chapter 6 exponents and practice 6.1.3 1. (2 4 3 = 2 4•3 ...chapter 6 exponents and polynomials...

Chapter 6 Exponents and Polynomials

Section 6.1 Properties of Exponents

Practice 6.1.11. 56•54 = 56+4 = 510

2. y6y7 = y6+7 = y13

3. w3w5w = w3+5+1 = w9

4. x2y5

5. x3 +x5

Practice 6.1.21. 6x4(3x3) = (6•3)(x4•x3) = 18(x4+3)= 18x7

2. -7m(3m4) = (-7•3)(m•m4) = -21(m1+4) = -21m5

3. 1/2 t5(4t8) = 1/2(4)(t5•t8) = 2(t5+8)= 2t13

4. 5x2(-7x3)(4x7)=(5•(-7)•4)(x2•x3•x7)=

-140(x2+3+7) = -140x12

5. 1/7 x5(-14x3)(-3x5) = 1/7•(-14)•(-3)(x5•x3•x5)=6(x5+3+5) = 6x13

6. 5c3(6c5 + 2c) = 5c3(6c5) + 5c3(2c) =5•6(c3•c5) + (5•2)(c3•c) = 30(c3+5) + 10(c3+1) = 30c8 + 10c4

7. 4s2(3s4 + 8s3) = 4s2(3s4) + 4s2( 8s3)=4•3(s2•s4) + 4•8(s2•s3) = 12(s2+4) +32(s2+3) = 12s6 + 32s5

8. -m(m2 + 4) = -m(m2) + (-m)(4) = -1(1)(m•m2) - 4m = -1(m1+2) - 4m-m3 - 4m

9. -4w(-6w3 + 5) = -4w(-6w3) + (-4w)(5) = -4(-6)(w•w3) + (-4)(5)w = 24w4 - 20w

Practice 6.1.31. (24)3 = 24•3 = 212

2. (d5)7 = d5•7 = d35

3. (a9)4 = a9•4 = a36

Practice 6.1.41. (4s)2 = 42•s2 = 16s2

2. (-v5)4 =(-1)4v5•4 = v20

3. (23v5)2 = 23*2•v5•2 = 26v10 = 64v10

4. (-2x4)4 = (-24)•x4•4 = 16x16

5. (m5n2)7 = m5•7n2•7 = m35n14

6. (3xy3)2 = 32•x2•y3•2= 9x2 y6

Practice 6.1.5

1.107

103 = 107 - 3 = 104

2.m 6

m = m 6 - 1 = m 5

3.12x 7

4 x 5 = 3 x 7 - 2 = 3 x 5

4.15t 15

− 5 t 5 = - 3 t 15 - 5 = - 3 t 10

5.x 3 y 5

x 2 y 3 = x 3 - 2 y 5 - 3 = xy2

6.m 7 n 4

m 2 n = m 7 - 2 n 4 - 1 = m5 n 3

Practice 6.1.6

1. ( 2 m

) 4 = 2 4

m 4 = 16

m 4

2. ( w 4

) 3 = w 3

4 3 = w 3

64

3. ( 3 w

y 2 ) 4 = 3 4 w 4

y 2 • 4 = 81w 4

y 8

4. ( 4 d 2

c 4 ) 3 = 4 3 d 2 • 3

c 4 • 3 = 256d 6

c 12

167CHAPTER 6 EXPONENTS AND POLYNOMIALS

Practice 6.1.71. m5(m2)6 =m5(m2•6) = m5 m12 =

m5+12 = m17

2.t 7 ( t 5 ) 2

t 3 = t

7 t 5 • 2

t 3 = t

7 t 10

t 3

= t 7 + 10

t 3 = t

17

t 3 = t 17 - 3 = t 14

3.( x 5 x 3 ) 2

x 2 = ( x 5 + 3 )

x 2 = x 8

x 2 = x 8 - 2 = x 6

4.( 5 x 2 ) 2 ( 2 y 8 ) 3

10x 2 ( 5 y ) = 25x 4 • 8 y 24

50x 2 y =

200x 4 y 24

50x 2 y = 4 x 2 y 23

5.16x 6 ( 3 y 5 ) 3

12x 3 ( 2 y 3 ) 2 = 16x 6 • 27y 15

12x 3 • 4 y 6 = 9 x 3 y 9

Exercise Set 6.1

1. 6263 =62+3 = 65

3. h3h = h3+1 = h4

5. y7y6 = y7+6 = y13

7. y5z5

9. m4m5m3=m4+5+3=m12

11. r2r7r8 = r2+7+8 = r17

13. x2y4x4y5 = x2+4y4+5 = x6y9

15. hh2jj3 = h1+2j1+3 = h3j4

17. 2n3(3n5) = 6n3+5 = 6n8

19. -3g2(-5g7) = 15g2+7 = 15g9

21. -3w(2w4)(4w)= -24w1+4+1=-24w6

23. 8x(-2xy3)(-3x3y8) = 48x1+1+3y3+6 =48x5y9

25. 2a2(3a3 + 4a) = 2a2(3a3) + 2a2(4a)=

6a2+3 + 8a2+1 =6a5 + 8a3

27. 5t2(6t5 + 4t4 - 3t) = 5t2•6t5 + 5t2•4t4

- 5t2•3t = 30t2+5 + 20t2+4 - 15t2+1 = 30t7 + 20t6 - 15t3

29. 2/3 w2(6w5)(-5w3) = -20w2+5+3 =-20w10

31. 2/3 d2(6d5 + 9d3) = 2/3 d2(6d5)+ 2/3 d2(9d3) =4d2+5+6d2+3 = 4d7+6d5

33. (c5)6 = c5•6 = c30

35. (k2)4 = k2•4 = k8

37. (4t)3 = 43t3 = 64t3

39. (5w4)2 = 52w4•2 = 25w8

41. (-4r3)4 = (-4)4r3•4 =256r12

43. (-3r4s)3 = (-3)3r3•4s 3 = -27r12s 3

45.5 4

5 2 = 5 4 - 2 = 5 2 = 25

47.y 6

y 2 = y 6 - 2 = y 4

49.r 8 p 4

rp= r 8 - 1 p 4 - 1 = r 7 p 3

51.s 6 t 4

s 4 t 3 = s 6 - 4 t 4 - 3 = s 2 t

53.x 7

y 5

55. ( 8 x

) 2 = 8 2

x 2 = 64

x 2

57. ( 7

t 3 ) 2 = 7 2

t 6 = 49

t 6

59. ( 2 x 5

y 4 ) 4 = 2 4 x 20

y 16= 16x 20

y 16

61. 3x2(2x)3 = 3x2•23x3 = 3x2•8x3 =24x5

168 CHAPTER 6 EXPONENTS AND POLYNOMIALS

63.2 x 2 ( 4 x 3 ) 2

16x 5 = 2 x 2 • 4 2 x 6

16x 5 =

2 x 2 • 16x 6

16x 5 = 2 x 8

x 5 = 2 x 3

65.8 x 3 ( 4 xy2 ) 2 ( 3 x 4 ) 2

24x 3 ( 6 x 2 y 3 ) = 8 x 3 • 16x 2 y 4 • 9 x 8

24• 6 x 5 y 3 =

8 x 13y 4

x 5 y 3 = 8 x 8 y

67. a)Total = c+ (c-80) = c+c-80 =2c-80b)Total = c + 1/4 c = 5/4 cc)Total = c+(c+60)=c+ c+ 60 =

2c+60

68. Let x = first angle; 1/4 x + 10 =second x + (1/4 x+10) = 90

4(x) + 4(1/4 x+10) = 4•904x + x + 40 = 3605x = 320x = 64 degrees1/4 (64) + 10 = 16 + 10 = 26 degrees

69. m =1 - ( - 2 )

1 - 3 = 1 + 2

- 2 = − 3

2

70. 9.5 ≥ x <-------------------------------<--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->x

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

71.

-10-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8910 x

Data #1

72. Let x = first integer; x + 1 = secondx+2 = the thirdx + (x + 1) +(x+2)= -9

3x + 3 = -92x = -12x = -6 ; -6 + 1 = -5; -6+2=-4

73. Yes, this would pass the verticalline test.

74. Yes, this would pass the verticalline test.

75. Eq 1: x + 7y = -3Eq 2: 2x + 6y = 10Solve eq 1 for x: x = -3 - 7ySubstitute in eq 2: 2(-3-7y) + 6y =

10-6 - 14y + 6y = 10-6 - 8y = 10-8y = 16y = -2Substitute to solve for x: x + 7(-2) = -3x - 14 = -3x = 11The solution is the ordered pair (11,-

2)

76. Eq 1: 2x - 5y = 7Eq 2: 3x + 2y = 12 times eq 1: 2(2x - 5y) = 2•75 times eq 2: 5(3x + 2y) = 5•1Eq 1 transformed: 4x - 10y = 14Eq 2 transformed: 15x + 10y = 5Add together 19x =19x = 1Substitute to solve for y: 2(1) - 5y =

72 - 5y = 7-5y = 5y = -1The solution is the ordered pair (1,-

1).


