CHAPTER Factoring Polynomials 7 Solutions Key · 2 ˜ 2 5 5 Variable term as a product x ˜ x...

Are you reADy?

1. B; a polynomial with two terms

2. A; a whole number greater than 1 that has more than two whole-number factors

3. F; a number that is multiplied by another number to get a product

4. C; the product of any number and a whole number

5. E; a whole number greater than 1 that has exactly two factors, itself and 1

6. 3, 6, 9, 12 7. 4, 8, 12, 16

8. 8, 16, 24, 32 9. 15, 30, 45, 60

10. Yes; 5 × 4 = 20

11. no; factors of 50: 1, 2, 5, 10, 25, 50

12. Yes; 8 × 15 = 120 13. Yes; 7 × 35 = 245

14. Prime 15. Prime

16. Composite; 17. Composite; 10 = 2 · 5 38 = 2 · 19

18. Composite; 19. Composite; 115 = 5 · 23 147 = 21 · 7

20. Prime 21. Composite; 93 = 3 · 31

22. 2(x + 5) 23. 3h(h + 1)2(x) + 2(5) 3h(h) + 3h(1)2x +10 3 h 2 + 3h

24. xy ( x 2 - x y 3 )

xy ( x 2 ) - xy (x y 3 )

x 3 y - x 2 y 4

25. 6m ( m 2 - 4m - 1)

6m ( m 2 ) - 6m (4m) - 6m (1) 6 m 3 - 24 m 2 - 6m

26. (x + 3)(x + 8)x(x) + x(8) + 3(x) + 3(8) x 2 + 8x + 3x + 24 x 2 + 11x + 24

27. (b - 7)(b + 1)b(b) + b(1) - 7(b) - 7(1) b 2 + b - 7b - 7 b 2 - 6b - 7

28. (2p - 5)(p - 1)2p(p) + 2p(-1) - 5(p) - 5(-1)2 p 2 - 2p - 5p + 52 p 2 - 7p + 5

29. (3n + 4)(2n + 3)3n(2n) + 3n(3) + 4(2n) + 4(3)6 n 2 + 9n + 8n + 126 n 2 + 17n + 12

FActors AnD GreAtest common FActors

CheCk it OUt!

1a. 40 = 2 · 2 · 2 · 5 = 2 3 · 5

b. 33 = 3 · 11

c. 49 = 7 2

d. 19 is a prime number.

2a. Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 16: 1, 2, 4, 8, 16 The GCF of 12 and 16 is 4.

b. 15 = 3 · 5 25 = 5 · 5 The GCF of 15 and 25 is 5.

3a. 18 g 2 = 2 · 3 · 3 · g · g 27 g 3 = 3 · 3 · 3 · g · g · g The GCF of 18 g 2 and 27 g 3 is 9 g 2 .

b. 16 a 2 = 2 · 2 · 2 · 2 · a · a 9b = 3 · 3 · b The GCF of 16 a 2 and 9b is 1.

c. 8x = 8 · x 7 v 2 = 7 · v · v The GCF of 8x and 7 v 2 is 1.

4. First find the GCF of 36 and 48 because the number of CDs per shelves must be a common factor of 36 and 48. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 The GCF of 36 and 48 is 12. The greatest possible number of CDs in each shelf is 12. Find the number of shelves if each shelf holds 12 CDs.

36 CDs by pop artists + 48 CDs by country artists

________________________________________ 12 CDs per shelf

= 7 shelves

think and disCUss

1. Use a factor tree or divide the number by prime factors until the quotient is 1.

2.

