chapter 7 factoring algebraic 3. 9(x - 5) 5 = 3 2 •(x - 5 ... 7 factoring algebraic expressions...

Chapter 7 Factoring Algebraic Expressions

Section 7.1 Greatest Common Factor (GCF)

Practice 7.1.11. 28 = 22•7

42 = 2• 3• 770 = 2•5•7GCF = 2•7=14

2. 72 = 23•32

96 = 25 •3144 = 24•32

GCF = 23•3 = 24

3. 36 = 22•32

72 = 23 •32

144 = 24•32

GCF = 22•32=36

Practice 7.1.21. 4r4 = 22• r4

16r2 = 24•r2

24r5 = 23•3•r5

GCF = 22•r2 = 4r2

2. -10t2 = -1•2•5•t2

-25t5 = -1•52•t5

-15t4 = -1•3•5•t4

GCF = -1•5•t2 = -5t2

3. -18r4s 4 = -1•-2•32•r4s 4

-12r9s 8 = -1•22•3•r9s 8

-36r3 = -1•22•32r3

GCF = -1•2•3•r3 = -6r3

4. 3a11b3 = 3•a11•b3

9a6b = 32•a6•b15a7b4 = 3•5•a7•b4

GCF = 3a6b

Practice 7.1.31. x(x-8) = x•(x-8)

y(x-8) = y•(x-8)GCF =(x-8)

2. 6(x - 1)3 = 2•3•(x-1)3

18(x-1)2 = 2•32•(x - 1)2 GCF = 2•3(x - 1)2 = 6(x -1)2

3. 9(x - 5)5 = 32•(x - 5)5

12(x - 5)6 = 22•3•(x - 5)6

GCF = 3(x - 5)5

4. 21(a + b)4 = 3•7•(a + b) 414(b + a)2 = 2•7•(b + a)2

GCF = 7(a + b)2

Exercise Set 7.1

1. 36 = 22•32

48 = 24•3GCF = 22•3 = 12

3. 24 = 23•342 = 2•3•7GCF = 2•3 = 6

5. 27 = 33

54 = 2•33

108 = 22•33

GCF = 33 = 27

7. 2x3y5 = 2•x3•y5

3x8y2 = 3•x8•y2

GCF = x3y3

9. 5y4 = 5•y4

15y7 = 3•5•y7

10y3 = 2•5•y3

GCF = 5y3

11. 9s6 = 32•s6

6s5 = 2•3•s5

15s4 = 3•5•s4

GCF =3s4

13. m3(m+5)2 = m3•(m+5)2 m2(m+5)8 = m2•(m+5)8

GCF = m2(m+5)2

15. m(n+1)8 = m•(n+1)8

5m3(n+1)4 = 5•m3•(n+1)4

GCF = m(n+1)4

17. 18m5(m+2) = 2•32•m5•(m+2)24m2(m+2)3 = 23•3•m2•(m+2)3 GCF = 2•3•m2•(m+2) =6m2(m+2)

19. 6x3(3x+7)3 = 2•3•x3(3x+7)3

15x4(3x+7)7 = 3•5•x4(3x+7)7

194 CHAPTER 7 FACTORING ALGEBRAIC EXPRESSIONS

GCF = 3•x3(3x+7)3 = 3x3(3x+7)3

21. 5t3(t + 12)2 = 5•t3•(t+12)2

15(t + 12) = 3•5•(t+12)GCF = 5(t+12)

23. 3+2x is a sum whose terms are 3and 2x. 2x is a product whose factorsare 2 and x.

24. 48 = 24•3

25. m = 1 - 2 7 - 7

= − 3 0

is undefined

The line is vertical so the equation is x = 7.

26.1 2

x - 1 3

( 3 - 9 x ) ≥ 2 x

6 • 1 2

x - 6 • 1 3

( 3 - 9 x ) ≥ 6 • 2 x

3x -2(3-9x) ≥12x3x - 6 + 18x ≥12x21x - 6 ≥ 12x9x ≥6x ≥6/9x ≥ 2/3

27. Let x = monthly cost of cheaper x + 15 =monthly cost of expensive27x = 18(x + 15)27x = 18x + 2709x = 270x = $30 a month for the cheapercomputer

28. C = 2πrTotal radius = inside radius +thickness of pipeLet x = thickness of pipe47.1 = 2•3.14(x + 2)47.1 = 6.28(x + 2)47.1 = 6.28x + 12.5634.54 = 6.28xx = 5.5 mm

29. f(x) = x2 - 5f(-3) = (-3)2 - 5f(-3) = 9-5 = 4

30.18y 4 − 6 y 3 − 6 y 2

6 y 2 =

18y 4

6 y 2 − 6 y 3

6 y 2 − 6 y 2

6 y 2 =

3y2 - y - 1

31. Eq 1: 2x - 3y = -7Eq 2: 21 + 6x = 9yEq 2 in standard form: -6x + 9y = 213 times Eq 1: 3(2x - 3y) = -

7(3)Eq 1 transformed: 6x - 9y = -21Eq 2: -6x + 9y = 21Add together 0 = 0The lines are the same. Solve for y: 2x - 3y = -7-3y = -7-2xy = 7/3 + 2/3 xThe solutions can be represented by(x, 7/3 + 2/3 x)

32. Eq 1: 3x + 5y = -12Eq 2: -2x + 3y = -112 times eq 1: 2(3x + 5y) = 2(-12)3 times eq 2: 3(-2x +3y) = 3(-11)Eq 1 transformed: 6x + 10y = -24Eq 2 transformed: -6x + 9y = -33Add together 19y = -57y = -3Substitute to solve for x:3x + 5(-3) = -123x - 15 = -123x = 3x = 1The solution is the ordered pair (1,-3).

33. -(-w2g5)2 = - (w4g10) = -w4g10

34. -4-2 - 2-4 = − 1

4 2 − 1

2 4 =

- 1 16

− 1 16

= - 2 16

= - 1 8

35. 3x(5x2 - 4x + 1) = 15x3 - 12x2 + 3x

36. (x - 3)(3x2 + 2x - 4) = x(3x2 + 2x - 4) - 3(3x2 + 2x - 4) = 3x3 + 2x2 - 4x - 9x2 - 6x + 12 = 3x3 - 7x2 - 10x + 12

37. Yes, for each input value there isone and only one value for the output.

195CHAPTER 7 FACTORING ALGEBRAIC EXPRESSIONS

38. a)Let x = number of quarters; y =number of dollarsy = x/4

b)

xnumber ofquarters

ynumber of

dollars

8 8/4 = 2

18 18/4 = 4.50

30 30/4 = 7.50

6•4 = 24 6

9.5•4 = 38 9.5

c)

123456789

101112

Number of quarters

Data #1y

x


Section 7.2 Factoring the GCF Out of Polynomials

Practice 7.2.11. The GCF of 10 and 15 is 5.

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

2. The GCF of 3 and 12 is 3: the GCFof x6 and x2 is x62 . 3x6 + 12x2 = 3x2 (x4) + 4•3x2 = 3x2 (x4 + 4)

3. The GCF of 64,16 and 8 is 8; there isno GCF for the variable factors.64r4 + 16r2 - 8 = 8(8r4) + 8•2r2 -1•8=8(8r4 + 2r2 -1)

4. The GCF of 45 and 15 is 15; GCF ofa4 and a3 is a3; GCF of b5 and b2 is

b2

45a4b5 - 15a3b2 =15a3 b2 (3ab3) -1(15a3 b2 ) =15a3 b2 (3ab3 -1)

5. The GCF of 27 and 54 is 27; GCF ofc7 and c3 is c3; GCF of d9 and d10 isd9

27c7d9 - 54c3d10 =27c3 d9 (c4) -2d 27c3 d9 =27c3 d9 (c4 -2d)

Practice 7.2.21. -25 = 1(-25) = -1(25)

2. 32 = 1(32) = -1(-32)

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

4. 4m - 2n = 1(4m - 2n) = -1(-4m + 2n)

5. 7v4 - 8v3 + 2 = 1(7v4 - 8v3 + 2) = -1(-7v4 + 8v3 - 2)

6. -12w3 + 15w2 + 11 = 1(-12w3 + 15w2 + 11) = -1(12w3 - 15w2 - 11)

7. 25 - w2 = 1(25 - w2) = -1(-25 + w2)

Practice 7.2.31. -w7 - 6w = -w(w6) + 6(-w) =

-w(w6 + 6)

2. -6m5 + 12m = -6m(m4) + -6m(-2) = -6m(m4 - 2)

3. -4k4 + 8k3 - 12k = -4k(k3) + -4k(-2k2) + -4k(3) =-4k(k3 - 2k2 + 3)

4. -6r5 + 18r4 - 12r = -6r(r4) + (-6r)(-3r3) + (-6r)(2) =-6r(r4 - 3r3 + 2)

Practice 7.2.41. GCF = (z+1)

5z(z+1) - 8(z + 1) = (z + 1) (5z - 8)

2. GCF = 2(t+ 9)2t(t + 9) + 6(t + 9) =t•2(t+9) + 3•2(t+9) = 2(t +9)(t +3)

3. GCF = 4(n2 - 8)4m2(n2 - 8) - 12(n2 - 8) = m2•4 (n2 - 8) - 3•4 (n2 - 8) = 4 (n2 - 8)(m2 - 3)

4. GCF = 7(x + y)7(x + y)4 - 14(x + y)2 + 21(x + y)=(x+y)3•7(x+y) -2(x+y)•7(x+y)+3•7(x+y) = 7(x + y)[(x+y)3 - 2(x + y) +3]

5. GCF = 4(m - n)12(m - n)5 +8(m -n)3 - 4(m- n) = 3(m-n)4•4(m- n) +2(m-n)2•4(m- n) -

1•4(m- n) =4(m- n)[3(m-n)4 + 2(m-n)2 - 1]


Practice 7.2.51. 6p2 + 4p + 3rp + 2r =

(6p2 + 4p) + (3rp + 2r) = 2p(3p + 2) + r(3p + 2)=(3p + 2)(2p + r)