Section 6.2 Exponents Involving 1,0, and Negative Integers

Practice 6.2.11. 3-4 The base is 3 and the exponentis

-4. This is equivalent to 1

3 4 = 1

81

2. 5w-7 The base is w and theexponent is -7. This is equivalent to5

w 7 .

3. 7-2m-6The first base is 7 and itsexponent is -2. The second base is

m and its exponent is -6. This is

equivalent to 1

7 2 m 6 = 1

49m 6

4.9

x − 5 The base is x and its exponent

is -5. This is equivalent to 9x5.

5.4 − 3

d − 5 The first base is 4 and its

exponent is -3. The second is d andits exponent is -5. This is

equivalent to d 5

4 3 = d 5

64

6.3 x − 2

y − 4 The first base is x and its

exponent is -2. The second is y andits exponent is -4. This is

equivalent to 3 y 4

x 2

7.2 + x − 1

y The base is x and the

exponent is -1. This is equivalent to

2 + 1 x

y

Practice 6.2.2

1. 3-1 + 4-1 = 1 3

+ 1 4

= 4 12

+ 3 12

= 7 12

2. 5 + 1

6 − 2 + 2 0 = 5+62+1 = 5 + 36 + 1 =

42

3. 4-1 + 1

3 − 2 =1/4 + 32 = 1/4 + 9 = 9 1/4

4. 3•2-3 - 6-1 =3

2 3 − 1

6 = 3

8 − 1

6 = 9

24− 4

24= 5

24

Practice 6.2.3

1. (z-5z2)3 = (z-3)3 = z -9 = 1

z 9

2. (b-6b5)6 = (b-1)6 = b-6 =1

b 6

3. ( c − 4

− 4 ) − 2 = c 8

( - 4 ) − 2 = ( - 4 ) 2 c 8

16c8

4. ( d − 5

− 3 d − 2 ) − 4 = d 20

( − 3 ) − 4 d 8 =

(-3)4d12 = 81d12

5.( 5 m 4 ) 4

5 m − 2 = 5 4 m 16

5 m − 2 = 5 3 m 16m 2


125m18

6.( 6 t − 4 ) 2

2 t − 6 = 6 2 t − 8

2 t − 6 = 36t 6

2 t 8 = 18

t 2

Practice 6.2.4

1. ( x 2 x 5

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

4 x 6 ) 2 = ( x

4 ) 2

x 2

16

2. ( 5 x − 2 x 4

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

x 8 ) 3 = ( 5

x 6 ) 3

5 3

x 18= 125

x 18

3.3 x 2 • 5 x 4

2 x − 3 ( x − 2 ) 2 = 15x 6

2 x − 3 ( x − 4 ) =

15x 6

2 x − 7 = 15x 6 x 7

2 = 15x 13

2

4.2 x 5 • 4 x − 3

12x 5 ( x 3 ) − 5 = 8 x 2

12x 5 x − 15=

2 x 2

3 x − 10= 2 x 2 x 10

3 = 2 x 12

3

Exercise Set 6.2

1. 3-6 The base is 3 and the exponent

is - 6. This is equivalent to 1

3 6

3. 12z-6 The base is z and theexponent is -6. This is equivalent to 12

z 6

5. (7x)--5 The base is 7x and theexponent is -5. This is equivalent to

1

( 7 x ) 5 = 1

7 5 x 5

7.2

d − 2 The base is d and the exponent

is -2. This is equivalent to 2d2.

9. 4-2v7 The first base is 4 and the

exponent is -2. The second is v andthe exponent is 7. This is equivalent

to v 7

4 2 = v 7

16

11. 2-3m-9 The first base is 2 and itsexponent is -3. The second is m andits exponent is -9. This is

equivalent to 1

2 3 m 9 = 1

8 m 9

13.3 − 2

k − 6 The first base is 3 and its

exponent is -2. The second is k andits exponent is -6. This is

equivalent

to k 6

3 2 = k 6

9

15.3 + x − 1

2 The base is x and its

exponent is -1. This is equivalent to

3 + 1 x

2

17. 3 - 2-1 = 3 - 1/2 = 6/2 - 1/2 = 5/2

19. 2 - 1

5 − 1 - 50 = 2 - 5 - 1 = -4

21. 5•2-1 - 3-1 = 5•1/2 - 1/3 = 5/2 - 1/3 = 15/6 - 2/6 = 13/6

23. (a-6a4) -5 = (a-2) -5 = a10

25. ( d − 3

2 − 3 ) − 2 = d 6

2 6 = d 6

64

27. ( 2 x − 3

3 x − 2 ) − 2 = 2 − 2 x 6

3 − 2 x 4 = 3 2 x 2

2 2 = 9 x 2

4

29.( 4 x 2 y 3 ) 2

( 2 xy4 ) − 3 = 4 2 x 4 y 6

2 − 3 x − 3 y − 12

16x4y6•23x3y12 = 16x4y6•8x3y12 =128x7y18

31. ( 12x 4 y − 2

18x − 5 y ) − 2 = ( 2 x 4 x 5

3 yy2 ) − 2 =


( 2 x 9

3 y 3 ) − 2 = 2 − 2 x − 18

3 − 2 y − 6 = 3 2 y 6

2 2 x 18= 9 y 6

4 x 18

33.( 8 x 2 y − 5 ) 4

( 16x − 3 y 2 ) 3 = 8 4 x 8 y − 20

( 2 • 8 ) 3 x − 9 y 6 =

8 4 x 8 x 9

2 3 • 8 3 y 6 y 20= 8 x 17

2 3 y 26= 8 x 17

8 y 26= x 17

y 26

35. ( 4 − 2 x − 3 y − 1

x − 4 y ) (

8 − 1 x − 2 y

x 3 y − 1 ) − 2 =

( x 4

4 2 x 3 yy) (

8 2 x 4 y − 2

x - 6 y 2 ) =

x

16y 2 •

64x 4 x 6

y 2 y 2 = 4 x 11

y 6

37.1 2

− 3 4

( 2 3

) 2 = 1 2

− 3 4

( 4 9

) =

1 2

− 1 3

= 3 6

− 2 6

= 1 6

38. 8x - 3(x - 1) = x + 38x - 3x + 3 = x + 35x + 3 = x + 34x = 0 x = 0

39.

-10-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8910 x

Data #2

40.

-10-9-8-7-6-5-4-3-2-101234567

y

-10-9-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8910 x

Data #3

41. f(x) = 2x + 7f(0) = 2•0 + 7 = 7

42. Eq 1 2x + y = 7Eq 2 x - 3y = -14Solve eq 1 for y: y = 7 - 2xSubstitute in eq 2: x - 3(7 - 2x) = -14x - 21 + 6x = -147z = 7x = 1Substitute to solve for y: 2•1 + y = 72 + y = 7y = 5The solution is the ordered pair

(1,5).

43. Let x = number not damaged; 85 - x= number damaged

x-15 = damaged that can be sold(85-x) - (x - 15) = damaged that

can’tbe sold85 - x - x + 15 = 100 - 2x representsthe number damaged that can’t sold

44. 45 minutes = 45/60 hr = 0.75 hrdistance = rate•time40 kph•0.75 hr = 30 kilometers

45. Let q = number of quarters29 - q = number of dimes0.25q = value of quarters0.10(29-q) = value of dimes0.25q + 0.10(29-q) = 5.000.25q + 2.90 - 0.10q = 5.000.15q = 2.10q = 14 quarters29 - 14 = 15 dimes


46. x + 2y = 32y = 3 - xy = 3/2 - x/2m = -1/2; y intercept is (0,3/2)

47. m = 1 - 1

− 6 - 0 = 0

− 6 = 0

y intercept is (0,1) so b = 1y = 0x + 1y = 1

48. a)Let x = number of inches; y =times growing (in years)

y = 12.5xb)

xstalctite length

(inches)

ytime growing

(years)

3.5 12.5(3.5)=43.75

5 12.5(5)= 62.5

18 12.5(18)= 225

125÷12.5=10 125

100÷12.5=8 8

c)

(3.5, 43.75)(5, 62.5)

(8, 100)

(10, 125)

(18, 225)

406080

100120140160180200220240

0 2 4 6 8 10 12 14 16 18 20Stalactite length (inches)

Data #4y

x


Section 6.3 Scientific Notation

Practice 6.3.11. 9.9 x 105 Yes

-16.8 x 10-3 No, the decimalnumber is smaller than -10.

4 2/3 x 103 No, not a decimal.-4.7 x 109 Yes0.36 x 10-8 No, the decimal

number is smaller than 1.