100x2

Coefficient100

Prime factorization of coefficient 2 · 2 · 5 · 5

Variable term as a product

x · x

Variable Termx2

Prime Factorization

of 100x2

22 · 52 · x2

Factoring PolynomialsSolutions Key

223 Holt McDougal Algebra 1

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7CHAPTER

CS10_A1_MESK710372_C07.indd 223 3/30/11 11:23:20 PM

exerCisesGuided Practice

1. Possible answer: the greatest number that is a factor of two given numbers

2. 20 = 2 · 2 · 5 = 2 2 · 5

3. 36 = 3 · 3 · 2 · 2 = 3 2 · 2 2

4. 27 = 3 · 3 · 3 = 3 3

5. 54 = 3 · 3 · 3 · 2 = 3 3 · 2

6. 96 = 2 · 2 · 2 · 2 · 2 · 3 = 2 5 · 3

7. 7

8. 100 = 2 · 2 · 5 · 5 = 2 2 · 5 2

9. 75 = 3 · 5 · 5 = 3 · 5 2

10. Factors of 12: 1, 2, 3, 4, 6, 12Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 20, 30, 60 The GCF of 12 and 60 is 12.

11. Factors of 14: 1, 2, 7, 14 Factors of 49: 1, 7, 49 The GCF of 14 and 49 is 7.

12. Factors of 55: 1, 5, 11, 55 Factors of 121: 1, 11, 121 The GCF of 55 and 121 is 11.

13. 6 x 2 = 2 · 3 · x · x 5 x 2 = 5 · x · x The GCF of 6 x 2 and 5 x 2 is x 2 .

14. 15 y 3 = 3 · 5 · y · y · y -20y = -1 · 2 · 2 · 5 · y The GCF of 15 y 3 and -20y is 5y.

15. 13 q 4 = 13 · q · q · q · q 2 p 2 = 2 · p · p The GCF of 13 q 4 and 2 p 2 is 1.

16. First find the GCF of 54 and 18 because the number of beads per necklace must be a common factor of 54 and 18. Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54 Factors of 18: 1, 2, 3, 6, 9, 18 The GCF of 54 and 18 is 18. The greatest possible number of beads in each necklace is 18. Find the number of necklaces if each necklace takes 18 beads to make.

54 glass beads + 18 clay beads

__________________________ 18 beads per necklace

= 4 necklaces

Practice and Problem SolvinG

17. 18 = 2 · 3 · 3 18. 64 = 2 · 2 · 2 · 2 · 2 · 2 = 2 · 3 2 = 2 6

19. 12 = 2 · 2 · 3 20. 150 = 2 · 3 · 5 · 5 = 2 2 · 3 = 2 · 3 · 5 2

21. 17 22. 226 = 2 · 113

23. 49 = 7 · 7 24. 63 = 3 · 3 · 7 = 7 2 = 3 2 · 7

25. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Factors of 63: 1, 3, 7, 9, 21, 63 The GCF of 36 and 63 is 9.

26. Factors of 14: 1, 2, 7, 14 Factors of 15: 1, 3, 5, 15 The GCF of 14 and 15 is 1.

27. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40 The GCF of 30 and 40 is 10.

28. 8 a 2 = 2 · 2 · 2 · a · a 11 = 1 · 11 The GCF of 8a 2 and 11 is 1.

29. 9s = 3 · 3 · s 63 s 3 = 3 · 3 · 7 · s · s · s The GCF of 9s and 63 s 3 is 9s.

30. -64 n 4 = -1 · 2 · 2 · 2 · 2 · 2 · 2 · n · n · n · n 24 n 2 = 2 · 2 · 2 · 3 · n · n The GCF of -64 n 4 and 24 n 2 is 8 n 2 .

31. First find the GCF of 72 and 108 because the number of tarts must be a common factor of 72 and 108. 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, 72 108: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108 The GCF of 72 and 108 is 36, so there will be 36 fruits in each tart. There will be a total of 5 tarts: 2 raspberry and 3 blueberry.

32. 5 · t = 5t 33. 2 · 2 · x · x = 4 x 2

34. 11 35. 2 · n = 2n

36. Possible answer: Even numbers greater than 2 all have 2 as a factor and thus are not prime.

37. No; An odd composite number and an even composite number can have no factors in common other than 1.

38a. Since the area of a rectangle is length times width, to find all possible whole number lengths, find 2 whole numbers that have the product 84.