2. 10q2 + 25rq + 14q + 35r =(10q2 + 25r) + (14q + 35r) =5q(2q + 5r) + 7(2q + 5r)+(2q + 5r)(5q + 7)

3. 6x2 + 2xy + 24xz + 8yz = 2(3x2 +xy +12xz + 4yz)2[(3x2 + xy) + (12xz + 4yz)] 2[x(3x + y) +4z((3x + y)] =2(3x + y)(x + 4z)

Practice 7.2.61. y2 + xy - y -x =

(y2 + xy) + (-y -x) = y(y + x) + (-1)(y + x) =(y + x)(y -1)

2. 3a2 + 6ab - 2a -4b =(3a2 + 6ab) + (- 2a -4b) =3a(a + 2b) +(-1)(2)(a + 2b) =(a + 2b)(3a -2)

3. 6x2 + 14xz - 9xy - 21yx = (6x2 + 14xz) + (- 9xy - 21yz) =2x(3x +7z) + (-1)(3y)(3x +7z) =(3x +7z)(2x - 3y)

4. 10a2 + 6ab - 35a - 21b = (10a2 + 6ab) + (- 35a - 21b) =2a(5a +3b) + -1(7)(5a +3b) =(5a +3b)(2a - 7)

Exercise Set 7.21. GCF = 12r4s 3

12r4s 3 + 36r6s 7 =12r4 s 3 (1) +12r4 s 3 (3r2s 4) =12r4 s 3 (1 + 3r2s 4)

3. GCF = 2xy4x6y3 - 16x2y4 + 2xy = 2xy(2x5y2) -2xy(8xy3) + 1•2xy =2xy(2x5y2 -8xy3 + 1)

5. GCF = 6x3y4

12x5y7 - 18x8y4 + 36x3y12 =6x3 y4 (2x2y3)-3x5(6x3 y4 )+6y8(6x3 y4 ) =

6x3 y4 (2x2y3 - 3x5 + 6y8)

7. GCF = 5f3g15f5g12 - 25f3g9 + 45f13g=5f3 g(3f2g11) - 5g8(5f3 g) +9f10(5f3 g)5f3 g(3f2g11 - 5g8 + 9f10)

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

11. 5t + u = 1(5t + u) = -1(-5t -u)

13. 4p4 - 3p3 + 9 = 1(4p4 - 3p3 +9) =-1(-4p4 + 3p3 - 9)

15. -5x3 + 7x -9 = 1(-5x3 + 7x -9) =-1(5x3 - 7x + 9)

17. 25 - w3 = 1(25 - w3) = -1(-25 + w3)

19. -x3 + 3x2 - 5x = (-x)(x2) +(-x)(-3x) +5•(-x) =-x(x2 -3x + 5)

21. GCF = -2xy-2xy5 + 4xy3 - 8xy2 + 6xy =-2xy(y4)+(-2xy)(-2y2)+(-2xy)(4y) +(-3)(-2xy) =(-2xy)(y4 - 2y2 +4y - 3)

23. GCF = -4rst-8r6s 3t5 + 4r3s 4t2 - 12rst =(-4rst)(2r5s 2t4) +(-4rst)(-r2s 3t) + (-4rst)(3) =(-4rst)(2r5s 2t4 -r2s 3t + 3)

25. GCF = r2 +32r(r2 + 2) - 3(r2 + 2) =(r2 + 2)(2r -3)

27. GCF = r(r + 3)6r(r + 3) - 5r2(r + 3) =r(r + 3)•6 - 5r• r(r + 3)r(r + 3)(6 - 5r)

29. GCF = 4m(m+5)16m3(m +5)2 - 12m(m +5)=4m(m+5)•4m2(m+5)- 3•4m(m+5)=4m(m+5)[4m2(m+5) -3]

31. GCF = 4(v + 4)4v(v + 4) - 12(v + 4) = v•4(v + 4) -3•4(v + 4) =


4(v + 4)(v -3)

33. GCF = 4m(m+2n)3

8m(m+2n)5+12m(m+2n)4-16m(m+2n)3 =4m(m+2n)3 •2m(m+2n)2

+3(m+n)•4m(m+2n)3 -4 =4m(m+2n)3 [2m(m+2n)2+3(m+n)-4]

35. mr + ms + nr + ns = (mr + ms) + (nr + ns) =m(r + s) + n(r + s) = (r + s)(m + n)

37. 2f2 - 14f + fg - 7g =(2f2 - 14f) + (fg - 7g) =2f(f -7) + g(f -7) =(f -7)(2f + g)

39. 15wt - 10t + 3w -2 =(15wt - 10t) + (3w -2) = 5t(3w -2) + 1(3w -2) =(3w -2)(5t + 1)

41. 12xz + 3x + 8yz + 2y =(12xz + 3x) + (8yz + 2y) =3x(4z +1) +2y(4z + 1) =(4z + 1)(3x + 2y)

43. 2r2 - 2r - 5s + 5sr = (2r2 - 2r) + (- 5s + 5sr) =2r(r -1) + 5s(r -1) =(r -1)(2r + 5s)

45. ab + 5b - 7a - 35 = (ab + 5b) + (-7a - 35) = b(a + 5) + -7(a + 5) =(a + 5)(b -7)

47. 8s2 - 12s - 18rs + 27r = (8s2 - 12s) + (-18rs + 27r) =4s(2s -3) + (-9r)(2s - 3) =(2s -3)(4s -9r)

49. 8x2 + 10xy - 4x - 5y =(8x2 + 10xy) + (-4x -5y) =2x(4x + 5y) +(-1)(4x + 5y) =(4x + 5y)(2x -1)

51. 15 -2(4•32/9) = 15 - 2(4•9/9) = 15 - 2(36/9) = 15 -2(4) = 15 -8 =7

52. 1287 = 32•11•13

53. a = 2(b + c) a = 2b + 2ca - 2b = 2c

c = a - 2 b

2

54. (6m +1)(2m + 3) =6m(2m) + 6m(3) + 1(2m) +1(3) =12m2 + 18m + 2m+ 3 =12m2 + 20m + 3

55. Eq 1: x - y = -3Eq 2: 2x +3y = 4Solve for x in eq 1: x = y - 3Substitute in eq 2: 2(y -3) + 3y =42y - 6 + 3y = 45y = 10 y = 2Substitute to solve for x: x - 2 = -3x = -1The solution is the ordered pair (-

1,2).

56.

-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

57. x +5[4x+3(x-2)-3] =x + 5[4x +3x -6 -3] =x + 5[7x -9] =x + 35x - 45 =36x - 45

58. The base is 3 and the exponent is 6.-36 = -729


59. 2 a 3 + a 2 + 4

4 a + 1 8 a 4 + 6 a 3 + a 2 + 16a + 4 8 a 4 + 2 a 3 Subtract 4 a 3 + a 2 4 a 3 + a 2 Subtract 0 + 16a + 4 16a + 4 Subtract 0

60. This is a fourth degree polynomial

61. Let x = number of nickelsx + 100 = number of quarters0.05x = value of nickels0.25(x + 100) = value of quarters28.30 = 0.05x +0.25(x + 100) 28.30= 0.05x + 0.25x + 2528.30 =0.30x + 253.30 = 0.30xx = 11nickels

62.

(-1, -4)

(0, 3)(2, 3)

(3, 9)

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

10

-10-9-8-7-6-5-4-3-2-1 0 1 2 3 4 5 6 7 8 910 x

Data #1f ( x )

63.( 7 . 4 x 103 ) ( 1 . 5 x 102 )

2 . 22x 104 =

11. 1 x 105

2 . 22x 104 = 5 x 10

64. There is no x intercept. The y

intercept is (0,4.5).

65. a)w(60) = -1/3(60) + 218 w(60) = -20 + 218 = 198Byron’s weight was initially 218. After 60 days he has lost 20 pounds,weighing 198 pounds.b)w(d) = 190 After d days Byronweighs 190190 = -1/3 d+ 218-28 = -1/3 dd = 84 daysc)w(30) = 208After 30 days Byron weighs 208pounds. He has lost 10 pounds.d)w(d) = w(18) -7w(18) = -1/3(18) +218= -6+218

=212212 - 7 = -1/3 d+ 218205 = -1/3 d + 218-13 = -1/3 dd = 39 daysFrom the 18th to the 39th day he lost an additional 7 pounds.


Section 7.3 Factoring Trinomials of the Form x2 + bx + c

Practice 7.3.11. x2 + 8x + 12

b = 8; c = 12(6)(2) = 12; 6 +2 = 8x2 + 8x + 12 = (x + 6)(x + 2)

2. x2 + 20x + 99b = 20; c = 99(9)(11) = 99; 9 + 11 = 20x2 + 20x + 99 = (x +9)(x + 11)

3. x2 - 2x - 15

b = -2; c = -15(-5)(3) = -15; -5 + 3 = -2x2 - 2x - 15 = (x - 5)(x + 3)

4. x2 + 2x - 99b = 2; c = -99(11)(-9) = -99; 11 + -9 = 2x2 + 2x - 99 = (x + 11)(x - 9)

5. 11x + 30 + x2

Standard form: x2 + 11x + 30b = 11; c = 30(5)(6) = 30; 5 + 6 = 11x2 + 11x + 30 = (x + 5)(x + 6)

6. -10x + 72 + x2 - 7xStandard form: x2 - 17x + 72b = -17; c = 72(-9)(-8) = 72; -9+ -8 = -17x2 - 17 + 72 = (x - 9)(x - 8)

Practice 7.3.21. x2 - 14x - 72

b = -14; c = -72Product Sum-1(72) 71-2(36) 34-3(24) 21-4(18) 144(-18) -14x2 - 14x - 72 = (x - 18)(x + 4)

2. y2 - 49b = 0; c = -49Product Sum-1(49) 48-7(7) 0y2 - 49 = (y -7)(y + 7)

3. r2 + 17r - 60b = 17; c = -60Product Sum-1(60) 59-2(30) 28-3(20) 17r2 + 17r - 60 =(r + 20)(r - 3)

4. 21x + x2 -100Standard form: x2 + 21x - 100b = 21; c = -100(25)(-4) = -100; 24 + -4 = 21x2 + 21x - 100 = (x + 25)(x -4)


5. -96 + 29x + x2

Standard form: x2 + 29x - 96Product Sum-1(96) 95-2(48) 46-3(32) 29x2 + 29x - 96 = (x - 3)(x + 32)

6. x2 + 7x + 15b = 7; c = 15Product Sum1(15) 163(5) 8There are no factors of 15 that add tobe 7. This cannot be factored.