2. -5.7 x 104 Yes7.08 x 10-6 Yes25.9 x 104 No, the decimal numberis greater than 10.7 1/4 x 105 No, not a decimal.0.78 x 102 No, the decimal number

is smaller than 1.

Practice 6.3.21. 78,023 = 7.8 x 104 since we moved

the decimal point 4 places to the left.

2. -6,001 = -6.001 x 103 (since wemoved the decimal point 3 places tothe left) -6,001x103 = -6.001 x103 x103 = -6.001x 106

3. 0.000256 = 2.56 x 10-4 since wemoved the decimal point 4 places tothe right.

4. -0.0035= -3.5 x10-3

-0.0035 x 10-4 = -3.5 x10-3x 10-4 = -3.5 x 10 -7

Practice 6.3.31. 5 x 10 5 = 500,000 since the

expontent tells us to move thedecimal point 5 places to the right.

2. -1.37 E4 = -13700 since this is thesame as -1.37 x 104 The exponent

tells us to move the decimal


point 4 places to the right.

3. 5.321 x 10-3 = 0.005321 since theexponent tells us to move the

decimal point 3 places tothe left.

4. 2.0 x 10-5 = 0.00002 since theexponent tells us to move the

decimal point 5 places to the left.

Practice 6.3.41. (3 x105)(2 x10-6)= 3•2x(105•10-6)=

6 x 10-1

2. (2 x 1012)(4 x 109) = 2•4x(1012•109) = 8 x1021

3. (78,000)(1,100,000,000,000,000) = (7.8 x 104) (1.1 x 1015) =7.8•1.1 x(104•1015) = 8.58 x 1019

4. (0.000036)(25000) =(3.6 x10-5)(2.5x 104) = 3.6•2.5x(10-5•104 ) = 9 x10-1

5. (-5.2 x 106)(1.4 x 10-8) = -5.2•1.4 x(106•10-8) = -7.28 x 10 -2

6.1 . 21 x10− 8

− 1 . 1 x 103 = - 1 . 1 x 10− 8 - 3 =

-1.1 x 10-11

7.9 . 8 x 104

( 5 . 0 x 10− 1 ) ( 2 . 5 x 103 ) =

9 . 8 x 104

12. 5 x 102 = 0 . 784 x102 =

7 . 84x 10− 1 x 102 = 7 . 84 x10

8.( 6 . 3 x 1012) ( 2 . 8 x 10− 30)

5 . 6 x 10− 4 =

6 . 3 ( 2 . 8 ) x ( 1012x 10− 30)

5 . 6 x 10− 4 =

17. 64x 10− 18

5 . 6 x 10− 4 = 3 . 15x 10− 18 - ( - 4 ) =

3.15 x 10-18+4 =3.15 x 10-14

9. 8.1 x10-1 + 2.914 x 102 = 0.81 +291.4 = 292.21 = 2.9221x102

Practice 6.3.51. 67,250,000 = 6.725 x 107

12,500 = 1.25 x 104

6 . 725 x107

1 . 25 x 104 = 5 . 38 x103 hours

2. 13,200,000,000,000,000,000,000 =1.32 x1022

2,000 = 2.0 x 103

1 . 32x 1022

2 . 0 x 103 = 0 . 66 x1019 =

6 . 6 x 10− 1 x 1019 = 6 . 6 x 1018 tons

Exercise Set 6.3

1. 2 x 10-3 Yes0.2 x 103 No, the decimal number

is less than 1.23.2 x 109 No, the decimal number

is greater than 10.8.01 x 108 Yes

3. 6,700,000 = 6.7 x 106 since wemoved the decimal point 6 places tothe left.

5. -32,510,000,000 = -3.251 x1010 sincewe moved the decimal point 10

places to the left.

7. 0.007003 = 7.003 x 10 -3 since wemoved the decimal point 3 places tothe right.

9. 0.52 x 10-3 = 5.2 x 10-1 x10-3 = 5.2 x 10-4

11. 273 x 10-3 = 2.73 x 102 x 10-3 = 2.73 x 10-1

13. 899,500,000 since the exponent tellus to move the decimal point 8 placesto the right.

15. -0.049 since the exponent tells us tomove the decimal point 2 places to

the left.

17. 8,000,000,000,000,000 since this isthe same as 8.0 x 1015 .


19. (7.5 x1012)(2.1 x 104) =7.5•2.1x(1012x104)= 15.75 x1016 =1.575 x 10 x1016 =1.575 x1017

21. (3.6 x 10-5)(5.3 x103)=3.6•5.3 x(10-4 x 103) = 19.08 x 10-1 =1.908 x 10 x 10-1=1.908

23.( - 3 . 6 x 103 ) ( 2 . 3 x 105

( 1 . 8 x 10− 3 ) − 3 . 6 • 2 . 3 x 103 x 105

1 . 8 x 10− 3 = − 8 . 28x 108

1 . 8 x 10− 3

= - 4 . 6 x 108 x 103 = - 4 . 6 x 1011

25.( 6 , 000, 000, 000, 000) ( 0 . 000104)

( 520, 000, 000, 000) ( 0 . 00002) =

( 6 . 0 x 1012) ( 1 . 04x 10− 4 )

( 5 . 2 x 1011) ( 2 x 10− 5 ) =

6 . 24x 108

10. 4 x 106 = 0 . 6 x 102 =

6 . 0 x 10− 1 x 102 = 6 . 0 x 10

27. 2.68x10-3 + 5x10-2 = 0.00268 + 0.05 = 0.05268 = 5.268 x 10-2

29. 5.1x104 - 1.5x102 = 51,000 - 150 = 50850 = 5.085 x 104

31. 11,000 = 1.1x 104

9 x10-28 x1.1x104 = 9.9 x10--24grams

33. 32,550 = 3.255x104

0.0778 = 7.78x 10-2

(3.255x104 )(7.78x10-2) = 25.3239 x102 = 2.53239x10x102 =2.53239 x 103 pounds

35. 0.00005 = 5.0x10-5

150,000 = 1.5x 105

(5.0x10-5 )(1.5x105) = 7.5 cm

37. 2(8-3)+ [23-2(5+3)]2 = 2•5 +[23 - 2•8]2 = 10 + [23 - 16]2=10 + [7]2 = 10 + 49 =59

38. 3x2 - (x - y) = 3(-2)2 - (-2 - (-3)) = 3•4 - (-2 + 3) = 12 - (1) = 11

39. 2(2 - x) - 3(x - 5) - (x - 3) = 4 - 2x - 3x + 15 - x + 3 = -6x + 22

40.1 4

x - 10 < 4 ( x 2

+ 1 )

1 4

x - 10 < 2 x + 4

4 • 1 4

x - 4 • 10 < 4 • 2 x + 4 • 4

x - 40 < 8x + 16-7x <56x > -8

41. Let x = number of months4.25x + 1.5(x ) = 2765.75x = 276x = 48 months

42. Let x = amount at 8%; 1100 - x =amount at 10%interest = principal• rate • time 0.08x + 0.1(1100-x) = 970.08x + 110 - 0.1x = 97 - 0.02x = -13x = $650 at 8%1100 - 650=$450 at 10%

43. Eq 1 y + 3 = 6xEq 2 -12x + 6 = -2ySolve eq 1 for y : y = 6x - 3Substitute in eq 2: -12x + 6 = -2(6x - 3)-12x + 6 = -12x + 6They are the same line. There are

an infinite number of solutions. Theycan be written as the following ordered

pairs (x,6x-3).

44. 1/2 m3(2m3•3m)m = 1/2 m3(6m4)m3m8

45.1

5 − 1 + 1

6 − 1 + 1

7 − 2 = 5 + 6 + 72 =

5 + 6 + 49 = 60

46. Eq 1: m= 2225 - 1875

1 - 0 = 350


y intercept is (0,1875) so b= 1875Eq 1: y = 350x + 1875

Eq 2: m = 3275 - 3000

1 - 0 = 275

y intercept is (0,3000) so b = 3000Eq 2: y = 275x + 3000Substitute eq 1 in eq 2: 350x + 1875 = 275x + 300075x = 1125x = 15 monthsy = 275(15) + 3000y = 4125 + 3000 = $7125At 15 months the cost of buying andleasing are the same. At that time

the cost is $7125.

47. x intercept is (-1,0)y intercept is (0,-3)

48. Yes, it passes the vertical line test.

Section 6.4 Introduction to Polynomials

Practice 6.4.11. This is a single term so it is a

monomial.

2. This has 4 terms so it is a four-termpolynomial.

3. This is not a polynomial since thereis

a variable in the denominator.

4. This is not a polynomial since theexponent on the first term is not awhole number.

Practice 6.4.21. The term is 7y2; the coefficient is 7.

2. The terms are: 5x2 with a coefficientof 5, -3x with a coefficient of -3 and 7

which is a constant term.

