 4 - x 3 + 2 x 2 - 2x

4

c. x - 3 1 x 3 - 2 x 2 + 1

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

x - 3

4

d. m + n

66. A = ℓw = (2x + 3)(4x) = 2x(4x) + 3(4x) = 8 x 2 + 12x The area is represented by 8 x 2 + 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) = 12 x 2 + 6x + 6x + 3 = 12 x 2 + 12x + 3 The area is represented by 12 x 2 + 12x + 3.

68. A = ℓw = (x - 5)(x - 5) = x(x) + x(-5) - 5(x) - 5(-5) = x 2 - 5x - 5x + 25 = x 2 - 10x + 25 The area is represented by x 2 - 10x + 25.

69a. A = ℓw = (2x)(x) = 2 x 2 The area is represented by 2 x 2 .

b. A = 2 x 2 = 2 (20) 2 = 2(400) = 800 The area is 800 m 2 .


CS10_A1_MESK710372_C06.indd 212 3/30/11 11:29:34 PM

70. (1.5 a 3 ) (4 a 6 )

= (1.5 · 4) ( a 3 · a 6 ) = 6 a 9

71. (2x + 5)(x - 6) = 2x(x) + 2x(-6) + 5(x) + 5(-6) = 2 x 2 - 12x + 5x - 30 = 2 x 2 - 7x - 30

72. (3g - 1)(g + 5) = 3g(g) + 3g(5) - 1(g) - 1(5) = 3 g 2 + 15g - g - 5 = 3 g 2 + 14g - 5

73. (4x - 2y)(2x - 3y) = 4x(2x) + 4x(-3y) - 2y(2x) - 2y(-3y) = 8 x 2 - 12xy - 4xy + 6 y 2 = 8 x 2 - 16xy + 6 y 2

74. (x + 3)(x - 3) = x(x) + x(-3) + 3(x) + 3(-3) = x 2 - 3x + 3x - 9 = x 2 - 9

75. (1.5x - 3)(4x + 2) = 1.5x(4x) + 1.5x(2) - 3(4x) - 3(2) = 6 x 2 + 3x - 12x - 6 = 6 x 2 - 9x - 6

76. (x - 10)(x + 4) = x(x) + x(4) - 10(x) - 10(4) = x 2 + 4x - 10x - 40 = x 2 - 6x - 40

77. x 2 (x + 3) = x 2 (x) + x 2 (3) = x 3 + 3 x 2

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

= x ( x 2 ) + x(2x) + 1 ( x 2 ) + 1(2x) = x 3 + 2 x 2 + x 2 + 2x = x 3 + 3 x 2 + 2x

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

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

= x (2 x 2 ) + x(x) + x(-6) - 4 (2 x 2 ) - 4(x) - 4(-6) = 2 x 3 + x 2 - 6x - 8 x 2 - 4x + 24 = 2 x 3 - 7 x 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)

= ( a 2 - ab + ab - b 2 ) (a - b)

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

81. (2p - 3q) 3 = (2p - 3q)(2p - 3q)(2p - 3q) = (2p(2p) + 2p(-3q) - 3q(2p) - 3q(-3q)) ( 2p - 3q)

= (4 p 2 - 6pq - 6pq + 9 q 2 ) (2p - 3q)

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

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

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

= 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.

84. A = ℓwh = (x + 5)(x)(x + 2) = (x(x) + 5(x)) (x + 2)

= ( 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


CS10_A1_MESK710372_C06.indd 213 3/30/11 11:29:37 PM

88. H

2a ( a 2 - 1)

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

89. D 3 x 3 y 2 z · x 2 yz

= (3) ( x 3 · x 2 ) ( y 2 · y) (z · z)

= 3 x 5 y 3 z 2 This has degree 5 + 3 + 2 = 10.