Practice 7.3.31. 5x2 + 40x + 75

GCF = 55x2 + 40x + 75 = 5(x2 + 8x + 15)b = 8; c = 15(3)(5) = 15; 3 + 5 = 85x2 + 40x + 75 = 5(x + 5)(x + 3)

2. 3x2 + 27x + 42GCF = 33x2 + 27x + 42 = 3(x2 + 9x + 14)b = 9; c = 14(2)(7) = 14; 2 + 7 = 93x2 + 27x + 42 = 3(x + 2)(x + 7)

3. x2 + 12x - 3b = 12; c = -3Product Sum-1(3) 2There are no factors of -3 that add to12. This is not factorable.

4. 60 - x2 - 7x ---->-x2 - 7x + 60GCF = -1-x2 - 7x + 60 = -1(x2 + 7x - 60)b = 7; c = -60(12)(-5) = -60; 12 + -5 = 7-x2 - 7x + 60 = -1(x + 12)(x - 5)

5. 4x3 + 160x + 52x2

Standard form: 4x3 + 52x2 + 160xGCF = 4x4x3+52x2+160x = 4x(x2 + 13x + 40)b = 13; c = 40(8)(5) = 40; 8 + 5 = 134x3 + 52x2 + 160x = 4x(x +8)(x+5)

6. 100x + 5x3 + 65x2 + 80xStandard form: 5x3 + 65x2 + 180xGCF = 5x5x3 +65x2 +180x = 5x(x2 + 13x +

36)b = 13; c = 36(4)(9) = 36; 4 + 9 = 135x3 + 65x2 + 180x = 5x(x + 4)(x +

9)

Exercise Set 7.31. x2 + 11x + 28

b = 11; c = 28(4)(7) = 28; 4 + 7 = 11x2 + 11x + 28 = (x + 4)(x+ 7)

3. x2 + 13x + 40b = 13; c = 40(8)(5) = 40; 8 + 5 = 13x2 + 13x + 40 = (x + 8)(x + 5)

5. x2 + 6x - 27b = 6; c = -27(9)(-3) = -27; 9 + -3 = 6x2 + 6x - 27 = (x + 9)(x - 3)

7. m2 - 9m - 36b = -9; c = -36(-12)(3) = -36; -12 + 3 =-9m2 - 9m - 36 = (m - 12)(m + 3)

9. x2 - 18x + 45b = - 18; c = 45(-15)(-3) = 45; -15+ -3 =-18x2 - 18x + 45 = (x - 15)(x - 3)

11. t2 - 17t + 72b = -17; c = 72(-9)(-8) = 72; -9 +-8 = -17t2 - 17t + 72 = (t - 9)(t - 8)

13. y2 - 27y + 72b = -27; c = 72Product Sum-1(-72) -73-2(-36) -38-3(-24) -27y2 - 27y + 72 = (y - 24)(y - 3)

15. m2 + 11m - 60b = 11; c = -60Product Sum-1(60) 59


-2(30) 28-3(20) 17-4(15) 11m2 + 11m - 60 = (m + 15)(m -4)

17. x2 - 32x - 144b = -32; c = -144Product Sum1(-144) -1432(-72) -703(-48) -454(-36) -32x2 - 32x - 144 = (x - 36)(x + 4)

19. y2 - 22y + 12b = -22; c = 12Product Sum-1(-12) -13-2(-6) -8-3(-4) -7There are no factors of 12 that add tobe -22. This is not factorable.

21. t2 - 28t + 192b = -28; c = 192Product Sum-1(-192) -193-2(-96) -98-3(-64) -67-4(-43) -47-6(-32) -38-8(-24) -32-12(-16) -28t2 - 28t + 192 = (t-12)(t - 16)

23. w2 - 8w + 19b = -8; c = 19Product Sum-1(-19) -20There are no factors of 19 that add tobe -8. This is not factorable.

25. 6x2 - 36x - 162GCF = 66x2 - 36x - 162 =6(x2 - 6x - 27)b = -6; c= -27(-9)(3) = -27; -9 +3 =-66x2 - 36x - 162 = 6(x - 9)(x + 3)

27. -x2 + x + 12GCF = -1-x2 + x + 12 = -1(x2 - x - 12)b = -1; c = -12

(-4)(3) = -12; -4 +3 =-1-x2 + x + 12 = -1(x - 4)(x + 3)

29. 5x2 - 4x + 2There are no common factors and

this does not fit the pattern x2 + bx + c. We can’t factor this.

31. 12 - x2 + x Standard form: -x2 + x + 12GCF = -1-x2 + x + 12 = -1(x2 - x - 12)b = -1; c = -12(-4)(3) = -12; -4 +3 =-1-x2 + x + 12 = -1(x - 4)(x + 3)

33. -x2 + 9x + 22GCF = -1-x2 + 9x + 22 = -1(x2 - 9x - 22)b = -9; c = -22(-11)(2) = -22; -11 + 2 =-9-x2 + 9x + 22 = -1(x - 11)(x + 2)

35. 18x2 + 165x - 3x3

Standard form: -3x3 + 18x2 + 165xGCF = -3x-3x3 + 18x2 + 165x = -3x(x2 - 6x -

55)b = -6; c = -55(-11)(5) = -55; -11 + 5 =-6-3x3 +18x2 +165x = -3x(x -11)(x +

5)

37. Let x = cost of popcornx + 1 = cost of drinkx + (x + 1) = 42x + 1 = 42x = 3x = $1.50 for popcorn

38. Let x = rate for $11000 investmentx -0.08 = rate for $4000 investmentInterest = principle• rate• time467.50 =11000•1•x+ 4000•1(x - 0.08)467.50 = 11000x + 4000x - 320787.50 = 15000xx = 0.053 = 5.3% for $11000x -0.08 =0.053-0.08= -0.027 = -2.7%She lost money on her $4000investment.

39. x + y = 0y = -x


slope = -1; y intercept is (0,0)

40. a)Let x = number of months; y =cost

Computer Depot: y = 100 + 50xPica Computer: y = 60xb)

050

100150200250300350400450500550600

Number of months

Data #4y

x

c) The intersection is (10,600). Ifrented for 10 months the cost is thesame ($600) for both companies. d)If rented less than 10 months Pica

is cheaper, if more than 10 monthsComputer Depot is cheaper.

41. Solve Eq 1 for y: x - 2y = -3-2y = -3 -xy = 3/2 +x/2The lines are the same.There are aninfinite number of solutions whichmay be written: (x, 3/2 + x/2)

-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 #5

42. Eq 1: 1/2 x + y = 2Eq 2: 2/3 x + 1/2 y = 62 times eq 1: 2(1/2 x + y) = 2•2Eq 1 transformed: x + 2y = 4

6 times eq 2: 6(2/3 x + 1/2 y) = 6•6Eq 2 transformed: 2•2x + 3y = 364x + 3y = 36Solve eq 1 for x: x = 4 - 2ySubstitute in eq 2: 4(4 - 2y) + 3y =

3616 - 8y + 3y = 36-5y = 20y = -4Substitute in eq 1 to solve for x:1/2 x + -4 = 21/2 x = -6x = -12The solution is the ordered pair (-12,-4).

43.64p 7 + 32p 5 − 28p 3

4 p =

64p 7

4 p + 32p 5

4 p − 28p 3

4 p =

16p 6 + 8 p 4 − 7 p 2

44. 3100 x 105 = 3.1 x 103 x 105 = 3.1 x 108

45. 12x2 = 22•3•x2

18xy = 2•32x•yGCF = 2•3•x = 6x

46. -2xy3 - 2y2

GCF = -2y2

-2y2(xy) + -2y2(1) =-2y2(xy +1)

47. -32 + 3-1 - 3(-1) = -9 + 1/3 +3 =-6 + 1/3 = -5 2/3 or -17/3

48. ( a 5

a − 3 ) • ( 2 a ) = ( a 5 a 3 ) ( 2 a ) = 2 a 9

49. (4 a 2 b 3

( a 3 b 2 ) − 2 ) 2 = ( 4 a 2 b 3

( a - 6 b - 4 ) ) 2 =

(4a2b3a6b4)2 = (4a8b7)2 =16a16b14

50. (4x2y - 2y) + (3x -y) + (4y + x2y) =5x2y + y + 3x

51. (2c -5)(3c -1) = 2c(3c) + (-1)(2c) +(-5)(3c) + (-5)(-

1)=6c2 - 2c - 15c + 5=


6c2 - 17c + 5

52. (2x -1)2 = (2x)2 +2(2x)(-1) + (-1)2

=4x2 - 4x + 1

53. 2n2(3n)2 = 2n2 • 9n2 = 18n4

54. (4b2 - 6b +1) -(2b + 8b2) - (b -5) =4b2 - 6b +1 -2b -8b2 - b + 5 =-4b2 -9b + 6

55. m =− 1 - ( - 5 )

0 - 4 = − 1 + 5

− 4

= 4 − 4

= - 1

56.x

years since1992

ySales

(dollars inthousands)

0 6.3

1 8.1

4 13.5

6 17.1

(0, 6.3)

(1, 8.1)

(4, 13.5)

(6, 17.1)

5

7.5

10

12.5

15

17.5

Years since 1992

Data #1y

x

Section 7.4 Factoring Trinomials of the Form ax2 + bx +c

Practice 7.4.11. 5x2 + 6x + 1

5x2 = (5x)(1x)1 = (1)(1)(5x +1)(1x + 1) = 5x(1x) +(1)(5x) +(1)(x) +(1)(1) =5x2 +5x + x + 1 = 5x2 + 6x + 1(5x +1)(x +1) is correct.