1 × 84; 2 × 42; 3 × 28; 4 × 21; 6 × 14; 7 × 12

b. P = 2(7 + 12) c. P = 2(1 + 84) = 2(19) = 38 ft = 2(85) = 170 ft

39. First find the GCF of 35 and 40, because the number of guards in each row must be a common factor of 35 and 40. 35: 1, 5, 7, 35 40: 1, 2, 4, 5, 8, 10, 20, 40 The GCF of 35 and 40 is 5. The greatest possible number of guards in each row is 5. Find the number of rows if each row has 5 guards.

35 Cavaliers + 40 Blue Devils _________________________ 5 Guards per row

= 15 rows

40. 11: 1, 11 12: 1, 2, 3, 4, 6, 12 14: 1, 2, 7, 14 12 and 14; GCF = 2

41. 8: 1, 2, 4, 8 20: 1, 2, 4, 5, 10, 20 63: 1, 3, 7, 9, 21, 63 8 and 20; GCF = 4


CS10_A1_MESK710372_C07.indd 224 3/30/11 11:23:21 PM

42. 16: 1, 2, 4, 8, 1621: 1, 3, 7, 2127: 1, 3, 9, 2721 and 27; GCF = 3

43. 32: 1, 2, 4, 8, 16, 32 63: 1, 3, 7, 9, 21, 63 105: 1, 3, 5, 7, 15, 21, 35, 105 63 and 105; GCF = 21

44. 25: 1, 5, 25 35: 1, 5, 7, 35 54: 1, 2, 3, 4, 6, 9, 16,18, 27, 54 25 and 35; GCF = 5

45. 35: 1, 5, 7, 35 54: 1, 2, 3, 6, 9, 18, 27, 54 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 54 and 72; GCF = 18

46. 2 4 · 3; possible answer: because 48 = 2 · 24 and 24 = 2 3 · 3, 48 = 2 · 2 3 · 3 = 2 4 · 3

From top to bottom, left to right: 47. 36; 2; 9; 3; 2 3 · 3 2 48. 27; 3; 9; 3 4 49. 105; 5; 7; 2 · 3 · 5 · 7 50. 2; 14; 7; 2 3 · 7 51. 2; 2; 27; 3; 2 2 · 3 3 52. 2; 34; 17; 2 3 · 17 53. 24; 2; 6; 3; 2 4 · 3 54. 2; 70; 5; 2 2 · 5 · 7 55. 2; 2; 10; 5; 2 3 × 5 56a. Use the given formula, d = vt + 1 __

2 a t 2

d = (2)t + 1 __ 2 (2) t 2

= 2t + t 2

b. 2t = 2 · t t 2 = t · t The GCF of 2t and t 2 is t.

teSt PreP

57. D; 16, 24, 48 has a GCF of 8.

58. F; the GCF of 48 and 12 is 12, and the GCF of 12 and 8 is 4.

59. 1 ft

24 ft

P = 50 ft12 ft

2 ft

P = 28 ft

8 ft

3 ft

P = 22 ft

6 ft

4 ft

P = 20 ftPatricia should make the pen 4 ft × 6 ft because these dimensions give the shortest perimeter and she will need to buy the least fencing.

challenGe and extend

60. 4 n 3 = 2 · 2 · n · n · n 16 n 2 = 2 · 2 · 2 · 2 · n · n 8n = 2 · 2 · 2 · n The GCF of 4 n 3 , 16 n 2 , and 8n is 4n.

61. 27 y 3 = 3 · 3 · 3 · y · y · y 18 y 2 = 2 · 3 · 3 · y · y 81y = 3 · 3 · 3 · 3 · y The GCF of 27 y 3 , 18 y 2 , and 81y is 9y.