90. 6 x 2 - 2 (3 x 2 - 2x + 4)

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

91. x 2 - 2x(x + 3) = x 2 - 2x(x) - 2x(3) = x 2 - 2 x 2 - 6x = - x 2 - 6x

92. x(4x - 2) + 3x(x + 1) = x(4x) + x(-2) + 3x(x) + 3x(1) = 4 x 2 - 2x + 3 x 2 + 3x = 7 x 2 + x

93a. A = ℓw = (x + 1)(x - 1) = x(x) + x(-1) + 1(x) + 1(-1) = x 2 - x + x - 1 = x 2 - 1 The area is represented by x 2 - 1.

b. A = ℓw = (x + 5)(x + 3) - (x + 1)(x - 1)

= x(x) + x(3) + 5(x) + 5(3) - ( x 2 - 1) = x 2 + 3x + 5x + 15 - x 2 + 1 = 8x + 16

94. A = s 2 = (8 + 2x) 2 = (8 + 2x)(8 + 2x) = 8(8) + 8(2x) + 2x(8) + 2x(2x) = 64 + 16x + 16x + 4 x 2 = 4 x 2 + 32x + 64

P = 4s

= 4 ( x 2 + 48)

= 4 ( x 2 ) + 4(48) = 4 x 2 + 192

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

6-6


CS10_A1_MESK710372_C06.indd 214 3/30/11 11:29:39 PM

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

(3 + 2 y 2 ) (3 - 2 y 2 ) = (3) 2 - (2 y 2 ) 2

= 9 - 4 y 4

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


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.


21. (a + b) 2 = a 2 + 2ab + b 2 (x + 3) 2 = (x) 2 + 2(x)(3) + (3) 2 = x 2 + 6x + 9


CS10_A1_MESK710372_C06.indd 215 3/30/11 11:29:42 PM

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


CS10_A1_MESK710372_C06.indd 216 3/30/11 11:29:45 PM

47. (a + b) 2 = a 2 + 2ab + b 2 (1 + x) 2 = (1) 2 + 2(1)(x) + (x) 2 = 1 + 2x + x 2

48. (a - b)(a + b) = a 2 - b 2 (1 - x)(1 + x) = (1) 2 - (x) 2 = 1 - x 2

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

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

2

= x 6 - 2 x 3 a 3 + a 6

50. (a + b)(a + b) = a 2 + 2ab + b 2 (5 + n)(5 + n) = (5) 2 + 2(5)(n) + (n) 2 = 25 + 10n + n 2

51. (a - b)(a + b) = a 2 - b 2 (6a - 5b)(6a + 5b) = (6a) 2 - (5b) 2 = 36 a 2 - 25 b 2

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

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

= r 2 - 8r t 4 + 16 t 8

a b (a - b) 2 a 2 - 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 a 2 + 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) a 2 - 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 = a 2 - 2ab - b 2 = 16 x 2 - 24x + c Therefore, a 2 = 16 x 2 = (4x) 2 . So a = ±4x.

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

-24x _____ ∓8x

= ∓8xb

_____ ∓8x

±3 = b So c = b 2 = (±3) 2 = 9.

63. Possible answer: The square of a difference is not the same as a difference of squares; a 2 - 2ab + b 2 .

64a.

x + 3

x - 3

b. A = ℓw = (x + 3)(x - 3) = (x) 2 - (3) 2 = x 2 - 9 The area is represented by x 2 - 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 = 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.


CS10_A1_MESK710372_C06.indd 217 3/30/11 11:29:48 PM


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

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

= ( x 2 + 8x + 16) (x - 4)

= (x - 4) ( x 2 + 8x + 16)

= x ( x 2 + 8x + 16) - 4 ( x 2 + 8x + 16)

= x ( x 2 ) + x(8x) + x(16) - 4 ( x 2 ) - 4(8x) - 4(16) = x 3 + 8 x 2 + 16x - 4 x 2 - 32x - 64 = x 3 + 4 x 2 - 16x - 64

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

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

= ( x 2 - 16) (x - 4) = x 2 (x) + x 2 (-4) - 16(x) - 16(-4) = x 3 - 4 x 2 - 16x + 64

73. Let x 2 + bx + c = x 2 + bx + (± √ c ) 2 since c = (± √ c ) 2 .

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

(x± √ c )(x± √ c ) = x 2 ± 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. 4 r 2 + 2 r 6 - 3r → 2 r 6 + 4 r 2 - 3r The leading coefficient is 2.

2. y 2 + 7 - 8 y 3 + 2y → -8 y 3 + y 2 + 2y + 7 The leading coefficient is -8.

3. -12 t 3 - 4t + t 4 → t 4 - 12 t 3 - 4t The leading coefficient is 1.

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

5. 2 + 3 x 3 → 3 x 3 + 2 The leading coefficient is 3.