2. 3x2 - x -23x2 = (3x)(1x)-2 = (-1)(2) or (1)(-2)(3x -1)(x + 2) = (3x)(1x) +(3x)(2) + (-1)(x) + (-

1)(2) =3x2 + 6x -x - 2 = 3x2 + 5x - 2This is not the correct factorization. We will try another combination.(3x +2)(1x -1) = (3x)(1x) + (3x)(-1) +2(1x) + (2)(-

1) =3x2 - 3x + 2x - 2 = 3x2 - x -2(3x + 2)(x -1) is correct .

3. 7x2 - 22x + 37x(1x) = 7x2

(-1)(-3) = 3Try: (7x - 3)(1x - 1) =7x(1x) - (1)(7x) - 3(1x) + -3(-1) =7x2 - 7x - 3x +3 =7x2 - 10x + 3 This is not correct.Try: (7x - 1)(1x - 3) = 7x(1x) + 7x(-3) - 1(1x) + -1(-3) = 7x2 - 21x - 1x + 3 = 7x2 - 22x + 3(7x -1)(x - 3) is correct .

Practice 7.4.21. 4w2 + 9w +5


a = 4; b = 9; c =5ac = 4•5 = 20; 4 + 5 = 9 =bRewrite as: 4w2 + 4w + 5w + 5Factor by grouping:4w(w+1)+5(w+1) =(w +1)(4w +5)

2. 5m2 + 11m + 6a =5; b =11; c = 6ac = 30; 5+6 = 11 =bRewrite as: 5m2 + 5m+6m+6Factor: 5m(m+1) +6(m+1) =(m+1)(5m+6)

3. 3x2 - 13x - 10a = 3; b = -13 ; c = -10ac = 3(-10) = -30; (-15)(2) = -30 -15 + 2 =-13Rewrite: 3x2 - 15x + 2x - 10Factor: 3x(x - 5) + 2(x - 5) =(x -5)(3x + 2)

4. 12w2 + 17w - 5a =12; b = 17; c = -5ac = (12)(-5) = -6020•(-3) = -60 20 - 3 = 17 = bRewrite: 12w2 + 20w - 3w - 5Factor: 4w(3w + 5) -1(3w +5) =(3w + 5)(4w -1)

5. 8z2 + 18z + 10GCF = 2Factor: 2(4z2 + 9z + 5)ac = (4)(5) = 20; 4 + 5 = 9 = bRewrite: 2[4z2 + 4z + 5z +5] Factor: 2[4z(z +1) +5(z + 1)] =2[(z +1)(4z + 5)]

6. 12m2 - 10m + 2GCF = 2Factor: 2(6m2 - 5m + 1)a =6; b = -5; c = 1ac = 6•1 = 6-3(-2) = 6 -3 + -2 = -5Rewrite: 2[6m2 - 3m - 2m + 1]Factor: 2[3m(2m-1) -1(2m -1)]2[(2m-1)(3m-1)]

7. 4x2 + 7x + 5a = 4; b = 7; c = 5ac = 4•5 = 20Product Sum1(20) 21

2(10) 124(5) 9There are no factors of 20 that addtobe 7. This cannot be factored.

Exercise Set 7.41. 2x2 + 5x + 2

2x(1x) = 2x2; 2(1) = 2Try: (2x +1)(1x + 2) = 2x(1x) + 2x(2) + 1x(1) + 2•1 =2x2 + 4x + 1x + 2 = 2x2 + 5x + 2(2x + 1)(1x + 2) is correct.

3. 35x2 + 3x -2(7x)(5x) = 35x2; -2= (-1)(2) or (1)(-

2)Try: (7x -1)(5x + 2) = (7x)(5x) +(7x)(2) +(5x)(-1) + (-

1)(2)= 35x2 + 14x - 5x - 2 = 35xv + 9x - 2This is not correct.Try: (7x + 2)(5x - 1) =(7x)(5x) + (7x)(-1) + 2(5x) + 2(-1)

=35x2 - 7x + 10x - 2 = 35xv + 3x - 2(7x + 2)(5x -1) is correct.

5. 3r2 -10r +73r(1r) = 3r2; (-7)(-1) = 7Try: (3r - 1)(r -7) = 3r2 -21r - r + 7=3r2 - 22r + 7 This is not correct.Try: (3r - 7)(r -1) = 3r2 -3r -7r +7 =3r2 - 10r + 7(3r -7)(r-1) is correct.

7. 12z2 + 20z + 3a = 12; b = 20; c = 3ac = 12•3 = 36 ; 18(2) = 36 18+2 = 20Rewrite: 12z2 + 18z + 2z + 3Factor: 6z(2z + 3) + 1(2z +3) =(2z + 3)(6z +1)

9. 6m2 - 11m + 3a = 6; b = -11; c = 3ac = 6•3 = 18 (-9)(-2) = 18-9 + -2 = -11Rewrite: 6m2 - 9m -2m + 3Factor: 3m(2m -3) -1(2m-3) =(2m -3)(3m -1)

11. 6y2 - 7y -20


a = 6; b = -7; c = -20ac = 6(-20) = -120Product Sum1(-120) -1192(-60) -583(-40) -374(-30) -265(-24) -196(-20) -148(-15) -7Rewrite: 6y2 + 8y - 15y - 20Factor: 2y(3y + 4) - 5(3y + 4)(3y +4)(2y -5)

13. 3r2 - 2r -9a = 3; b = -2; c = -9ac = 3(-9) = -27Product Sum1(-27) -263(-9) -6There are no factors of -27 that add

to be -2. This cannot be factored.

15. 12y2 - 7y -10a = 12; b = -7; c = -10ac = -120Product Sum1(-120) -1192(-60) -583(-40) -374(-30) -265(-24) -196(-20) -148(-15) -7Rewrite: 12y2 + 8y - 15y - 10Factor: 4y(3y +2) - 5(3y + 2) =(3y +2)(4y -5)

17. 6 + 4w2 - 11wStandard form: 4w2 - 11w +6a =4; b = -11; c =6ac = 4•6 = 24 -8(-3) = 24-8 + -3 = -11Rewrite: 4w2 - 8w -3w + 6Factor: 4w(w - 2) -3(w -2) =(w -2)(4w -3)

19. 12t2 - 13t + 3a = 12; b = -13; c = 3ac = 12•3 = 36 -9(-4) = 36-9 + -4 = -13Rewrite: 12t2 - 9t -4t + 3Factor: 3t(4t - 3) -1(4t -3) =

(4t -3)(3t -1)

21. 12x2 - 5x -2a = 12; b = -5; c = -2ac = 12(-2) = -24 -8(3) = -24-8 + 3 = -5Rewrite: 12x2 -8x + 3x - 2Factor: 4x(3x - 2) + 1(3x -2)(3x -2)(4x + 1)

23. 50x2 - 55x + 15GCF = 5Factor: 5(10x2 - 11x + 3)a = 10; b = -11 ; c = 3ac = 10•3 = 30 (-5)(-6) = 30-5 + -6 = -11Rewrite: 5[10x2 - 5x - 6x + 3]Factor: 5[5x(2x-1) -3(2x -1)] = 5[(2x-1)(5x-3)]

25. 6 + 21m + 15m2

Standard form: 15m2 + 21m + 6GCF = 3Factor 3(5m2 + 7m + 2)a = 5; b = 7; c = 2ac = 5•2 = 10 5 + 2 = 7Rewrite: 3[5m2 + 5m + 2m + 2]=Factor: 3[5m(m + 1) + 2(m + 1) ] =3[(m+1)(5m +2)]

27. 8m2 + 4m - 40GCF = 4(2m2 + m -10)a = 2; b = 1; c= -10ac = 2(-10) = -20 5(-4) = -205 + -4 = 1Rewrite: 4[2m2 + 5m - 4m - 10]Factor: 4[m(2m + 5) - 2(2m + 5)]

=4[(2m +5)(m -2)]

29. -2 +3y2 - 4yStandard form: 3y2 - 4y - 2a = 3; b = -4 ; c = -2ac = 3(-2) = -6Product Sum1(-6) -52(-3) -1There are no factors of -6 that add to

- 4. This cannot be factored.

31. 8x - (7 -x) + 2(3 - 8x) = 8x - 7 + x + 6 - 16x= -7x - 1


32. 4 - 2(5 -x) = 2x - 3(x -3)4 -10 + 2x = 2x - 3x + 92x - 6 = -x + 93x = 15x = 5

33. 5x -2 = 3(1 - 2x) -55x -2 = 3 - 6x - 55x - 2 = -2 - 6x11x = 0x = 0

34. 6x2y5 - 3xy4

GCF = 3xy4

3xy4(2xy ) - 3xy4(1) = 3xy4(2xy -1)

35. The domain is all reals and the rangeis all reals.

36. f(x) = 2x3 - 5x + 7f(2) = 2(2)3 - 5(2) + 7 = 2•8 - 10 + 7f(2) = 16 - 10 + 7 = 13

37. Eq 1: x - 5y = 3Eq 2: 3x + y = 95 times eq 2: 5(3x + y) = 5•9Simplified; 15x + 5y = 45Eq 2 x - 5y = 3Add together 16x = 48x = 3Substitute in eq 1 to solve for y:3 - 5y = 3-5y = 0y = 0The solution is the ordered pair

(3,0).

38. Eq 1: 8x - 5y = 5Eq 2: 3x + 3y = -3Divide eq 2 by 3: x + y = -15 times eq 2: 5(x + y) = -1•5Simplify: 5x + 5y = -5Eq 1 8x - 5y = 5Add together 13x = 0x = 0Substitute to solve for y:8(0) - 5y = 5-5y = 5y = -1The solution is the ordered pair (0,-

1).

39.