62. 100 = 2 · 2 · 5 · 5 25 s 5 = 5 · 5 · s · s · s · s · s 50s = 2 · 5 · 5 · s The GCF of 100, 25 s 5 , and 50s is 25.

63. 2 p 4 r = 2 · p · p · p · p · r 8 p 3 r 2 = 2 · 2 · 2 · p · p · p · r · r 16 p 2 r 3 = 2 · 2 · 2 · 2 · p · p · r · r · r The GCF of 2 p 4 r, 8 p 3 r 2 , and 16 p 2 r 3 is 2 p 2 r.

64. 2 x 3 y = 2 · x · x · x · y 8 x 2 y 2 = 2 · 2 · 2 · x · x · y · y 17x y 3 = 17 · x · y · y · y The GCF of 2 x 3 y, 8 x 2 y 2 , and 17x y 3 is xy.

65. 8 a 4 b 3 = 2 · 2 · 2 · a · a · a · a · b · b · b 4 a 3 b 3 = 2 · 2 · a · a · a · b · b · b 12 a 2 b 3 = 2 · 2 · 3 · a · a · b · b · b The GCF of 8 a 4 b 3 , 4 a 3 b 3 , and 12 a 2 b 3 is 4 a 2 b 3 .

66. 1 × 20; 2 × 10; 4 × 5; 20 × 1; 10 × 2; 5 × 4

67. Possible answer: 21, 35, 49; 7, 14, 84

68. Possible answer: 6, 35, 143 6 = 2 × 3; 35 = 5 × 7; 143 = 11 × 13

FActorinG by GcF

CheCk it OUt!

1a. 5b = 2 · b 9 b 3 = 3 · 3 · b · b · b The GCF of 5b and 9 b 3 is b.5(b) + 9 b 2 (b)b(5 + 9 b 2 )

b. 9 d 2 = 3 · 3 · d · d 8 2 = 2 · 2 · 2 · 2 · 2 · 2 The GCF of 9d 2 and 8 2 is 1; it cannot be factored.

c. 18 y 3 = 2 · 3 · 3 · y · y · y 7 y 2 = 7 · y · y The GCF of 18 y 3 and 7 y 2 is y 2 .-1[18y( y 2 ) + 7( y 2 )]- y 2 (18y + 7)

d. 8 x 4 = 2 · 2 · 2 · x · x · x · x 4 x 3 = 2 · 2 · x · x · x 2 x 2 = 2 · x · x The GCF of 8x 4 , 4x 3 and 2x 2 is 2x 2 4 x 2 ( 2x 2 ) + 2x (2x 2 ) - 1( 2x 2 )2 x 2 (4x 2 + 2x - 1)

2. A = 2 x 2 + 4x = x(2x) + 2(2x) = 2x(x + 2)

Possible expression for the dimensions of the solar panel are 2x cm and (x + 2) cm.

3a. 4s(s + 6) - 5(s + 6)(4s - 5)(s + 6)

b. 7x(2x + 3) + (2x + 3)7x(2x + 3) + 1(2x + 3)(7x + 1)(2x + 3)

c. 3x(y + 4) - 2y(x + 4)There are no common factors.


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d. 5x(5x - 2) - 2(5x - 2)(5x - 2)(5x - 2)(5x - 2 ) 2

4a. 6 b 3 + 8 b 2 + 9b + 12( 6b 3 + 8b 2 ) + (9b + 12) 2b 2 (3b + 4) + 3 (3b + 4) ( 2b 2 + 3) (3b + 4)

b. 4r 3 + 24r + r 2 + 6( 4r 3 + 24r) + ( r 2 + 6)4r( r 2 + 6) + 1( r 2 + 6)(4r + 1)( r 2 + 6)

5a. 15x 2 - 10x 3 + 8x - 12(15 x 2 - 10 x 3 ) + (8x - 12)5 x 2 (3 - 2x) + 4(2x - 3)5 x 2 (3 - 2x) + 4(-1)(2x - 3)5 x 2 (3 - 2x) - 4(3 - 2x)(5 x 2 - 4)(3 - 2x)

b. 8y - 8 - x + xy(8y - 8) + (-x + xy)8(y - 1) - x(1 - y)8(y - 1) - x(-1)(-1 + y)8(y - 1) + x(y - 1)(8 + x)(y - 1)

think and disCUss

1. Possible answer: when you know the GCF of the monomials in a polynomial, you can factor out the GCF from each monomial to factor the polynomial.