6. -3 a 2 + 16 + a 7 + a → a 7 - 3 a 2 + a + 16 The leading coefficient is 1.

7. Degree: 3 Terms: 3 2 x 3 + 5x - 4 is a cubic trinomial.

8. Degree: 2 Terms: 1 5 b 2 is a quadratic monomial.

9. Degree: 4 Terms: 4 6 p 2 + 3p - p 4 + 2 p 3 is a quartic polynomial.

10. Degree: 2 Terms: 3 x 2 + 12 - x is a quadratic trinomial.

11. Degree: 7 Terms: 4 -2 x 3 - 5 + x - 2 x 7 is a 7th-degree polynomial.

12. Degree: 4 Terms: 4 5 - 6 b 2 + b - 4 b 4 is a quartic polynomial.

13. C(x) = x 3 - 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. (10 m 3 + 4 m 2 ) + (7 m 2 + 3m)

= 10 m 3 + (4 m 2 + 7 m 2 ) + 3m = 10 m 3 + 11 m 2 + 3m

15. (3 t 2 - 2t) + (9 t 2 + 4t - 6)

= (3 t 2 + 9 t 2 ) + (-2t + 4t) + (-6) = 12 t 2 + 2t - 6

16. (12 d 6 - 3 d 2 ) + (2 d 4 + 1) = 12 d 6 + 2 d 4 - 3 d 2 + 1

17. (6 y 3 + 4 y 2 ) - (2 y 2 + 3y)

= (6 y 3 + 4 y 2 ) + (-2 y 2 - 3y)

= 6 y 3 + (4 y 2 - 2 y 2 ) + (-3y)

= 6 y 3 + 2 y 2 - 3y

18. (7 n 2 - 3n) - (5 n 2 + 5n)

= (7 n 2 - 3n) + (-5 n 2 - 5n)

= (7 n 2 - 5 n 2 ) + (-3n - 5n) = 2 n 2 - 8n

19. ( b 2 - 10) - (-5 b 3 + 4b)

= ( b 2 - 10) + (5 b 3 - 4b) = 5 b 3 + b 2 - 4b - 10

20. P = (2 s 3 + 4) + (4 s 2 + 1) + (5s) = 2 s 3 + 4 s 2 + 5s + (4 + 1) = 2 s 3 + 4 s 2 + 5s + 5

21. 2 h 3 · 5 h 5

= (2 · 5) ( h 3 · h 5 ) = 10 h 8

22. ( s 8 t 4 ) (-6s t 3 )

= (-6) ( s 8 · s) ( t 4 · t 3 ) = -6 s 9 t 7

23. 2ab (5 a 3 + 3 a 2 b)

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

= (2 · 5) (a · a 3 ) (b) + (2 · 3) (a · a 2 ) (b · b) = 10 a 4 b + 6 a 3 b 2

24. (3k + 5) 2 = (3k + 5)(3k + 5) = 3k(3k) + 3k(5) + 5(3k) + 5(5) = 9 k 2 + 15k + 15k + 25 = 9 k 2 + 30k + 25

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

= 2 x 3 (4 x 2 ) + 2 x 3 (y) + 3y (4 x 2 ) + 3y(y) = 8 x 5 + 2 x 3 y + 12 x 2 y + 3 y 2


CS10_A1_MESK710372_C06.indd 218 3/30/11 11:29:50 PM

26. ( p 2 + 3p) (9 p 2 - 6p - 5)

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

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

= 9 p 4 - 6 p 3 - 5 p 2 + 27 p 3 - 18 p 2 - 15p = 9 p 4 + 21 p 3 - 23 p 2 - 15p

27. A = bh = (x + 7)(x - 3) = x(x) + x(-3) + 7(x) + 7(-3) = x 2 - 3x + 7x - 21 = x 2 + 4x - 21 The area is represented by ( x 2 + 4x - 21) square

units.