3 x 3 − 2 x 2 + x + 1

x + 1 3 x 4 + x 3 − x 2 + 2 x + 1 3 x 4 + 3 x 3 Subtract

− 2 x 3 − x 2 − 2 x 3 − 2 x 2 Subtract

x2 + 2 x x 2 + x Subtract

x + 1 x + 1 Subtract

0

40. 4x2(x +2)3 and 6x2(x +2)4

GCF = 2x2(x + 2)3

41. 30min-utes =0.5 hrs

15min-utes=0.25hrs

Let x =numberof standard models

y = numberof extended modelsEq 1: 1x + 2y = 140Eq 2: 0.5x + 0.25y = 40Solve eq 1 for x: x = 140 -2ySubstitute in eq 2: 0.5(140-2y) +0.25y = 4070 -y + 0.25y = 40-0.75y = -30 y = 40 extended unitsSubstitute to solve for x: x + 2•40 =140x + 80 = 140x = 60 standard units

42. 1 hr 15 minutes = 1.25 hrs1 hr 30 minutes = 1.5 hrsdistance = rate •timeDistances are the same.Let r = rate going to Rentonr - 10 = rate returning1.25r = 1.5(r -10)1.25r = 1.5r - 15


-0.25r = -15r = 60 mph to Renton

43. Using (0,-8) and (6,0)

m = 0 - ( - 8 )

6 - 0 = 8

6 = 4

3 The y intercept is (0,-8) so b = -8.y = 4/3 x -8

44. For every increase of 1 in thenumber of dimes, the valueincreases by 10 cents.

45.

-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

46. A line perpendicular to y = -2 (ahorizontal line) would be x = someconstant (a vertical line). Since theline passes through (4,1) the

equation is x =4.

47. Line 1 m= 5 - ( - 1 ) − 2 - 3

= 5 + 1 − 5

= - 6 5

Line 2 m= − 3 - 1 8 - 0

= − 4 8

= − 1 2

The lines are neither parallel norperpendicular.

48.( c 2 d - 3 ) 2

c − 3 •

d 4

c − 2 = c 4 d − 6

c − 3 •

d 4

c − 2 =

c 4 c 3

d 6 •c2 d 4 = c 9

d 2

49. ( 6 m

m 3 n 7 ) 2 (

m 5

n − 3 ) 2 =

( 6

m 2 n 7 ) 2 ( m 5 n 3 ) 2 =

36

m 4 n 14•m10n 6 = 36m 6

n 8

50. 8y(6y3 - 4y2 + 2) = 8y(6y3 ) + 8y(-4y2) + 2•8y =48y4 - 32y3 + 16y

51. (2y -1)(4y2 + 2y + 1) =2y(4y2 + 2y + 1) - 1(4y2 + 2y +1) =8y3 + 4y2 + 2y - 4y2 - 2y - 1 =8y3 - 1

Section 7.5 Factoring Special


Products and General Factoring

Practice 7.5.11. m2 - 81 = (m)2 - (9)2 =

(m + 9)(m - 9)

2. 81w2 - 4 = (9w)2 - (2)2 = (9w -2)(9w + 2)

3. 4r2 - 9s2 = (2r)2 - (3s)2

(2r + 3s)(2r - 3s)

4. 36q2 - 25d2 = (6q)2 - (5d)2 = (6q - 5d)(6q + 5d)

Practice 7.5.21. 100a2 - 81b2 = (10a)2 - (9b)2 =

(10a + 9b)(10a - 9b)

2. 9z2 - 49v2 = (3z)2 - (7v)2 =(3z - 7z)(3z + 7v)

3. This is the sum of two squares andis not factorable.

4. 2m2 - 32 = 2(m2 - 16) = 2[(m)2 - (4)2] = 2(m + 4)(m -4)

5. x2 +2x - 1 This is not the differenceof two squares. There is a middleterm 2x.

6. 25x4 -9 = (5x2)2 - (3)2 = (5x2 -3)(5x2 + 3)

7. -1 + 9x2

Standard form: 9x2 -1 =(3x)2 - (1)2 = (3x + 1)(3x -1)

8. -36x2 + 25 = -1(36x2 - 25) =-1[(6x)2 - (5)2] = -1(6x + 5)(6x - 5)

Practice 7.5.31. y2 + 4y + 4

y2 = (y)2; 4 = (2)2

4y = 2•2•yy2+ 4y +4 = (y +2)2

2. z2 - 2z + 1z2 = (z)2; 1 = (-1)2

-2z = 2(-1)z

z2 - 2z + 1 = (z -1)2

3. 25m2 - 10m + 125m2 = (5m)2 ; (-1)2 = 12(-1)(5m) = -10m25m2 - 10m + 1 = (5m -1)2

4. 25x2 - 40x + 16 (5x)2 = 25x2; (-4)2 = 162(-4)(5x) = -40x25x2 - 40x + 16 = (5x -4)2

5. 100c2 + 40c + 16GCF = 44(25c2 + 10c + 4)25c2 = (5c)2; 4 = (2)2

2(2)(5c = 20cThis does not fit the pattern.100c2+40c+16 = 4(25c2+10c+4)

Practice 7.5.41. 6y2 + 3y2 - 4 = 9y2 -4

This is the difference of two squares.(3y)2 - (2)2 =(3y -2)(3y +2)

2. 2z2 + 72 - 26zStandard form: 2z2 - 26z + 72 GCF = 22(z2 - 13z + 36)b = -13; c = 36 (-9)(-4) = 36 -9 + -4 = -132(z -9)(z -4)

3. 36 - m2 Standard form:-m2 + 36GCF = -1-1(m2 - 36)-1[(m)2 -(6)2] = -1(m + 6)(m -6)

4. 3m2 - 12m + 2mn - 8n3m(m -4) + 2n(m -4) = (m -4)(3m + 2n)

5. x4 - 11x3 + 24x2

GCF = x2

x2(x2 - 11x + 24)-8(-3) = 24 -8 +-3 = -11x2(x -8)(x -3)

6. 4x2 + 7x + 2x2 -3 =6x2 + 7x -3a = 6; b =7; c = -3ac = 6(-3) = -189(-2) = -189 + -2= 7


Rewrite: 6x2 + 9x - 2x - 3Factor: 3x(2x + 3) -1(2x+3) =(2x +3)(3x -1)

Exercise Set 7.51. c2 - d2 = (c)2 -(d)2 =

(c +d)(c -d)

3. 4t2 - 121 = (2t)2 - (11)2 =(2t + 11)(2t -11)

5. 25y2 - 1 = (5y)2 - (1)2 =(5y -1)(5y +1)

7. 16 - 9x2

Standard form = -9x2 + 16GCF = -1-1(9x2 - 16) = -1[(3x)2 - (4)2]-1(3x + 4)(3x -4)

9. 25x2 - 144y2 =(5x)2 - (12y)2 = (5x + 12y)(5x - 12y)

11. 12n2 - r2 This is not the differenceof two squares since 12 is not a

perfect square. It cannot befactored.

13. y2 - 25z2 = (y)2 - (5z)2 =(y + 5z)(y - 5z)

15. 9x2 -5 This is not the difference oftwo squares since 5 is not a perfectsquare. It cannot be factored.

17. 36z4 - 49 = (6z2)2 - (7)2 =(6z2 +7)(6z2 -7)

19. 9 -100d2

Standard form: -100d2 + 9GCF =-1-1(100d2 - 9)= -1[(10d)2 - (3)2]=-1(10d +3)(10d-3)

21. 49y2 + 28y + 449y2 = (7y)2 4 = (2)2

2•2•7y = 28y the middle term49y2 + 28y + 4 = (7y + 2)2

23. z2 + 3z +1z2 =(z)2 1 = (1)2

2•1•z = 2z ≠ 3zThis is not the square of a binomial.It cannot be factored.

25. 16m2 + 40m + 25(4m)2 = 16m2 (5)2 = 252•5•4m = 40m the middle term16m2 + 40m + 25 = (4m +5)2

27. m2 + 8m +16(m)2 = m2 (4)2 = 162•4•m = m the middle termm2 + 8m +16 = (m+4)2

29. r2 + 10r + 36(r)2 = r2 (6)2 = 362•6•r = 12r ≠10r This is not the square of a binomial.

It cannot be factored.

31. 9x2 -12x - 4(3x)2 = 9x2 (-2)2 = 4 ≠ -4This is not the square of a binomial.

33. 4x2 - 96 + 20xStandard form: 4x2 + 20x - 96GCF = 44(x2 + 5x - 24)b = 5; c = -24(8)(-3) = -24 8 +-3 = 54(x + 8)( x -3)

35. 2m2 + 8 + 8m Standard form: 2m2 + 8m +8GCF = 22(m2 + 4m +4)(m)2 = m2 (2)2 = 4 2•2•m =4m2(m2 + 4m+4) = 2(m +2)2

37. 60r2 + 10r - 24rs - 4s GCF = 22(30r2 +5r - 12rs -2s) =2[5r(6r + 1) - 2s(6r + 1)] =2(6r +1)(5r -2s)

39. y2 + 75 - 4y2 = -3y2 + 75 =-3(y2 - 25) = -3[(y)2 - (5)2]-3(y -5)(y +5)

41. 3x2 -6x + 5x2 - 5 = 8x2 - 6x -5a = 8; b = -6; c = -5ac = 8(-) = -40 -10(4) = -40-10 +4 =-6Rewrite: 8x2 - 10x + 4x - 5 =Factor: 2x(4x -5) +1(4x -5) =(4x - 5)(2x +1)


43.x 2

+ 3 ( 6 - x ) = 8 - 5 2

( x - 4 )

2 • x 2

+ 2 • 3 ( 6 - x ) = 2 • 8 - 2 • 5 2

( x - 4 )

x + 6(6 -x) = 16 - 5(x -4)x + 36 - 6x = 16 -5x + 20-5x + 36 = 36 - 5x36 = 36This is an identity. The solutions

are all real numbers.

44. 8-0.5(x- 2) ≥ 5-0.3x8- 0.5x +1 ≥ 5 - 0.3x9 - 0.5x ≥ 5-0.3x-0.2x ≥ -4x ≤ 20

45.