3. The terms are x2 with a coefficient of1 and -x with a coefficient of -1.

4. The terms are -x with a coefficient of -1 and 1 which is a constant term.


Practice 6.4.31. This is a polynomial.

Degree of first term is 1.Degree of second term is 0.Degree of polynomial is 1.

2. This is a polynomialDegree of first term is 2Degree of second term is 1Degree of third term is 0Degree of polynomial is 2

3. This is not a polynomial. Theexponents are not whole numbers.

Practice 6.4.41. Yes -7s9 + 5s7 +12s5+ 23

2. Yes 5 d 4

9 + 3 d 2

4 − d

9

3. Yes -0.3a2 + 2.75a + 1.9

4. Yes -4.9h2 -6.7h + 2.6

5. No. There is a variable in thedenominator.

6. No. There is a variable in thedenominator.

Exercise Set 6.4

1. Yes, it is a monomial because thereis one term. It is also called a constant

polynomial.

3. Yes, it is a trinomial because thereare 3 terms.

5. No. The exponent is not a wholenumber.

7. Yes, it is a trinomial because thereare 3 terms.

9. 2/3 n3 is the term with a coefficientof 2/3.

11. The first term is 3y3 with a

coefficient of 3; the second is -4y2

with a coefficient of -4; the third is2y with a coefficient of 2 and thefourth term is - 4 which is aconstant term.

13. The first term is 2/7 d3 with acoefficient of 2/7; the second is -3/7 dwith a coefficient of -3/7 and the thirdterm is 2 which is a constant term.

15. -15y + 9 is a polynomial of degree 1. The coefficient of the first term is -

15. 9 is a constant term.

17. -x + 2 is a polynomial of degree 1. The coefficient of the first term is -1and 2 is a constant term.

19. This is not a polynomial since theexponent -2 is not a whole number.

21. This is not a polynomial since thereis

a variable in the denominator.

23. -3x3 + 2x2 + 5x - 7

25.5 6

y 3 − 2 3

y 2 + 5 6

27.− 4 w 7

3 + 2 w 3 + 8 w 2 + 15

29. This is not a polynomial since the exponent -3 is not a whole number.

31. -4.9h2 -6.7h + 2.6

33. 12 - 5(3•22/6) = 12 - 5(3•4/6) = 12 - 5(12/6) = 12 - 5•2 = 12 - 10 = 2

34.1 2

( x - 4 ) + 2 3

x = 5 - ( x 2

− 3 )

6 • 1 2

( x - 4 ) + 6 • 2 3

x = 6 • 5 - 6 • ( x 2

− 3 )

3(x-4) + 4x = 30 - 3x + 183x - 12 + 4x = 48 - 3x7x - 12 = 48 - 3x10x = 60x = 6


35. 0.1(x-3) - (x -0.2) <1-0.9x0.1x - 0.3 - x + 0.2 < 1 -0.9x-0.9x - 0.3 < 1- 0.9x-0.3 <1This is an identity . The solution is

all real numbers.

36. f(x) = 5x2 - 2x + 1f(-1) = 5(-1)2 - 2(-1) + 1f(-1) = 5•1 + 2 + 1f(-1) = 5 + 2 + 1 = 8

37. Eq 1 -11x + y = 8Eq 2 8x - y =-5Add together -3x = 3x = -1Substitute to solve for y:-11(-1) + y = 811 + y = 8y = -3The solution is the ordered pair (-1,-3).

38.( 3 y − 1 a 2 ) 2

( 6 y − 5 a − 2 ) = 3 2 y − 2 a 4

6 y − 5 a − 2 =

9 a 4 a 2 y 5

6 y 2 = 3 a 6 y 3

2

39. The first base is 3 and its exponentis 1. The second is y and its exponentis 5.

40. Yes

41. Let x = the original purchase pricex - 0.15x = 1614150.85x = 161415x = $189,900

42. m=2 - 2

8 - ( - 3 ) = 0

11= 0

43. y = x---> y = 1x + 0 m = 1 and y intercept is (0,0).

44. The domain is {-1,0,2,3} and therange is {1,4,5}.

45. x f(x) -3 3 -1 7

-2 or 0 61 3

-5 or 3 -9

46. There were originally 200 papertowels in the box and eaxh day 3 arebeing used.

Section 6.5 Adding and Subtracting Polynomials

Practice 6.5.11. [-12t2 + 9] + [7t2 - 8t + 5] =

-12t2 + 9 + 7t2 - 8t + 5 =-5t2 - 8t + 14

2. [4xw2 - 12w + 15] + [5xw2 - 2] = 4xw2 - 12w + 15 + 5xw2 - 2 = 9xw2 - 12w + 13


3. [5a3 - 9a2 + a -32] + [-9a3 + a2 - 7]= 5a3 - 9a2 + a - 32 + -9a3 + a2 - 7 =

-4a3 - 8a2 + a - 39

4. (7u3-4u2+3u+2)+(6u2-12u+5)+(4u3-9) =11u3 +2u2-9u - 4

Practice 6.5.21. 3r2 - 5r + 6

-7r2 + 12-4r2 - 5r + 18

2. -12w3 + 7w2 - 8w + 5 4w2 + 9w - 7-12w3 + 11w2 + w - 2

3. -15s4 - 8s3 + 6s2 - 4 24s4 + 11s2 - 15 9s4 - 8s3 + 17s2 - 19

Practice 6.5.31. (4r3 - 3r2 + 9) - (6r3 - 9r + 5) =

4r3 - 3r2 + 9 - 6r3 + 9r - 5-2r3 - 3r2 + 9r + 4

2. (t2 - 12t + 7) - (4t2 - 3t + 12) = t2 - 12t + 7 - 4t2 + 3t - 12-3t2 - 9t - 5

3. (6s - 12) -(-5s2 + 7s - 3)6s - 12 + 5s2 - 7s + 35s2 - s - 9

4. (15x2 - 3x + 6) -(-3x2 + 2x + 5) +(4x2 - 8) = 15x2 - 3x + 6 + 3x2 - 2x - 5 + 4x2 - 8= 22x2 - 5x - 7

5. (8x2y + 5xy2) - (2x2y - 7xy2 + 1) = 8x2y + 5xy2 - 2x2y + 7xy2 - 1 = 6x2y + 12xy2 - 1

Practice 6.5.51. Total profit = (x2 - 3x + 2) +(5x - 3)

Total profit = x2 + 2x - 1When x = 12, total profit =

(12)2 + 2(12) - 1 =144 + 24 - 1 = 167

2. a)Profit = revenue - costProfit = (0.1x2 + 2x) - (7x + 125) Profit = 0.1x2 + 2x - 7x - 125Profit = 0.1x2 - 5x - 125b)Profit on 50 = 0.1(50)2- 5(50) -

125Profit = 250 - 250 - 125 =-125The company is losing money ($125)when 50 are sold.c)Profit on 100=0.1(100)2-5(100)-

125Profit = 1000 - 500 - 125 = 375The company is making $375 when100 are sold.

Exercise Set 6.51. (12x2 + 11) + (4x2 - 3x + 7) =

16x2 - 3x + 18

3. (1 3

t 2 − 1 4

t + 7 ) + ( − 3 4

t 2 + 9 )

4 12

t 2 − 1 4

t + 7 + − 9 12

t 2 + 9

= − 5 12

t 2 − 1 4

t + 16

5. [1.7a3 - 6.3a2 + 4.5a - 3.2] +[-6.5a3

+ 4.1a2 - 3.6] = -4.8a3 -2.2a2 + 4.5a - 6.8

7. [-9r3 + 4r2 + 7r - 5]+[6r3 + 3r - 9]+[r3 - 5r2] =-2r3 - r2 + 10r - 14

9. -8r2 - 6r + 12-12r2 + 5-20r2 - 6r + 17

11. -9u2 + 3u - 8 6u2 - 5u - 9-3u2 - 2u - 17

17. (d2 - 3d + 4) - (-9d2 - 5d + 11) = d2 - 3d + 4) + 9d2 + 5d - 11 = 10d2 + 2d - 7


19. (6s2 + 3s - 9) - (7s2 - 4s + 8) = 6s2 + 3s - 9 - 7s2 + 4s - 8 = -s2 + 7s - 17

21. (7w3 + 4w2 -11) - (-2w3 + 6w2 -15)=

7w3 + 4w2 -11 + 2w3 - 6w2 + 15 = 9w3 - 2w2 + 4

23. (4v2-3v+2)-(5v3+3v2 +5v-8)+(3v3-8v2+3)=4v2-3v+2- 5v3-3v2 - 5v+ 8+ 3v3 -8v2+3=-2v3 - 7v2 - 8v + 13