2. Factoring by GCF

1. Find the greatest common factor.

4. Check by multiplying.

2. Write each term as a product using the GCF.

3. Use the Distributive Property to factor out the GCF.


1. 15a = 3 · 5 · a 5 a 2 = 5 · a · a The GCF of 15a and 5 a 2 is 5a.15a - 5 a 2 3(5a) - a(5a)5a(3 - a)

2. 10 g 3 = 2 · 5 · g · g · g 3g = 3 · g The GCF of 10 g 3 and 3g is g.10 g 3 - 3g10 g 2 (g) - 3(g)g(10 g 2 - 3)

3. 35x = 5 · 7 · x 42 = 6 · 7 The GCF of 35x and 42 is 7.-35x + 42-5x(7) + 6(7)7(-5x + 6)

4. 4 x 2 = 2 · 2 · x · x 6x = 2 · 3 · x The GCF of 4 x 2 and 6x is 2x.-(4 x 2 + 6x)- (2x(2x) + 3(2x)) -2x(2x + 3)

5. 12 h 4 = 2 · 2 · 3 · h · h · h · h 8 h 2 = 2 · 2 · 2 · h · h 6h = 2 · 3 · h The GCF of 12h 4 , 8 h 2 and 6h is 2h.12 h 4 + 8 h 2 - 6h6 h 3 (2h) + 4h(2h) - 3(2h)2h(6 h 3 + 4h - 3)

6. 3x 2 = 3 · x · x 9x = 3 · 3 · x 3 = 3 The GCF of 3 x 2 , 9x and 3 is 3.3 x 2 - 9x + 3 x 2 (3) - 3x(3) + 1(3)3( x 2 - 3x + 1)

7. 9 m 2 = 3 · 3 · m · m m = m The GCF of 9 m 2 and m is m.9 m 2 + m9m(m) + 1(m)m(9m + 1)

8. 14 n 3 = 2 · 7 · n · n · n 7n = 7 · n 7 n 2 = 7 · n · n The GCF of 14 n 3 , 7n and 7 n 2 is 7n.14 n 3 + 7n + 7 n 2 2 n 2 (7n) + 1(7n) + n(7n)7n(2 n 2 + 1 + n)

9. 36f = 2 · 2 · 3 · 3 · f 18 f 2 = 2 · 3 · 3 · f · f 3 = 3 The GCF of 36f, 18 f 2 and 3 is 3.36f + 18 f 2 + 312f(3) + 6 f 2 (3) + 1(3)3(12f + 6 f 2 + 1)

10. 15 b 2 = 3 · 5 · b · b 7b = 7 · b The GCF of 15 b 2 and 7b is b.-15 b 2 + 7b-15b(b) + 7(b)b(-15b + 7)

11. -16 t 2 + 320t -t(16t) + 20(16t) 16t(-t + 20) Using the factored form of the expression can tell you when the rocket will land again.