28. (a + b) 2 = a 2 + 2ab + b 2 (d + 9) 2 = (d) 2 + 2(d)(9) + (9) 2 = d 2 + 18d + 81

29. (a + b) 2 = a 2 + 2ab + b 2 (3 + 2t) 2 = (3) 2 + 2(3)(2t) + (2t) 2 = 4 t 2 + 12t + 9

30. (a + b) 2 = a 2 + 2ab + b 2 (2x + 5y) 2 = (2x) 2 + 2(2x)(5y) + (5y) 2 = 4 x 2 + 20xy + 25 y 2

31. (a - b) 2 = a 2 - 2ab + b 2 (m - 4) 2 = (m) 2 - 2(m)(4) + (4) 2 = m 2 - 8m + 16

32. (a - b) 2 = a 2 - 2ab + b 2

33. (a - b) 2 = a 2 - 2ab + b 2 (3w - 1) 2 = (3w) 2 - 2(3w)(1) + (1) 2 = 9 w 2 - 6w + 1

34. (a + b)(a - b) = a 2 - b 2 (c + 2)(c - 2) = (c) 2 - (2) 2 = c 2 - 4

35. (a + b)(a - b) = a 2 - b 2 (5r + 6)(5r - 6) = (5r) 2 - (6) 2 = 25 r 2 - 36

36. S = 4π r 2 = 4π (x - 5) 2

= 4π ( (x) 2 - 2(x)(5) + (5) 2 )

= 4π ( x 2 - 10x + 25)

= 4π ( x 2 ) + 4π(-10x) + 4π(25) = 4π x 2 - 40πx + 100π The area is represented by

(4π x 2 - 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. -2 p 3 q -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 ___ m 2

14. b c 0 = b · c 0 = b · 1 = b

15. - 1 __ 2 x -2 y -4 = - 1 __

2 · x -2 · y -4

= - 1 __ 2 · 1 __

x 2 · 1 __

y 4

= - 1 _____ 2 x 2 y 4

16. 2 b 6 ___ c -4

= 2 · b 6 · 1 ___ c -4

= 2 · b 6 · c 4 = 2 b 6 c 4

17. 3 a 2 c -2 ______ 4 b 0

= 3 __ 4 · a 2 · c -2 · 1 __

b 0

= 3 __ 4 · a 2 · 1 __

c 2 · 1 __

1

= 3 a 2 ___ 4 c 2

6-1


CS10_A1_MESK710372_C06.indd 219 3/30/11 11:29:53 PM

18. q -1 r -2

______ s -3

= q -1 · r -2 · 1 ___ s -3

= 1 __ q · 1 __ r 2

· s 3

= s 3 ___ q r 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 = ( z 10 ) 1 __ 5

= z 10 · 1 __

5 = z 2

24. 3

√ 125 x 6

= (125 x 6 ) 1 __ 3

= (125 ) 1 __ 3

· ( x 6 )

1 __ 3

= ( 5 3 ) 1 __ 3

· ( x 6 )

1 __ 3

= ( 5 3 · 1 __

3 ) · ( x

6 · 1 __ 3 )

= ( 5 1 ) · ( x 2 ) = 5 x 2

25. √ x 8 y 6

= ( x 8 y 6 ) 1 __ 2

= ( x 8 · 1 __

2 ) · ( y

6 · 1 __ 2

)

= ( x 4 ) · ( y 3 ) = x 4 y 3

26. 3

√ m 6 n 12

= ( m 6 n 12 ) 1 __ 3

= ( m 6 · 1 __

3 ) · ( n

12 · 1 __ 3 )

= ( m 2 ) · ( n 4 ) = m 2 n 4

PoLynomiaLs

27. 5 = 5 x 0 Degree: 0

28. 8s t 3 = 8 s 1 t 3 Degree: 1 + 3 = 4

29. 3 z 6 Degree: 6

30. 6h = 6 h 1 Degree: 1

31. 2n - 4 + 3 n 2 → 3 n 2 + 2n - 4 The leading coefficient is 3.

32. 2a - a 4 - a 6 + 3 a 3 → - a 6 - a 4 + 3 a 3 + 2a The leading coefficient is -1.