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10

y

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

Data #2

46. Eq 1: 2x -y = -2Eq 2: 3x + 5y = -35 times eq 1: 5(2x -y) = 5(-2)Simplified: 10x -5y = -10Eq 2 3x + 5y = -3

13x = -13x = -1Substitute to solve for y: 2(-1) - y =

-2-2 - y = -2-y = 0y = 0The solution is the ordered pair (-

1,0).

47.− 60t 7 − 24t 6 − 12t 4 + 36t 3

12t 3 =

-60t 7

12t 3 − 24t 6

12t 3 − 12t 4

12t 3 + 36t 3

12t 3 =

-5t4 - 2t3 - 1t + 3

48. ( 1 2

) − 2 − ( 4 3

) − 2 = ( 2 1

) 2 − ( 3 4

) 2 =

4 - 9 16

= 3 7 16

49. x(4x-1)(8x2) = 32x2

50.( 5 t ) − 3

( 5 t ) − 1 • (

5 t

) − 2 = ( 5 t ) 1

( 5 t ) 3 • (

t 5

) 2 =

1

( 5 t ) 2 • (

t 5

) 2 = 1

25t 2 •

t 2

25= 1

625

51. (3a +b)(9a2 - 3ab + b2) =3a(9a2 - 3ab + b2) +b(9a2 -3ab

+b2)=27a3 -9a2b +3ab2 + 9a2b - 3ab2 + b3

= 27a3 + b3

52. ( 1 2

m 2 + 1 4

m + 1 8

) + ( 3 2

m 2 − 1 16

) =

1 2

m 2 + 1 4

m + 1 8

+ 3 2

m 2 − 1 16

=

4 2

m 2 + 1 4

m + 2 16

− 1 16

=

2 m 2 + 1 4

m + 1 16

53. (0.002x3+0.54x2-3)- (0.2x4+2.5x3 -0.07x2 +1x+1) =0.002x3+0.54x2-3 -0.2x4-2.5x3 +0.07x2 -x -1 = -0.2x4 -2.498x3+0.61x2 -x -4

54. 10a3 + 18ab3 = 2a(5a2 + 9b3)

55. x2 - 14x + 48(-6)(-8) = 48 -6 +-8 =-14x2 - 14x + 48 = (x -8)(x - 6)

56. 3x2 + 21x + 30 = 3(x2 + 7x + 10)(5)(2) = 10 5 +2 = 73(x +5)(x +2)

57. 2x2 - 2xy + 3x - 3y =2x(x - y) + 3(x - y) =


(x -y)(2x +3)

58. (11.8 x10-7) (2.12 x 10-5) = 25.016 x 10-12 = 2.5016 x10 x10-12 = 2.5016 x 10-11

59. a)biker = 1/3 sb)biker = s+3c)biker = s -3d)biker = s-1

60. He sold 114 child ticketsLet x = adult tickets114(2.50) + 4.75 x = 1163.75285 + 4.75x = 1163.754.75x = 878.75x = 185 adult tickets

61. Let x = first odd integer x +2 = secondx +(x +2) = 2(x +2)x + x + 2 = 2x + 42x +2 = 2x + 42 = 4This is a contradiction. There is nosolution.

62. m =0 - ( - 1 )

5 - 2 = 1

3

63. 2x +3y +6 =0Solve for y : 3y = -2x -6y = -2/3 x -2m = -2/3 y intercept is (0,-2)

64. m = 2 - 0

− 3 - 0 = - 2

3 y intercept is (0,0)y = -2/3 x + 0 or y = -2/3 x

65. Yes, it passes the vertical line test.

66. m = 8 . 77 - 4 . 27

50 - 20= 4 . 5

30= 0 . 15

For every increase of one dollar invalue of the bill, the life of the billincreases by 0.15 of a year or about

55 days.


Section 7.6 Solving Polynomial Equations by Factoring

Practice 7.6.11. (y -12)(y + 5) = 0

y - 12 = 0 or y + 5 = 0y = 12 or y = -5

2. 12x2 - 36x = 012x(x - 3) = 012x = 0 or x - 3 = 0x = 0 or x = 3

3. x2 - 2x - 15 = 0(-5)(3) = -15; -5 + 3= -2(x - 5)(x + 3) = 0x - 5 = 0 or x + 3 = 0x = 5 or x = -3

4. 25t2 - 36 = 0(5t)2 - (6)2 = 0(5t + 6)(5t -6) = 05t + 6 = 0 or 5t - 6 = 05t = -6 or 5t = 6t = -6/5 or t = 6/5

Practice 7.6.21. z2 - 13z = -36

Standard form: z2 - 13z + 36 =0

-9(-4) = 36; -9 +-4 =-13(z - 9)(z - 4) = 0z - 9 = 0 or z - 4 = 0z = 9 or z = 4

2. 4(x2 - 24) = 20xSimplify: 4x2 - 96 = 20xStandard form: 4x2 - 20x - 96 = 04(x2 - 5x - 24) = 04(x - 8)(x + 3) = 0x - 8 = 0 or x + 3 = 0x = 8 or x = -3

3. 6y2 + y = 4 - yStandard form: 6y2 + 2y - 4 = 02(3y2 + y - 2) = 02(3y - 2)(y + 1) = 03y - 2 = 0 or y + 1 = 03y = 2 or y = -1y = 2/3 or y = -1

4. 3 - 4a = 6a2 + 3aStandard form: 6a2 + 7a - 3 = 0ac =6(-3)=-18; 9(-2) = -18; 9+ -2

=7Rewrite: 6a2 + 9a - 2a - 3 = 03a(2a + 3) - 1(2a + 3) = 0(3a - 1)(2a + 3) = 03a -1 = 0 or 2a +3 =03a = 1 or 2a =-3a = 1/3 or a = -3/2

Practice 7.6.31. Let x = first integer

x + 1 = secondx(x + 1) = 156x2 + x = 156x2 + x - 156 = 0Product Sum-1(156) 155-2(78) 76-3(52) 49-4(39) 35-6(23) 17-12(13) 1x2 - 12x + 13x - 156 = 0x(x - 12) + 13(x - 12) = 0(x - 12)(x + 13) = 0x - 12 = 0 or x + 13 =0x = 12 or x = -13When x = 12, x + 1 = 13. When x = -13, x + 1 = -12.


2. Let x = first number2x -1 = other numberx(2x -1) = 912x2 - x - 91 = 0ac = 2(-91) = -182Product Sum1(-182) -1812(-91) -897(-26) -1913(-14) -12x2 + 13x - 14x - 91 = 0x(2x + 13) - 7(2x + 13) = 0(2x + 13)(x - 7) = 02x + 13 = 0 or x - 7 = 02x = -13 or x = 7x = -13/2 = -6.5 or x = 72(-6.5) -1=-13-1=-14 2(7)-1=14-1=13When x =-6.5 the other number is -

14.When x = 7 the other number is 13.

Practice7.6.41.

x mph

x + 4 mph

20 miles

distance = rate • timedistance slow = x(1) = xdistance fast = 1(x + 4) = x +4By the Pythagorean theorem:(x)2 + (x +4)2 = (20)2

x2 + x2 + 8x + 16 = 4002x2 + 8x - 384 = 02(x2 + 4x - 192) = 0Product Sum-1(192) 191-2(96) 94-3(64) 61-4(48) 44-6(32) 26-8(24) 16-12(16) 42(x2 - 12x + 16x - 192) = 02[x(x -12) + 16(x - 12)] =02[(x - 12)(x + 16)] =0x -12 = 0 or x + 16 =0

x = 12 mph x = -16 Not validx + 4 = 16 mph for fast

2.

x+7 mph

x mph

97 miles

(x)2 +(x+7)2 = (97)2

x2 + x 2 + 14x + 49 = 94092x2 + 14x - 9360 =02(x2 + 7x - 4680) =0Product Sum-1(4680) 4679-2(2340) 2338-3(1560) 1557-4(1170) 1166-10(468) 458 -65(72) 72(x2 - 65x + 72x - 4680) =02[x(x -65) + 72(x - 65)] =02(x -65) (x + 72) = 0x - 65 = 0 or x + 72 = 0x = 65 mph x = -72 not validSpeed was 65 mph when going east.

Practice 7.6.51. a3 - 7a2 + 10a = 0

a(a2 - 7a + 10) =0a(a - 5)(a - 2) = 0a = 0 a -5 = 0 a - 2 =0a = 0 a = 5 a = 2

2. m3 + 4m2 = 21mm3 + 4m2 - 21m = 0m(m2 + 4m - 21) = 0m(m +7)(m -3) = 0m = 0 m+7 =0 m -3 =0m = 0 m = -7 m = 3

3. 4x3 - 25x = 0x(4x2 - 25) = 0

x(2x+5)(2x-5) = 0

x = 0 2x +5 =0 2x - 5 = 0

x = 0 2x = -5

2x = 5


x = 0 x = -5/2 x = 5/2

4. x3 + 12x2 = -2x2 - 48xx3 + 14x2 + 48x =0x(x2 + 14x + 48) = 0x(x + 6)(x + 8) =0x = 0 x + 6 = 0 x + 8 = 0x = 0 x = -6 x = -8

5. x3 - 2x2 = 48xx3 - 2x2 - 48x = 0x(x2 - 2x -48) =0x(x -8)(x + 6) = 0x = 0 x - 8 = 0 x + 6 =0x = 0 x = 8 x = -6

Exercise Set 7.61. (2x + 3)(x -1) = 0

2x +3 = 0 or x - 1 =02x = -3 or x = 1x = -3/2 or x = 1

3. (2a -8)(a -7) =02a - 8 = 0 or a - 7 = 02a = 8 or a = 7a = 4 or a = 7

5. 9x(7x -1) = 09x = 0 or 7x -1 = 0 x = 0 or 7x = 1x = 0 or x = 1/7

7. x2 + 5x + 4 = 0(x + 4)(x + 1) = 0x + 4 = 0 or x + 1= 0x = -4 or x = -1

9. x2 + 4x - 21 = 0(x + 7)( x- 3) =0x + 7 = 0 or x - 3 = 0x = -7 or x = 3

11. x2 - 144 = 0(x + 12)(x - 12) = 0x + 12 = 0 or x - 12 = 0x = -12 or x = 12

13. 8y2 + 4y = 04y(2y + 1) = 04y = 0 or 2y + 1 = 0y = 0 or 2y = -1y = 0 or y = -1/2