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12. 5(m - 2) - m(m - 2) (5 - m)(m - 2)

13. 2b(b + 3) + 5(b + 3) (2b + 5)(b + 3)

14. 4(x - 3) - x(y +2) Cannot be factored

15. x 3 + 4 x 2 + 2x + 8( x 3 + 4 x 2 ) + (2x + 8) x 2 (x + 4) + 2 (x + 4) ( x 2 + 2)(x + 4)

16. 6 x 3 + 4 x 2 + 3x + 2(6 x 3 + 4 x 2 ) + (3x + 2) 2 x 2 (3x + 2) + 1 (3x + 2) (2 x 2 + 1)(3x + 2)

17. 4 b 3 - 6 b 2 +10b - 15(4 b 3 - 6 b 2 ) + (10b - 15)2 b 2 (2b - 3) + 5(2b - 3)(2 b 2 + 5)(2b - 3)

18. 2 m 3 + 4 m 2 + 6m + 12(2 m 3 + 4 m 2 ) + (6m + 12)2 m 2 (m + 2) + 6(m + 2)2(m + 2)( m 2 + 3)

19. 7 r 3 - 35 r 2 + 6r - 307 r 3 - 35 r 2 + 6r - 307 r 2 (r - 5) + 6(r - 5)(7 r 2 + 6)(r - 5)

20. 10 a 3 + 4 a 2 + 5a + 2(10 a 3 + 4 a 2 ) + (5a + 2)2 a 2 (5a + 2) + 1(5a + 2)(2 a 2 + 1)(5a + 2)

21. 2 r 2 - 6r + 12 - 4r(2 r 2 - 6r) + (12 - 4r)2r(r - 3) - 4(r - 3)(2r - 4)(r - 3)2(r - 2)(r - 3)

22. 6 b 2 - 3b + 4 - 8b(6 b 2 - 3b) + (4 - 8b) 3b(2b - 1) - 4(2b - 1)(3b - 4)(2b - 1)

23. 14 q 2 - 21q + 6 - 4q(14 q 2 - 21q) + (6 - 4q) 7q(2q - 3) - 2(2q - 3)(7q - 2)(2q - 3)

24. 3r - r 2 + 2r - 63r - r 2 + 2r - 6r(3 - r) - 2(3 - r)(r - 2)(3 - r)

25. 2 m 3 - 6 m 2 + 9 - 3m(2 m 3 - 6 m 2 ) + (9 - 3m) 2 m 2 (m - 3) - 3(m - 3)

(2 m 2 - 3)(m - 3)

26. 6 a 3 - 9 a 2 - 12 + 8a(6 a 3 - 9 a 2 ) + (-12 + 8a)3 a 2 (2a - 3) + 4(2a - 3)

(3 a 2 + 4)(2a - 3)


27. 9 y 2 = 3 · 3 · y · y 45y = 3 · 3 · 5 · y The GCF of 9 y 2 and 45y is 9y.9 y 2 + 45yy(9y) + 5(9y)9y(y + 5)

28. 36 d 3 = 2 · 2 · 3 · 3 · d · d · d 24 = 2 · 2 · 2 · 3 The GCF of 36 d 3 and 24 is 12.36 d 2 + 243 d 3 (12) + 2(12)12(3 d 3 + 2)

29. 14 x 4 = 2 · 7 · x · x · x · x 5 x 2 = 5 · x · x The GCF of 14 x 4 and 5x 2 is x 2 .-14 x 4 + 5 x 2 -14 x 2 ( x 2 ) + 5( x 2 ) x 2 (-14 x 2 + 5)

30. 15f = 3 · 5 · f 10 f 2 = 2 · 5 · f · f The GCF of 15f and 10 f 2 is 5f.-15f - 10 f 2 3(-5f) + 2f(-5f)-5f(3 + 2f)

31. 4 d 4 = 2 · 2 · d · d · d · d d 3 = d · d · d 3 d 2 = 3 · d · d The GCF of 4 d 4 , d 3 , and 3 d 2 is d 2 .-1 (4 d 4 - d 3 + 3 d 2 )

- d 2 ( 4d 2 - d + 3)

32. 14 x 3 = 2 · 7 · x · x · x 63 x 2 = 3 · 3 · 7 · x · x 7x = 7 · x The GCF of 14 x 2 , 63 x 2 and 7x is 7x.14 x 3 + 63 x 2 - 7x2 x 2 (7x) + 9x(7x) - 1(7x)7x(2 x 2 + 9x - 1)