25. (4x2+5x+11)-(3x2 - 2x + 5)-(2x +4)= 4x2+5x+11 - 3x2 + 2x - 5 - 2x -4 =

x2 + 5x + 2

27. (1.3r2- 2.2r + 6.5) - (-3.1r2 + 2.4r +5.1) + (4.2r2 - 8) = 1.3r2- 2.2r + 6.5 + 3.1r2 - 2.4r - 5.1+4.2r2 - 8 = 8.6r2-4.6r -6.6

29. Perimeter = 2l + 2wPerimeter = 2(2x2 + 5x+ 3) +

2(x2+3) Perimeter = 4x2+10x + 6 +2x2 + 6Perimeter = 6x2 + 10x + 12When x = 4, perimeter = 6(4)2 + 10(4) + 126•16 + 40 + 12 = 96 +40 +12=148

31. a) 1980-1960= 20years = x0.21(20)2 + (-2•20) + 20.8=0.21•400 - 40 + 20.8 = 84 - 40 + 20.80 = 64.8 billion dollarsb)-0.0005(20)4 + 0.031(20)3 -0.554(20)2+ 3.51(20) + 4.83 = -0.0005(160000)+ 0.031(8000)-0.554(400) + 70.2 +4.83-80 + 248 - 221.6 + 70.2 + 4.83 =21.43 billion dollarsc)64.8 -21.43 = 43.37 billion dollarsd)(0.21x2 + (-2x) + 20.8) -(-

0.0005x4 + 0.031x3 - 0.554x2+ 3.51x

+ 4.83) =0.21x2 + (-2x) + 20.8+ 0.0005x4 -0.031x3 + 0.554x2- 3.51x - 4.83=0.0005x4 - 0.031x3+ 0.754x2 - 5.51x+ 15.97For 1980:0.0005(20)4-0.031(20)3 +0.754(20)2

-5.51(20) + 15.97 = 0.0005(160000) - 0.031(8000)+0.764(400) - 110.2 + 15.97 = 80 -248 +305.6 -110.2 +15.97 = 43.37 billion dollars

33. m = -1 since the lines areperpendiculary = mx + b0 = -1(6) + b0 = -6 + bb = 6y = -x + 6

34.6034

÷ 1651

= 6034

• 5116

=

4 • 152 • 17

• 3 • 174 • 4

= 458

35. 5x + 8y is a sum whose terms are 5xand 8y; 5x is a product whose

factors are 5 and x; 8y is a productwhose factors are 8 and y.

36. 225 = 3•3•5•5

37. y = 3x + 8y - 8 = 3xy/3 - 8/3 = x

38. Yes, each value for the input givesone and only one value for the output.

39. t + 3[t - 4(t + 6) + 5] = t + 3[t - 4t - 24 + 5] = t + 3[-3t - 19] = t - 9t - 57 = -8t - 57

40. Eq 1: 8x - 2y = 14Eq 2: 5x + 3y = 133 times eq 1: 3(8x - 2y) = 3•142 times eq 2: 2(5x + 3y) = 2•13Eq 1 transformed: 24x - 6y = 42Eq 2 transformed: 10x + 6y = 26Add together 34x = 68


x = 2Substitute to solve for y: 8(2) - 2y = 1416 - 2y = 14 - 2y = -2y = 1The solution is the ordered pair

(2,1).

41. Eq 1: 5x + 2y = 1Eq 2: 2x + y = 1-2 times eq 2: -2(2x + y) = -2•1Eq 1: 5x + 2y = 1Eq 2 transformed: -4x - 2y = -2Add together x = -1Substitute to solve for y:5(-1) + 2y = 1-5 + 2y = 12y = 6y = 3The solution is the ordered pair (-

1,3).

42. Let x = width: x + 2 = lengthOriginal perimeter = 2x + 2(x + 2) =2x + 2x + 4 = 4x + 4New width = 2xNew length =x+2+10 = x + 12New perimeter = 2(2x) + 2(x + 12) New perimeter = old perimeter + 262(2x) + 2(x + 12) = 4x + 4 + 264x + 2x + 24 = 4x + 306x +24 = 4x + 302x = 6x = 3 inches for the width

43. 204 = senior tickets: adult =3(204) = 6128.50(612) + 204(8.50-2.25) = 5202 + 204(6.25) = 5202 + 1275 = $6477

44. Let x = amount of Mix Ay = amount of Mix BEq 1: 0.25x+ 0.60y = 410 g of protein0.05x + 0.20y = 110 grams of fat-3 times eq 2: -3(0.05x +0.2y)=-

3•110Eq 1: 0.25x+ 0.6y = 410Eq 2 transformed:-0.15x -0.6y = -330Add together 0.10x = 80x = 800 grams of Mix A

Substitute to solve for y: 0.25(800) + 0.60y = 410200 + 0.60y = 4100.60y = 210y = 350 grams of Mix B

45.x

years since1984

ynumber oftravelers

(in millions)

0 10.992

4 13.463

7 15.042

10 15.759

(0, 11)

(4, 13.5)(7, 15) (10, 15.8)

02468

101214161820

Number of years since 1984

Data #5y

x


Section 6.6 Multiplying Polynomials

Practice 6.6.11. 4t(3t - 9) = 12t2 - 36t

2. -9m(6m - 5) = -54m2 + 45m

3. 3m2(4m2 -2m +9)=12m4 -6m3+ 27m2

4. -5b(4b2 -7b +11) =-20b3+ 35b2 - 55b

5. (4x2 - 3x + 5)(3x) = 12x3 - 9x2 + 15x

6. (9v2 - 5v - 10)(-2v) =-18v3+ 10v2+ 20v

Practice 6.6.21. (m - 2)(4m -3) = m(4m -3) - 2(4m-3)

= 4m2 - 3m - 8m + 6 = 4m2 - 11m + 6

2. (7d + 3)(3d - 5) = 7d(3d - 5)+3(3d-5)

= 21d2 - 35d + 9d - 15 = 21d2 - 26d - 15

3. (4m -n) (3m2 - 3mn + 4n2) = 4m(3m2-3mn+4n2) -n(3m2-

3mn+4n2) =4m(3m2) -4m(3mn)+4m(4n2) -n(3m2) - n(-3mn) -n(4n2) =12m3-12m2n+16mn2-3m2n+3mn2-4n3

=12m3 - 15m2n + 19mn2 - 4n3

4. (3r -s)(5r2+6rs-3s2) = 3r(5r2 + 6rs - 3s2) -s(5r2+6rs-3s2)=15r3 + 18r2s - 9rs2 - 5r2s - 6rs2 + 3s3

15r3 + 13r2s - 15rs2 + 3s3

Practice 6.6.31. (4m - 1)2 = (4m-1)(4m-1) =

4m(4m-1) -1(4m-1) = 16m2 - 4m - 4m + 1 = 16m2 - 8m + 1

2. (5r + s)2 = (5r + s)(5r +s) = 5r(5r + s) + s(5r +s) = 25r2 + 5rs + 5rs + s2 = 25r2 + 10rs + s2

3. (6y2 - 3w)2 = (6y2 - 3w)(6y2 - 3w) =

6y2(6y2 - 3w) - 3w (6y2 - 3w) = 36y4 - 18y2w - 18y2w + 9w2 =36y4 - 36y2w + 9w2

4. (4x + 1)3 = (4x + 1)(4x + 1)(4x +1)=

(4x + 1)[(4x + 1)(4x + 1)] = (4x + 1)[4x(4x + 1) + 1 (4x + 1)]=(4x + 1)[16x2 + 4x + 4x + 1]=(4x + 1)[16x2 + 8x + 1]=4x(16x2 + 8x +1)+1(16x2 + 8x + 1)

=64x3 + 32x2 + 4x + 16x2 + 8x + 1 = 64x3 + 48x2 + 12x + 1

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

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

5b)]=(2a +5b)[4a2 + 10ab + 10ab + 25b2]

= (2a + 5b)[4a2 + 20ab + 25b2] = 2a(4a2+20ab+25b2)+5b(4a2+20ab+25b2) = 8a3+40a2b+50ab2+20a2b+

100ab2+125b3

8a3 + 60a2b + 150ab2 + 125b3

Practice 6.6.41. Total number of accidents = Number

of accidents per million • number ofmillions(525x - 25) (1.5x + 158.7) = (525x)(1.5x+158.7)-25(1.5x+158.7)=787.5x2 + 83317.5x -37.5x - 3967.5=787.5x2 + 83280x - 3967.5


2. 1990-1980= 10years = x787.5(10)2 + 83280(10) - 3967.5 =787.5•100 + 832800 -3967.578750 +832800 -3967.5= about 907583 alcohol relatedaccidents in 1990.