33. Degree: 1 Terms: 2 2s - 6 is a linear binomial.

34. Degree: 5 Terms: 1 -8 p 5 is a quintic monomial.

35. Degree: 4 Terms: 3 - m 4 - m 2 - 1 is a quartic trinomial.

36. Degree: 0 Terms: 1 2 is a constant monomial.

adding and suBtraCting PoLynomiaLs

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

38. 4 x 5 - 6 x 6 + 2 x 5 - 7 x 5 = -6 x 6 + 4 x 5 + 2 x 5 - 7 x 5 = -6 x 6 - x 5

39. - h 3 - 2 h 2 + 4 h 3 - h 2 + 5 = - h 3 + 4 h 3 - 2 h 2 - h 2 + 5 = 3 h 3 - 3 h 2 + 5

40. (3m - 7) + (2 m 2 - 8m + 6) = 2 m 2 + (3m - 8m) + (-7 + 6) = 2 m 2 - 5m - 1

41. (12 + 6p) - (p - p 2 + 4)

= (12 + 6p) + (-p + p 2 - 4)

= p 2 + (6p - p) + (12 - 4) = p 2 + 5p + 8

42. (3z - 9 z 2 + 2) + (2 z 2 - 4z + 8)

= (-9 z 2 + 2 z 2 ) + (3z - 4z) + (2 + 8) = -7 z 2 - z + 10

43. (10g - g 2 + 3) - (-4 g 2 + 8g - 1)

= (10g - g 2 + 3) + (4 g 2 - 8g + 1)

= (- g 2 + 4 g 2 ) + (10g - 8g) + (3 + 1)

= 3 g 2 + 2g + 4

44. (-5 x 3 + 2 x 2 - x + 5) - (-5 x 3 + 3 x 2 - 5x - 3)

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

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

= - x 2 + 4x + 8

muLtiPLying PoLynomiaLs

45. (2r)(4r) = (2 · 4)(r · r) = 8 r 2

46. (3 a 5 ) (2ab)

= (3 · 2) ( a 5 · a) (b) = 6 a 6 b

47. (-3xy) (-6 x 2 y)

= (-3 · (-6)) (x · x 2 ) (y · y)

= 18 x 3 y 2

48. (3 s 3 t 2 ) (2s t 4 ) ( 1 __ 2 s 2 t 8 )

= (3 · 2 · 1 __ 2 ) ( s 3 · s · s 2 ) ( t 2 · t 4 · t 8 )

= 3 s 6 t 14

49. 2 ( x 2 - 4x + 6)

= 2 ( x 2 ) + 2(-4x) + 2(6) = 2 x 2 - 8x + 12

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

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

= (-3)(a · a)(b · b) + (-3)(- 2) (a · a 2 ) (b · b) + (-3 · 5)(a · a)(b)

= -3 a 2 b 2 + 6 a 3 b 2 - 15 a 2 b

6-2

6-3

6-4

6-5


CS10_A1_MESK710372_C06.indd 220 3/30/11 11:29:56 PM

51. (a + 3)(a - 6) = a(a) + a(-6) + 3(a) + 3(-6) = a 2 - 6a + 3a - 18 = a 2 - 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)

= x 2 - 2x - 10x + 20

= x 2 - 12x + 20

54. (t - 1)(t + 1) = t(t) + t(1) - 1(t) - 1(1) = t 2 + t - t - 1 = t 2 - 1

55. (2q + 6)(4q + 5) = 2q(4q) + 2q(5) + 6(4q) + 6(5) = 8 q 2 + 10q + 24q + 30 = 8 q 2 + 34q + 30

56. (5g - 8)(4g - 1) = 5g(4g) + 5g(-1) - 8(4g) - 8(-1) = 20 g 2 - 5g - 32g + 8 = 20 g 2 - 37g + 8

sPeCiaL ProduCts of BinomiaLs

57. (a - b) 2 = a 2 - 2ab + b 2 (p - 4) 2 = (p) 2 - 2(p)(4) + (4) 2 = p 2 - 8p + 16

58. (a + b) 2 = a 2 + 2ab + b 2 (x + 12) 2 = (x) 2 + 2(x)(12) + (12) 2 = x 2 + 24x + 144