15. 8d2 + 4d = 04d(2d + 1) = 04d = 0 or 2d + 1 = 0d = 0 or 2d = -1d = 0 or d = -1/2

17. 5t2 = 15t5t2 - 15t = 05t(t -3) = 05t = 0 or t - 3 = 0t = 0 or t = 3

19. (x + 6)(x - 4) = -9x2 + 6x - 4x - 24 + 9 = 0x2 + 2x - 15 = 0(x + 5)(x - 3) = 0x + 5 = 0 or x - 3 = 0x = -5 or x = 3

21. w2 + 12w = -35 w2 + 12w + 35 = 0(w + 7)(w +5) = 0w + 7 = 0 or w + 5 = 0w = -7 or w = -5

23. a2 - 5a = 24a2 - 5a - 24 = 0(a - 8)(a + 3) = 0a - 8 = 0 or a + 3 = 0a = 8 or a = -3

25. 20x2 + 3x = 220x2 + 3x - 2 = 0ac = -40 8(-5) = -40; 8+ -5 =320x2 + 8x - 5x - 2 = 04x(5x + 2) -1(5x + 2) = 0(5x + 2)(4x - 1) = 05x + 2 = 0 or 4x - 1 =05x = -2 or 4x = 1x = -2/5 or x = 1/4

27. 2x2 -72 =0 =

2(x2 - 36) = 02(x + 6)(x -6 ) = 0x + 6 = 0 or x - 6 = 0x = -6 or x = 6

29. x2 - 9x = -x -15x2 - 8x + 15 = 0(x - 5)(x - 3) = 0x - 5 = 0 or x - 3 = 0x = 5 or x = 3


31. 2b2 - 8b = -22b + 882b2 + 14b - 88 =02(b2 + 7b - 44) =02(b + 11)(b - 4) = 0b + 11 = 0 or b - 4 = 0b = -11 or b = 4

33. 12m2 - 13m = -10 + 10m12m2 - 23m + 10 = 0ac = 120 ; -15(-8) = 120; -15 +-8 =-

2312m2 - 15m -

8m + 10 = 03m(4m - 5) -

2(4m - 5) = 0(4m -5)( 3m - 2) = 04m - 5 = 0 or 3m - 2 = 04m = 5 or 3m = 2m = 5/4 or m = 2/3

35. Let w = the width; w +8 = lengthw(w +8) = 240w2 + 8w - 240 = 0Product Sum-1(240) 239-4(60) 56-6(20) 14-8(30) 22-12(20) 8(w - 12)(w + 20) = 0w - 12 = 0 or w + 20 = 0w = 12 or w = -20 not valid12 + 8 = 20The dimensions are 12 inches by 20inches.

37. Let n = the number(n -3)(n + 5) = 65n2 + 5n - 3n - 15- 65 = 0n2 + 2n - 80 = 0(n + 10)( n - 8) = 0n + 10 = 0 or n - 8 = 0n = -10 or n = 8

39. Let w = the width; w + 2 = length

w

w +2

10

a2 + b2 = c2

(w)2 + (w+ 2)2 = (10)2

w2 + w2 + 2•2w + 4 - 100 = 02w2 + 4w - 96 = 02(w2 + 2w - 48) = 02(w + 8)(w - 6) = 0w + 8 = 0 or w - 6 = 0w = -8 not valid or w = 6w + 2 = 6 + 2 = 8The rectangle is 6 inches by 8

inches.

41. Let r = Jerry’s rate; r +1 = Ben’s

2r

2(r +1)

10 miles

distance = rate •time

(2r)2 + [2(r + 1)]2 = (10)2 4r2 + 4(r + 1)2 = 1004r2 + 4(r2 + 2r + 1) - 100 =04r2 + 4r2 + 8r + 4 - 100 =08r2 + 8r - 96 =08(r2 + r - 12) = 08(r + 4)(r - 3) = 0r + 4 =0 or r - 3 = 0r = -4 not valid or r = 3 mph

43. 36x3 = x36x3 - x = 0x(36x2 - 1) = 0x(6x - 1)(6x + 1) = 0x = 0 6x + 1 = 0 6x - 1 = 0x = 0 6x = -1 6x = 1x = 0 x = -1/6 x = 1/6

45. a3 - 3a2 - 4a = 0a(a2 - 3a - 4) = 0a(a - 4)(a + 1) = 0a = 0 a - 4 = 0 a + 1 = 0a = 0 a = 4 a = -1

47. b3 + 4b2 = 25b2 + 100bb3 + 4b2 - 25b2 - 100b =0b3 - 21b2 - 100b = 0b(b2 - 21b - 100) = 0b(b - 25)(b + 4) = 0b = 0 b - 25 = 0 b + 4 = 0b = 0 b = 25 b = -4

49. 12w3 - 10w = 7w2

12w3 - 7w2 - 10w = 0


w(12w2 - 7w - 10) = 0ac=12(-10)=-120; (8)(-15)=-1208+ -15=-7w(12w2+8w -15w -10) =0w(4w(3w+2)-5(3w+2)) =0w(3w +2)(4w -5) =0w =0 or 3w +2 =0 or 4w -5 = 0w = 0 3w = -2 4w = 5w = 0 w = -2/3 w = 5/4

51. x3 + 7x2 = -7x2 - 49xx3 + 7x2 + 7x2 + 49x = 0x(x2 + 14x + 49) = 0x(x + 7)2 = 0x = 0 or x + 7 = 0x = 0 or x = -7

53. 4w4 = 25w2

4w4 - 25w2 = 0w2(4w2 - 25) = 0w2(2w + 5)(2w -5) = 0w2 = 0 2w + 5 = 0 2w - 5 = 0w = 0 2w = -5 2w = 5w = 0 w = -5/2 w = 5/2

55. 45 questions worth 5 points; 65 - 45 = 20 worth 10 points1/4(20) = 5 10 point questions

correct5 + 18 = 23 5 point questions

correct5(10) + 23(5) = 50 + 115 = 165

points

56. Let x = cost for RV site; x - 8= tent cost; x + 4 = RV with electricity18x + 27(x - 8) + 16(x + 4) = 915.5018x + 27x - 216 + 16x + 64 = 915.5061x -152 =915.5061x = 1067.50x = $17.50 for RV site17.50 -8 = $9.50 for tent site17.50+4.00=$21.50 RV with electricity

57. Let x = amount of 2% solution5 0(0.18) + 0.02x = 0.12(50 + x)9 + 0.02x = 6 +0.12x-0.10x = -3x = 30 gallons of the 2% mixture

58.6 . 0 x 1024

7 . 4 x1022= 0 . 811x 102

8.11 x 10-1 x 102 = 8.11 x 10 = 81.1 moons

59. Using (0,3) and (3,0)

m =0 - 3 3 - 0

= − 3 3

= - 1

y intercept is (0,3) so b = 3y = -1x + 3

60. The domain is {-4 ≤ x≤ 8}. Therange is {-4 ≤ y ≤5}

61. Eq 1: 4x - 6y = -1Eq 2: -8x + 12y = 22 times Eq 1: 2(4x - 6y ) = 2(-1)Simplified 8x - 12 y = -2Eq 2 -8x+ 12y = 2Add together 0 = 0This is an identity. There are aninfinite number of solutions.Solve for y: -8x + 12y = 212y = 2 + 8xy = 2/12 +8/12 xy = 1/6 + 2/3 xThe solutions may be written as(x,1/6 +2/3 x).

62. Eq 1: 3x - y = -10Eq 2: x - 5y = -8Solve x in Eq 2: x = -8 + 5ySubstitute in Eq 1: 3(-8 +5y) -y = -

10-24 + 15y - y = -10-24 + 14y = -1014y = 14y = 1Substitute to solve for x: 3x - (1) = -

103x - 1 = -103x = -9x = -3The solution is the ordered pair (-

3,1).


63.

2 x 2 + 4 x + 2 - 1 x - 4

x - 4 2 x 3 − 4 x 2 − 14x - 9 2 x 3 − 8 x 2 Subtract

4 x 2 − 14x 4 x 2 − 16x Subtract

2 x - 9 2 x - 8 Subtract

- 1

64. x < -2.5 <-------------- <--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|-->x

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

65. 18x3(3x -2)5 and 24x2(3x -2)4

GCF = 6x2(3x -2)4

66. g(x) = 10x -3g(1/2) = 10(1/2) - 3 = 5 - 3 = 2

67. x2 + 2x + xy + 2y=x(x + 2) + y(x + 2) =(x + 2)(x + y)

68. 48 + x2 - 16xStandard form: x2 - 16x + 48(x

-12)(x- 4)

69.5x2 +15x -20 =

5(x2 +3x - 4) =

5(x + 4)(x - 1)

70. 20x2 - 3x -2ac = 20(-2) = -40; -8(5) = -40; -8 +5=-320x2 - 8x + 5x - 2 =4x(5x - 2) + 1(5x - 2) =(5x - 2)(4x + 1)

71. 3x(x -7) -7x(4 - 2x) = 3x2 - 21x - 28x + 14x2 = 17x2 -49x

72. (8y -1)(y -8)=8y(y) -8(8y) -1y -1(-8)=

8y2 - 64y - y + 88y2 - 65y + 8

73. -x2(x3y)2(-2x)3 = -x2(x6y2)(-8x3) =8x11y2

74.( c 7 d ) 2 ( d 7 c ) − 2

cd= c 14d 2 d − 14c − 2

cd=

c 12d − 12

cd= c 11

dd12= c 11

d 13

75. 4t + (3t - t2) +(1 -t -2t2 + 3t3) =4t + 3t - t2 + 1 - t - 2t2 + 3t3 =3t3 -3t2 + 6t + 1