Practice 6.6.51. (2x + 1)(3x + 8) =

2x(3x)+2x(8)+1(3x) + 1(8) =6x2 + 16x + 3x + 8 = 6x 2 + 19x + 8

2. (5m +8)(3m - 2) = (5m)(3m) +5m(-2) +8(3m) + 8(-2)=15m2 - 10m + 24m - 16 = 15m 2 + 14m - 16

3. (2x -y)(3x -y) = (2x)(3x) - (2x)(y) - y(3x) - y(-y) = 6x2 - 2xy - 3xy + y 2 = 6x2 - 5xy + y 2

4. (3w -y)(4w + y) =(3w)(4w) + (3w)(y) - y(4w) - y(y)

= 12w 2 + 3wy - 4wy - y 2 = 12w 2 - wy - y 2

Practice 6.6.61. (w + 4)2 = (w)2 + 2(w)(4) + (4)2 =

w 2 + 8w + 16

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

3. (w - 9)2 = (w)2 - 2(w)(9) + (-9)2=w 2 - 18w + 81

4. (2r + 11)2 = (2r)2+ 2(2r)(11) +(11)2=

4r 2 + 44r + 121

Practice 6.6.71. (x + 7)(x - 7) = (x)2 - (7)2 = x2 -49

2. (r + 6m)(r - 6m) = (r)2 - (6m)2 = r 2 - 36m2

3. (4x - 5y)(4x + 5y) = (4x)2 - (5y)2 = 16x 2 - 25y 2

4. (7t - 9s)(7t + 9s) = (7t)2 - (9s)2 = 49t 2 - 81s 2

Practice 6.6.81. (m + 2)(m + 5) = m2+ (2+5)m +2•5= m 2 + 7m + 10

2. (r - 8)(r - 4) = r2 + (-8-4)r +(-8)•(-4)=

r2 - 12r + 32

3. (x +7)(x -12) = x2+(7 - 12)x +7(-12)=

x2 - 5x - 84

4. (y -2)(2y +1 =y(2y)+1•y -2(2y) -2•1=

2y2 + y - 4y - 2 =2y2 - 3y - 2

Exercise Set 6.6

1. 5t3(4t - 12) = 20t4 - 60t3

3. − 1 4

m 4 ( 12m 2 − 8 ) = - 3 m 6 + 2 m 4

5. 3y3(4y4 + 2y2 - 1) = 12y7 + 6y5 - 3y3

7. 6r2(3r2 - 5r + 7) = 18r4 - 30r3 + 42r2

9. (2x2 - 4x + 1) (5x2) = 10x4 - 20x3 + 5x2

11. (3w2 - 8w + 7)(4w) = 12w3 - 32w2 + 28w

13. (4x + 7)(2x - 8) = 4x(2x) - (4x)(8) + 7(2x) - 7(8)=8x2 - 32x + 14x - 56 = 8x2 - 18x - 56

15. (2x-8)(3x-12) = 2x(3x)-2x(12) - 8(3x) - 8(-12)=6x2 -24x-24x +96=6x2-48x +96

17. (2x + 1)(4x2 - 2x + 1) =


2x(4x2 - 2x + 1) +1(4xv - 2x + 1) =8x3 - 4x2 + 2x + 4x2 - 2x + 1=8x3 + 1

19. (4m +3n)(5m2 - 2mn - 6n2) = 4m(5m2-2mn- 6n2)+3n(5m2- 2mn-

6n2)= 20m3-8m2n-24mn2+15m2n -6mn2-

18n3 = 20m3 + 7m2 n - 30mn2-18n3

21. (8x2 + x -3)(x2+ 2x- 6) = 8x2(x2+2x-6)+ x(x2+2x-6) - 3(x2+2x- 6)=8x4+16x3-48x2+x3+2x2-6x-3x2 -6x+18= 8x4 + 17x3 - 49x2-12x + 18

23. (3x + 7)2 =(3x + 7)(3x + 7) = 3x(3x + 7) +7(3x + 7) =9x2+21x + 21x + 49 = 9x2 + 42x + 49

25. (3m + 5n)2 = (3m + 5n)(3m + 5n) = 3m (3m + 5n) + 5n (3m + 5n) = 9m2 + 15mn + 15mn + 25n2 = 9m2 + 30mn + 25n2

27. (3z - w)2=(3z - w)(3z - w) = 3z(3z - w)-w(3z - w) = 9z2 - 3zw - 3zw + w2 = 9z2 - 6zw + w2

29. (x + 3)3 =(x + 3)(x + 3)(x + 3) = (x + 3)[(x + 3)(x + 3)] = (x + 3)[x(x + 3) + 3(x + 3)]=(x + 3)[x2 + 3x + 3x + 9] =(x + 3) [x2 + 6x + 9] =x(x2 + 6x + 9) + 3(x2 +6x + 9) =x3 + 6x2 + 9x + 3x2 + 18x + 27 = x3 + 9x2+ 27x + 27

31. (2x - y)3 = (2x - y)(2x - y)(2x - y) = (2x-y)[(2x - y)(2x - y)] = (2x - y) [2x(2x - y) - y(2x - y) ] =(2x - y) [4x2 - 2xy - 2xy + y2] = (2x - y) [4x2 - 4xy + y2] = 2x(4x2 -4xy +y2 ) -y(4x2 - 4xy +

y2)=8x3 - 8x2y + 2xy2 - 4x2 y+4xy2 - y3=8x3 -12x2y + 6xy2 - y3

33. (4g - 9)(3g - 7) = (4g)(3g) - (4g)(7) -9(3g) -9(-7) = 12g2 - 28g - 27g + 63 = 12g2 - 55g + 63

35. (7s + 3)(4s - 5) = 7s(4s) - 7s(5) + 3(4s) - 3(5) = 28s2 - 35s + 12s - 15 = 28s2 - 23s - 15

37. (9r -s)(4r-s) =9r(4r) - (9r)(s) - s(4r) - s(-s) =36r2 - 9rs - 4rs + s2 = 36r2 - 13rs + s2

39. (5m + 3n)(4m + 2n) = 5m(4m) + 5m(2n) +3n(4m) +

3n(2n)= 20m2 + 10mn + 12mn + 6n2 = 20m2 + 22mn + 6n2

41. (9a + 7b)(3a - 4b) = 9a(3a) - 9a(4b) + 3a(7b) - (7b)(4b)

=27a2 - 36ab + 21ab - 28b2 = 27a2 - 15ab - 28b2

43. (5m - 4)2 = (5m)2 - 2(5m)(4) + (-4)2

= 25m2 - 40m + 16

45. (4x - 5y)2= (4x)2 -2(4x)(5y) +(-5y)2

= 16x2 - 40xy + 25y2

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

49. (a - 4b)(a + 4b) = (a)2 - (4b)2 = a2 - 16b2

51. (3s + 5t)(3s - 5t) = (3s)2 - (5t)2 = 9s2 - 25t2

53. ( 1 2

x - 2 3

) ( 1 2

x + 2 3

) =


( 1 2

x ) 2 − ( 2 3

) 2 =

1 4

x 2 − 4 9

55. (y + 7)(y + 8) = y2 + (7+8)y + 7•8 =y2 + 15y + 56

57. (r -8)(r -5) = r2 + (-8 -5)r + (-8)(-5)=

r2 - 13r + 40

59. (w + 7)(w + 5) = w2 +(7+5)w + 7•5=

w2 + 12w + 35

61. Multiplication distributes overaddition

62.1026

• 3940

= 102 • 13

• 3 • 134 • 10

= 3 8

63. y = 2/3 x + 0

-10-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-10 1 2 3 4 5 6 7 8910 x

Data #6

64. -( − 1 2

x 4 y 2 ) 4 = − ( - 1 2

) 4 x 16y 8 =

− 1 16

x 16y 8

65. (-1,-4)

-10-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x

Data #7

66. Eq 1: 5x - 3y = 5Eq 2: x - y = 1Solve eq 2 for x: x = y + 1Substitute in eq 1: 5(y + 1) - 3y = 55y + 5 - 3y = 52y = 0y =0Substitute to solve for x: x - 0

= 1x = 1The solution is the ordered pair

(1,0).

67. Time for second couple = 4 hrsTime for first couple = 4.5 hrsLet x = rate of first couplex + 0.5 = rate for seconddistance = rate • timeThe distances of the two couples areequal.4.5x = 4(x + 0.5) 4.5x = 4x + 20.5x = 2x = 4 mph for the first4 + 0.5 = 4.5 mph for the second

68. Line 1 m = 8 - 5

2 - ( - 2 ) = 3

2 + 2 = 3

4

Line 2 m=9 - 6 7 - 3

= 3 4

The slopes are equal so lines areparallel.

69.2 . 2 x 1041

2 x 1030= 1 . 1 x 1011


70. This is a second degree polynomialbecause on the term 4pq the

exponent on each variable is 1 and 1+1 = 2.