59. (a + b) 2 = a 2 + 2ab + b 2 (m + 6) 2 = (m) 2 + 2(m)(6) + (6) 2

= m 2 + 12m + 36

60. (a + b) 2 = a 2 + 2ab + b 2 (3c + 7) 2 = (3c) 2 + 2(3c)(7) + (7) 2

= 9 c 2 + 42c + 49

61. (a - b) 2 = a 2 - 2ab + b 2 (2r - 1) 2 = (2r) 2 - 2(2r)(1) + (1) 2

= 4 r 2 - 4r + 1

62. (a - b) 2 = a 2 - 2ab + b 2 (3a - b) 2 = (3a) 2 - 2(3a)(b) + (b) 2

= 9 a 2 - 6ab + b 2

63. (a - b) 2 = a 2 - 2ab + b 2 (2n - 5) 2 = (2n) 2 - 2(2n)(5) + (5) 2

= 4 n 2 - 20n + 25

64. (a - b) 2 = a 2 - 2ab + b 2 (h - 13) 2 = (h) 2 - 2(h)(13) + (13) 2

= h 2 - 26h + 169

65. (a - b)(a + b) = a 2 - b 2 (x - 1)(x + 1) = (x) 2 - (1) 2

= x 2 - 1

66. (a + b)(a - b) = a 2 - b 2 (z + 15)(z - 15) = (z) 2 - (15) 2

= z 2 - 225

67. (a - b)(a + b) = a 2 - b 2

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

= c 4 - d 2

68. (a + b)(a - b) = a 2 - b 2

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

= 9 k 4 - 49

chaPtEr tEst

1. ( 1 __ 3 b)

-2

= ( 1 __ 3 (12))

-2

= 4 -2

= 1 __ 4 2

= 1 ____ 4 · 4

= 1 ___ 16

2. (14 - a 0 b 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. 2 r -3 = 2 · r -3

= 2 · 1 __ r 3

= 2 __ r 3

4. -3 f 0 g -1 = -3 · f 0 · g -1

= -3 · 1 · 1 __ g

= - 3 __ g

5. m 2 n -3 = m 2 · n -3

= m 2 · 1 ______ n 3

= m 2 ___ n 3

6. 1 __ 2 s -5 t 3 = 1 __

2 · s -5 · t 3

= 1 __ 2

· 1 __ s 5

· t 3

= t 3 ___ 2 s 5

7. S = 3.14 r 2 + 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


CS10_A1_MESK710372_C06.indd 221 3/30/11 11:29:58 PM

10. √ 25 y 8 = (25 y 8 ) 1 __ 2

= (25 ) 1 __ 2

· ( y 8 )

1 __ 2

= √ 25 · ( y 8 · 1 __

2 )

= 5 · y 4 = 5 y 4

11. 5

√ 3 5 t 10

= ( 3 5 t 10 ) 1 __ 5

= ( 3 5 · 1 __

5 ) · ( t

10 · 1 __ 5 )

= ( 3 1 ) · ( t 2 ) = 3 t 2

12. 3a - 4b + 2a = 3a + 2a - 4b = 5a - 4b

13. (2 b 2 - 4 b 3 ) - (6 b 3 + 8 b 2 )

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

= (-4 b 3 - 6 b 3 ) + (2 b 2 - 8 b 2 ) = -10 b 3 - 6 b 2

14. -9 g 2 + 3g - 4 g 3 - 2g + 3 g 2 - 4 = -4 g 3 - 9 g 2 + 3 g 2 + 3g - 2g - 4 = -4 g 3 - 6 g 2 + g - 4

15. -5 ( r 2 s - 6)

= -5 ( r 2 s) - 5(-6) = -5 r 2 s + 30

16. (2t - 7)(t + 4) = 2t(t) + 2t(4) - 7(t) - 7(4) = 2 t 2 + 8t - 7t - 28 = 2 t 2 + t - 28

17. (4g - 1) (4 g 2 - 5g - 3)

= 4g (4 g 2 - 5g - 3) - 1 (4 g 2 - 5g - 3)

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

= 16 g 3 - 20 g 2 - 12g - 4 g 2 + 5g + 3 = 16 g 3 - 24 g 2 - 7g + 3

18. (a + b) 2 = a 2 + 2ab + b 2 (m + 6) 2 = (m) 2 + 2(m)(6) + (6) 2 = m 2 + 12m + 36

19. (a - b)(a + b) = a 2 - b 2 (3t - 7)(3t + 7) = (3t) 2 - (7) 2 = 9 t 2 - 49

20. (a - b) 2 = a 2 - 2ab + b 2

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

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

= 9 x 4 - 42 x 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) = x 2 - 4x + 3x - 12 = x 2 - x - 12 The area is represented by x 2 - x - 12.

b. A = x 2 - x - 12 = (4.5) 2 - (4.5) - 12 = 20.25 - 4.5 - 12 = 3.75 The area is 3.75 in 2 .


CS10_A1_MESK710372_C06.indd 222 3/30/11 11:30:00 PM

Documents

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