76. a)Using (68.8,76.5) and (71.8,78.7)

m = 78. 7 - 76. 5 71. 8 - 68. 8

= 2 . 2 3

= 0 . 73

y = mx + b76.5 = 0.73(68.8) + b76.5 = 50.2 + bb = 26.3 or about 26 roundingy = 0.73x + 26b)Y intercept of 26 says when menhave a life expectancy of 0 years,women have a life expectancy of

about 26 years. For eachadditional year for men, women’s lifeexpectancy increases by 0.73 years.

c)y = 0.73(75) + 26y = 54.8 +26 = 80.8 or about 81

yearsd)96 = 0.73x + 2670 =0.73xx = 95.9 or about 96 years

Chapter 7 Review Exercises

1. 32 = 25; 40 =23•5; 96 =25•3GCF = 23 = 8

2. 36 = 22•32 ; 72 =23•32; 108 =22•33

GCF = 22•32 = 4•9 =36

3. 20x2y3 = 22•5x2y3; 40x4y = 23•5x4y;50x6y4 =2•52x6y4

GCF = 2•5x2y = 10x2y

4. 24x5y3z4 = 23•3x5y3z4; 30x3y4z2 =2•3•5x3y4z2

42x8y6z7 =2•3•7x8y6z7


GCF = 2•3x3y3z2

5. 12x3(5x +1)7; 18x8(5x +1)4

GCF = 6x3(5x +1)4

6. 8x3y8(x -2)9 ; 20x2y5(x -2)7

GCF = 4x2y5(x -2)7

7. -9x2 + 5x + 1 = -1(9x2 - 5x -1)

8. 3x2 - 2x + 1 = -1(-3x2 + 2x - 1)

9. -8x4 + 3x2 -5 = -1(8x4 - 3x2 + 5)

10. -24x3y2 +16x2y3 = -8x2y2(3x -2y)

11. 45x4 - 9x2 = 9x2(5x2 -1)

12. 8x(x-3)-4x2(x-3) = 4x(x-3)(2 - x) or-4x(x -3)(x - 2)

13. 12x2(2x -1)3 + 2x(2x -1)5 =2x(2x-1)3(6x + (2x-1)2)

14. 15x2 - 6x - 10xy + 4y =3x(5x -2) - 2y(5x -2) =(5x -2)(3x - 2y)

15. wx + wy - 5hx - 5hy =w(x + y) - 5h(x + y) = (x + y) (w - 5h)

16. x2 - x - 12 = (x -4)(x + 3)

17. 10 + x2 - 7x = x2 - 7x + 10 =(x - 5)(x - 2)

18. x2 - 7x - 60 = (x - 12)(x + 5)

19. -64 +12x + x2 =x2 + 12x - 64 = (x + 16)(x -4)

20. 4x2 + 12x - 16 =4(x2 + 3x - 4)4(x + 4)(x - 1)

21. 5x2 - 5x - 60 = 5(x2 - x - 12) =5(x -4)(x + 3)

22. 18x2 - 45x - 8ac = 18(-8) = -144Product Sum1(-144) -1432(-72) -70

3(-48) -4518x2 + 3x - 48x - 8 =3x(6x + 1) -8(6x + 1) =(6x + 1)(3x - 8)

23. -15x2 +39x - 18 = -3(5x2 - 13x +6) ac = 5•6 = 30 -10(-3) = 30;-10 +-3= -13-3(5x2 - 10x -3x + 6) = -3[5x(x - 2) -3(x -2)] =-3[(x -2)(5x -3)]

24. 4x2 - 9y2 = (2x - 3y)(2x + 3y)

25. 25x2 - 1 = (5x + 1)(5x -1)

26. x2 - 10x + 25 = (x -5)2

27. 4x2 + 28x + 49(2x)2 = 4x2; (7)2 = 49; 2•7•2x =28x4x2 + 28x + 49 = (2x + 7)2

28. 8x3 + 16x2 - 8x2y - 16xy =8x(x2 + 2x - xy - 2y)=8x[x(x + 2) - y(x + 2)] =8x[(x + 2)(x -y)]

29. 75x3 - 3x = 3x(25x2 -1) = 3x(5x -1)(5x+1)

30. 40x3 - 45xy2 = 5x(8x2 - 9y2)

31. 10x2 - 5x = 05x(2x -1) = 05x = 0 or 2x -1 =0x = 0 or 2x = 1x = 0 or x = 1/2

32. 0 = 2x2 + x -3ac = 2(-3) = -6; -2(3) = -6; 3 + -2 =12x2 + 3x - 2x - 3 = 0x(2x + 3) -1(2x + 3) = 0(2x + 3)(x -1) = 02x + 3 =0 or x -1 = 02x = -3 or x = 1x = -3/2 or x = 1

33. 40 + 6x = 4x2

4x2 - 6x - 40 = 02(2x2 - 3x - 20) = 0ac = 2(-20) = -40; -8(5) = -40; -8 +5 =-3


2(2x2 - 8x + 5x - 20 ) = 02[2x(x - 4) + 5(x - 4)] = 02(x - 4)(2x + 5) = 0x - 4 =0 or 2x + 5 = 0x = 4 or 2x = -5x = 4 or x = -5/2

34. 9x2 = -10 - 21x9x2 + 21x + 10 = 0ac = 10•9 = 90; 15•6 = 90; 15+ 6= 219x2 + 15x + 6x + 10 = 03x(3x + 5) + 2(3x + 5) = 0(3x + 5)(3x + 2) = 03x + 5 = 0 or 3x + 2 = 03x = -5 or 3x = -2x = -5/3 or x = -2/3

35. x3 - x2 = 6xx3 - x2 - 6x = 0x(x2 - x - 6) = 0x(x -3)(x + 2) = 0x = 0 or x - 3 = 0 or x + 2 = 0x = 0 or x = 3 or x = -2

36. x4 - 4x2 = 0x2(x2 - 4) = 0x2(x + 2)(x - 2) = 0x2 = 0 or x + 2 = 0 or x - 2 = 0x = 0 or x = -2 or x = 2

37. Let x = first; x + 1 = secondx(x + 1) = 342x2 + x - 342 = 0Product Sum-1(342) -341-2(171) -169-3(114) -111-6(57) -51-18(19) 1x2 - 18x + 19x - 342 =0x(x - 18) + 19(x - 18) = 0(x - 18)(x + 19) = 0x - 18 = 0 or x + 19 = 0x = 18 or x = -19When x = 18 then x + 1 = 19.When x = -19 then x + 1 = -18.

38. Let x = the first number; x + 17 =second number.x(x + 17) = -42x2 + 17x + 42 = 0(x + 14)(x + 3) = 0x + 14 = 0 or x + 3 = 0

x = -14 or x = -3When x = -14 the x + 17 = 3. When x = -3 then x + 17 = 14.

39.

2r

2(r +3)

15 miles

(3r)2 + [2(r + 3)]2 = (15)2

9r2 + 4(r +3)2 =2259r2 + 4(r2 + 6r + 9)= 2259r2 + 4r2 + 24r + 36= 225

13r2 + 24r - 189 =0ac = 13(-189) = -2457Product Sum-1(2457) 2456-3(819) 816-9(273) 264-27(91) 64-39(63) 2413r2 - 39r + 63r - 189 = 013r(r -3) + 63(r - 3) =0(r - 3) (13r + 63) = 0r - 3 = 0 or 13r + 63 = 0r = 3 or 13r = -63 r = 3 or r = -63/13 not a valid

answerThe speed of the slower ship is 3

mph.


Chapter 7 Test

1. 28x4y6z2 = 22•7x4y6z2

70y8z5 = 2•5•7y6z5

98y3z8 = 2•72y3z8

GCF = 2•7y3z2 = 14y3z2

2. 4x6 - 3x4 - 2x2 - 3 = -1(-4x6 + 3x4+ 2x2 +3)

3. 12x4y3 - 3xy2 = 3xy2(4x3y - 1)

4. 6x4(3x + 2)5 - 15x3(3x + 2)4 = 3x3(3x + 2)4[2x(3x +2) - 5]

5. 4x2 - 5x - 8xy + 10y =x(4x - 5) - 2y(4x - 5) =(4x - 5)(x - 2y)

6. x2 - 5x - 14 = (x - 7)(x + 2)

7. 6x2 - 60x + 126 = 6(x2 -10x + 21)6(x - 7)(x - 3)

8. 5x2 - 9x - 18ac = 5(-18) = -90Product Sum1(-90) -892(-45) -433(-30) -275(-18) -136(-15) -65x2 + 6x - 15x - 18x(5x + 6) - 3(5x + 6) = (5x + 6)(x -3)

9. 49x2 - 9 = (7x + 3)(7x - 3)

10. 7x2 - 7xy - 21x + 21y =7(x2 - xy - 3x + 3y) = 7[x(x - y) - 3(x - y)] =7[(x - y)(x - 3)]

11. 20 + 8x2 - 68x + 100 = 8x2 - 68x + 120 =4(2x2 - 17x + 30)ac = 2•30 = 60Product Sum-1(-60) -61-2(-30) -32-3(-20) -23-4(-15) -19-5(-12) -17

4(2x2 - 5x - 12x + 30) = 4[x(2x -5) -6(2x - 5)] = 4(2x - 5)(x - 6)

12. 0 = 15x2 - 6x0 = 3x(5x - 2)3x = 0 or 5x - 2 = 0x = 0 or 5x = 2x = 0 or x = 2/5

13. 2x2 + 12x = -162x2 + 12x + 16 = 02(x2 + 6x + 8) = 02(x + 4)(x + 2) = 0x + 4 = 0 or x + 2 = 0x = -4 or x = -2

14. Let x = first integer; x+1 = secondx(x + 1) = x + (x + 1) + 5x2 + x = 2x + 6x2 - x - 6 = 0(x - 3)(x + 2) = 0x - 3 = 0 or x + 2 = 0x = 3 or x = -2When x = 3 then x + 1 = 4. When x

= -2 then x + 1 = -1.


chapter 7 factoring algebraic 3. 9(x - 5) 5 = 3 2 •(x - 5 ... 7 factoring algebraic expressions...

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