71. Domain {-9 ≤ x≤1} Range {-8 ≤ y≤8}

72. Using (-1,0) and (0,-8):

m = − 8 - 0 0 - ( - 1 )

= − 8 1

= - 8

y intercept is (0,-8) so b =-8y = -8x - 8

Section 6.7 Dividing Polynomials

Practice 6.7.1

1.14w 8 - 21w 5 + 7 w 4

7 w 3 =

14w 8

7 w 3 − 21w 5

7 w 3 + 7 w 4

7 w 3 =

2w5 - 3w2 + w

2.− 28a 6 + 20a 5 + 12a 3

4 a 2 =

- 28a 6

4 a 2 + 20a 5

4 a 2 + 12a 3

4 a 2 =

-7a4 +5a3+3a

3.35b 7 − 15b 4 + 10b

5 b 5 =

35b 7

5 b 5 − 15b 4

5 b 5 + 10b

5 b 5 =

7b2 - 3 b

+ 2

b 4

4.− 36d 9 + 18d 5 + 9 d 3

18d 4 =

− 36d 9

18d 4 + 18d 5

18d 4 + 9 d 3

18d 4 =

-2d5 + d + 1 2 d

Practice 6.7.21.

2 x 2 - x- 15

x + 2 2 x 3 + 3 x 2 - 17x - 30 2 x 3 + 4 x 2 S ubtract - x 2 - 17x - x 2 - 2 x Subtract − 15x - 30 - 15x - 30 Subtract 0


2.

12y 2 + 11y - 5 + 2

y - 1

y - 1 12y 3 − y2 − 16y + 7 12y 3 − 12y 2 Subtract 11y 2 − 16y 11y 2 − 11y Subtract - 5 y + 7 - 5 y + 5 Subtract 2

2 r 2 − r - 6 + 1

2 r + 5

2 r + 5 4 r 3 + 8 r 2 − 17r - 29 4 r 3 + 10r 2 Subtract - 2 r 2 - 17r - 2 r 2 - 5 r Subtract − 12r - 29 − 12r - 30 Subtract 1

3.

4.

4 x 2 + 8 x + 16

2 x - 4 8 x 3 + 0 x 2 + 0 x - 64 8 x 3 − 16x 2 Subtract 16x 2 + 0 x 16x 2 − 32x Subtract 32x - 64 32x - 64 Subtract 0

Exercise Set 6.7

1.4 m 2 − 8 m

4 m = 4 m 2

4 m − 8 m

4 m = m - 2

3.5 y 7 + 10y 4 + 25y

5 y 3 =

5 y 7

5 y 3 + 10y 4

5 y 3 + 25y

5 y 3 =

y 4 + 2 y + 5

y 2

5.15m 9 − 10m 7 + 5 m 4

15m 8 =

15m 9

15m 8 − 10m 7

15m 8 + 5 m 4

15m 8 =

m - 2 3 m

+ 1

3 m 4

7.45t 6 + 3 t 4 − 6 t 3

9 t 5 =

45t 6

9 t 5 + 3 t 4

9 t 5 − 6 t 3

9 t 5 =

5 t + 1 3 t

− 2

3 t 2

9.

3 x - 7

x + 2 3 x 2 − x - 14 3 x 2 + 6 x Subtract − 7 x - 14 − 7 x - 14 Subtract

0

11.

2 y - 8

5 y + 6 10y 2 - 28y - 48 10y 2 + 12y Subtract - 40y - 48 - 40y - 48 Subtract 0

13.

6 x 2 + 7 x - 3

x - 1 6 x 3 + x 2 - 10x + 3

6 x 3 - 6 x 2 Subtract

7 x 2 - 10x

7 x 2 - 7 x Subtract

- 3 x + 3 - 3 x + 3 Subtract 0


15.

2 x 2 - x + 1

2 x - 5 4 x 3 - 12x 2 + 7 x - 5 4 x 3 - 10x 2 Subtract - 2 x 2 + 7 x - 2 x 2 + 5 x Subtract 2 x - 5 2 x - 5 Subtract 0

17.

19.

5 x 2 − 7 x + 12 − 17

x + 2

x + 2 5 x 3 + 3 x 2 − 2 x + 7 5 x 3 + 10x 2 Subtract

- 7 x 2 - 2 x - 7 x 2 - 14x Subtract

12x + 7 12x + 24 Subtract

- 17

2 x 3 + 6 x 2 + 15x + 45 + 137

x - 3

x - 3 2 x 4 + 0 x 3 - 3 x 2 + 0 x + 2 2 x 4 - 6 x 3

6 x 3 - 3 x 2 6 x 3 - 18x 2

15x 2 + 0 x 15x 2 - 45x

45x + 2 45x - 135 137

21. x< 0 <---------------------

<--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->x

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

22.

-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x

Data #8

23.

− 2 − 4 − 1516

+ 3 0 = - 1

2 4 − 15

16+ 1 =

− 1 16

− 1516

+ 1 = − 1616

+ 1 = 0

24.( w 4 g ) − 2

( w − 2 ) 4 g = w − 8 g − 2

w − 8 g = 1

g•g2 = 1

g 3

25. Eq 1: 3x - y = 4Eq 2: 2x - 3y = -2-3 times Eq 1: -3(3x - y) = -3•4Eq 1 transformed: -9x + 3y = -12Eq 2: 2x - 3y = -2Add together -7x = -14x = 2Substitute to solve for y: 3(2) - y = 46 - y = 4-y = -2y = 2The solution is the ordered pair

(2,2).

26. The lines are parallel. There is nosolution.


-10-9-8-7-6-5-4-3-2-10123456789

10

y

-10-9-8-7-6-5-4-3-2-101 2 3 4 5 6 7 8 910 x

Data #1

27. m = -1 and the y intercept is (0,3).

28. (2xy - 3y) - (-8xy + 5x - y) = 2xy - 3y + 8xy - 5x + y=10xy - 2y - 5x

29. (6 x 104)(2 x 10-2) = 12 x 102 =1.2 x 10 x 102 = 1.2 x103

30. 135 = 33 • 5

31. a)f = fast train; slow = f + 115b)slow = 1/2 fc)slow = 1/4 fd)slow = f + 80

32. Let x = full pricex - 0.15x = 3400.85x = 340x = $400

33. m = 168 - 140

8 - 15= 28

− 7 = − 4

1 She is selling 4 boxes per house. (With every house her remainingboxes decrease by 4).

Chapter 6 Review Exercises

1. m6n4n3m = m6+1n4+3= m7n7

2. 9b9b2 = 81bb2 = 81b3

3. -a(-4a2 + 3a) = 4a3 - 3a2

4.1 5

y 8 ( - 15y 2 ) ( 1 2

y ) = − 3 2

y 11


5. (-x3y)4 = (-x3)4y4 = x12y4

6. -(a2b)2 = -a4b2

7.( b 2 b 4 ) 2 ( 2 b ) 3

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

2 b 4 =

b 12( 8 b 3 )

2 b 4 = 8 b 15

2 b 4 = 4 b 11

8.4 x − 2

x − 5 = 4 x 5

x 2 = 4 x 3

9.6 xy− 2

3 x − 2 y − 5 = 6 xx2 y 5

3 y 2 = 2 x 3 y 3

10.8 c 3 y 2

12c − 2 y 5 = 2 c 3 c 2

3 y 3 = 2 c 4

3 y 3

11.4 − 1 + 3 − 2

2 − 3 =

1 4

+ 1

3 2

1

2 3

=

( 1 4

+ 1 9

) ÷ 1

2 3 = ( 9

36+ 4

36) • 2 3 =

1336

• 8 = 269

12. 2-3 - 3-2 + 2-2 + 40 = 1

2 3 − 1

3 2 + 1

2 2 + 1 =

1 8

− 1 9

+ 1 4

+ 1 =

9 72

− 8 72

+ 1872

+ 1 = 1972

+ 1 = 1 1972

13. ( − 8 x 3 y

2 xy2 ) 2 ( x − 2 y − 2 ) =

( − 4 x 2

y ) 2 (

1

x 2 y 2 ) = 16x 4

y 2 (

1

x 2 y 2 )

= 16x 2

y 4

14. − 2 t ( − 2 t − 2

( - 2 t ) 2 ) = - 2 t (

− 2 t − 2

4 t 2 )

- 2 t ( − 2

4 t 2 t 2 ) = - 2 t (

− 1

2 t 4 ) = 1

t 3

15.8 x − 3 x 10

( 2 x 2 ) 2 = 8 x 7

4 x 4 = 2 x 3

16.12x 2 ( 2 xy3 ) 2

3 x ( 2 x − 3 ) 2 y = 12x 2 ( 4 x 2 y 6 )

3 x ( 4 x − 6 ) y 48x 4 y

12x − 5 y = 48x 4 x 5 y 6

12y = 4 x 9 y 5

17. -0.00083 = -8.3 x 10-4 since wemoved the decimal point 4 places tothe right.

18. 350 x 107 =3.5 x 102 x107 = 3.5 x109

19. Already in scientific notation

20. Already in scientific notation

21.1 . 0 x10− 10

0 . 010= 1 . 0 x10− 10

1 . 0 x10− 2 =

1 . 0 x10− 10x 102 = 1 . 0 x10− 8

22.( 7 . 2 x 10− 10) ( 1 x10− 3 )

1 . 6 x 10− 11=

7 . 2 x10− 13

1 . 6 x 10− 11= 4 . 5 x 10− 13x 1011 =

4 . 5 x10− 2

23. 1,000,000 =106

6.7x10-10 x 1.0 x106 = 6.7 x10-4 Thisis not as heavy as a grape. To findhow many specks of dust it wouldtake divide the weight of a speck ofduct into the weight of a grape.

3 x 10− 3

6 . 7 x10− 10= 0 . 4477612x 10− 3 x 1010=

4 . 477612x 10 − 1 x 10 7 = 4 . 477612x 10 6

This is about 4.48 million specks ofdust.

24.8 . 9 x 103

1 . 7 = 5 . 24x 103 or about 5,240

people

25. Yes


26. No, the negative exponent is not awhole number.

27. This is a second degree polynomial.

28. This is a fifth degree polynomialsince the x is third degree andthe y is second degree in the firstterm.

29. -3x2 + 5x +2

30.(-5xy+2x)+(3x-2xy)+(4x2-xy+1) = 4x2 -8xy + 5x + 1

31. ( 1 3

a 3 b + 1 2

ab3 ) + ( 3 4

a 2 b 2 + 3 4

ab3 )

+ ( 5 3

a 3 b - 1 2

a 2 b 2 ) =

4 12

a 3 b + 2 4

ab3 + 3 4

a 2 b 2 + 3 4

ab3

+ 2012

a 3 b - 2 4

a 2 b 2 =

2412

a 3 b + 5 4

ab3 + 1 4

a 2 b 2 =

2 a 3 b + 5 4

ab3 + 1 4

a 2 b 2

32. (-x2+ 5x) - (5x2 + 7x - 3) = -x2 + 5x - 5x2 - 7x + 3 = -6x2 - 2x + 3

33. -4m2(3m2 - 2m + 4) = -12m4 + 8m3 - 16m2

34. x(x - y + 1) + y(x - y + 1) = x2 - xy + x + xy - y2 + y = x2 + x + y - y2

35. (4x -y) - (y - 4x) - (x -4y) - (-4x - y)=

4x - y - y + 4x - x + 4y + 4x + y = 11x + 3y

36. (3m - 1)(6m + 5) = 3m(6m) + 3m(5) - 1(6m) -1(5) = 18m2 + 15m - 6m - 5 = 18m2 + 9m - 5

37. (k+1)(k3 - k2 - k) =

k (k3 - k2 - k) + 1 (k3 - k2 - k) = k4 - k3 - k 2 + k3 - k2 - k = k4 - 2k2 - k

38. (2y -3)3 = (2y - 3)(2y - 3)(2y - 3) = (2y - 3)[(2y - 3)(2y - 3)] = (2y - 3)[2y(2y - 3) - 3(2y - 3)]=(2y - 3)[4y2 - 6y - 6y + 9] = (2y - 3)[4y2 - 12y + 9] =2y(4y2 - 12y + 9] - 3(4y2 - 12y + 9)

=8y3 - 24y2 + 18y - 12y2 + 36y - 27 = 8y3 - 36y2 + 54y - 27

39. (3a -2)(5a + 9) = 3a(5a) + 3a(9) - 2(5a) -2(9) = 15a2 + 27a - 10a - 18 =15a2 +17a - 18

40. (z - 8)(2z - 11) = z(2z) - 11(z) - 8(2z) - 8(-11) =2z2 - 11z - 16z + 88 = 2z2 - 27z + 88

41. (5a - 4)2 = (5a - 4)(5a - 4) = 5a(5a - 4) - 4(5a - 4) = 25a2 - 20a - 20a + 16 = 25a2 - 40a + 16

42. (9d -5)(9d +5) = (9d)2 - (5)2 = 81d2 - 25

43. (6y - 1)(6y + 1) = (6y)2 - (1)2 = 36y2 - 1

44.− 20r 7 − 16r 5 + 8 r 3

− 4 r 2 =

− 20r 7

− 4 r 2 − 16r 5

− 4 r 2 + 8 r 3

− 4 r 2 = 5 r 5 + 4 r 3 − 2 r

45.51d 6 + 57d 5 − 93d 4

3 d 4 =

51d 6

3 d 4 + 57d 5

3 d 4 − 93d 4

3 d 4

17d2 + 19d - 31


46. 3 y 2 + 3 y - 6 + 2

y + 4

y + 4 3 y 3 + 15y 2 + 6 y - 223 y 3 + 12y 2

3 y 2 + 6 y 3 y 2 + 12y

- 6 y - 22 − 6 y - 24 2

47.

6 y 3 - 1 - 1

2 y - 5

2 y - 5 12y 4 − 30y 3 − 2 y + 4 12y 4 − 30y 3

- 2 y + 4 − 2 y + 5 - 1


Chapter 6 Test1. a3ba2b2 = a5b3

2. -1/3 z(2z5)(-6z12) = 4z18

3.( 3 x 3 y 2 ) 2 ( xy3 ) 2

9 ( xy) 3 = 9 x 6 y 4 ( x 2 y 6 )

9 x 3 y 3 =

x 3 y ( x 2 y 6 ) = x 5 y 7

4.5 m − 3

m − 6 = 5 m 6

m 3 = 5 m 3

5. 4 • 3 − 1 − 3 • 4 − 1 = 4 • 1 3

− 3 • 1 4

=

4 3

− 3 4

= 1612

− 9 12

= 7 12

6. − ( 1 3

) 4 + 3 0 + ( - 3 ) − 4 =

- 1 81

+ 1 + 1

( - 3 ) 4 = − 1

81+ 1 + 1

81= 1

7. ( 3 m − 8

( 2 m ) − 2 ) 2 = 9 m − 16

( 2 m ) − 4 =

9 ( 2 m ) 4

m 16= 9 • 16m 4

m 16= 144

m 12

8.6 x − 2 ( 3 y 3 ) 2

x − 3 y ( 18xy− 2 ) = 6 x 3 • 9 y 6 y 2

x 2 y ( 18x ) =

54x 3 y 8

18x 3 y = 3 y 7

9. 3c2(5c2 - c - 3) = 15c4 - 3c3 - 9c2

10. (5a - 9)(5a - 4) = (5a)(5a) - (5a)(4) - 9(5a) - (9)(-4)=25a2 - 20a - 45a + 3625a2 - 65a + 36

11. (2r + 3s - t)(r - t) = (2r + 3s - t)r + (2r + 3s - t)(-t) =

2r2 + 3rs - rt - 2rt - 3st + t2

12. (5x - 2)2 = (5x)2 + 2(5x)(-2) + (-2)2 25x2 - 20x + 4

13. 49,000,000 = 4.9 x 107 since wemoved the decimal point 7 places tothe left.

14. 8.01 x 10-4 = 0.000801 since thenegative exponent tells us to move

the decimal point 4 places to the left.

15.( 2 . 5 x 10− 3 ) ( 3 . 2 x 108 )

1 . 6 x 10− 4 =

2 . 5 ( 2 ) x 105 x 104 = 5 . 0 x 109

16. 4500 = 4.5 x 103

72,000,000 =7.2 x107

1000 = 1.0 x103

The weight of 1000 elephants = (4.5x 103) x (1.0x 103) = 4.5 x106

1000 elephants does not weigh asmuch as an ocean liner. To find howmany elephants it would take:7 . 2 x 107

4 . 5 x 103 = 1 . 6 x104 or 16000

elephants

17. No, there is a variable in thedenominator.

18. This is a third degree polynomial.

19. (-8x2 - 3x + 1) + (5x2 + 3x - 6) = -3x2 - 5

20. -x4 + 12x3 + 7x2 - 3

21. (-12c4+11d2 -11e)-(16c4-14d2 -11e)=

-12c4 +11d2 - 11e -16c4 +14d2 +11e=-28c4 + 25d2

22. (3w2-5w -2)-(4w -6w -5)-(8w2+6w- 1)=3w2-5w-2-4w2+6w+5-8w2-6w+1=-9w2 - 5w + 4

23.15q 3 − 5 q 2 − 15q

5 q =


15q 3

5 q − 5 q 2

5 q − 15q

5 q = 3 q 2 − q - 3

24.

h3 − 4 h 2 + h + 4

h + 6 h 4 + 2 h 3 − 23h 2 + 10h + 24h 4 + 6 h 3

− 4 h 3 − 23h 2 − 4 h 3 − 24h 2

h2 + 10h h 2 + 6 h

4 h + 24 4 h + 24 0


chapter 6 exponents and practice 6.1.3 1. (2 4 3 = 2 4•3 ...chapter 6 exponents and polynomials...

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