33. 21 c 2 = 3 · 7 · c · c 14c = 2 · 7 · c The GCF of 21 c 2 and 14c is 7c.21 c 2 + 14c3c(7c) + 2(7c)7c(3c + 2)

34. 33 d 3 = 3 · 11 · d · d · d 22d = 2 · 11 · d 11 = 11 The GCF of 33d 3 , 22d and 11 is 11.33 d 3 + 22d + 113 d 3 (11) + 2d(11) + 1(11)11(3 d 3 + 2d + 1)


CS10_A1_MESK710372_C07.indd 227 3/30/11 11:23:27 PM

35. 5 g 3 = 5 · g · g · g15 g 2 = 3 · 5 · g · gThe GCF of 5 g 3 and 15 g 2 is 5 g 2 .-5 g 3 - 15 g 2 g(-5 g 2 ) + 3(-5 g 2 )-5 g 2 (g + 3)

36. S = P + Prt = 1(P) + rt(P) = P(1 + rt)P(1 + rt) is the factored expression for P + Prt.

37. Cannot be factored

38. -4x(x + 2) + 9(x + 2)(x + 2)(-4x + 9)

39. 6y(y - 7) + (y - 7)6y(y - 7) + 1(y - 7)(6y + 1)(y - 7)

40. a(x - 3) + 2b(x - 3)(a + 2b)(x - 3)

41. -3(2 + b) + 4b(b + 2)(-3 + 4b)(b + 2)

42. 5(3x - 2) + x(3x - 2)(5 + x)(3x - 2)

43. 2 a 3 - 8 a 2 + 3a - 12(2 a 3 - 8 a 2 ) + (3a - 12)2 a 2 (a - 4) + 3(a - 4)(2 a 2 + 3)(a - 4)

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

45. 6 x 3 + 18 x 2 + x + 3(6 x 3 + 18 x 2 ) + (x + 3)6 x 2 (x + 3) + 1(x + 3)(6 x 2 + 1)(x + 3)

46. 7 x 3 + 2 x 2 + 28x + 8(7 x 3 + 2 x 2 ) + (28x + 8) x 2 (7x + 2) + 4(7x + 2)( x 2 + 4)(7x + 2)

47. n 3 - 2 n 2 + 5n - 10( n 3 - 2 n 2 ) + (5n - 10) n 2 (n - 2) + 5(n - 2)( n 2 + 5)(n - 2)

48. 10 b 3 - 16 b 2 + 25b - 40(10 b 3 - 6 b 2 ) + (25b - 40)2 b 2 (5b - 8) + 5(5b - 8)(2 b 2 + 5)(5b - 8)

49. 2 m 3 - 2 m 2 + 3 - 3m(2 m 3 - 2 m 2 ) - (3m - 3)2 m 2 (m - 1) - 3(m - 1)(2 m 2 - 3)(m - 1)

50. 2 d 3 - d 2 - 3 + 6d(2 d 3 - d 2 ) + (6d - 3) d 2 (2d - 1) + 3(2d -1)( d 2 + 3)(2d - 1)

51. 6 f 3 - 8 f 2 + 20 - 15f(6 f 3 - 8 f 2 ) - (15f - 20)2 f 2 (3f - 4) - 5(3f - 4)(2 f 2 - 5)(3f - 4)

52. 5 k 2 - k 3 + 3k - 15(5 k 2 - k 3 ) + (3k - 15) k 2 (5 - k) + 3(k - 5) k 2 (5 - k) - 3(5 - k)( k 2 - 3)(5 - k)

53. b 3 - 2b - 8 + 4 b 2 ( b 3 + 4 b 2 ) - (2b + 8) b 2 (b + 4) - 2(b + 4)( b 2 - 2)(b + 4)

54. 20 - 15x - 6 x 2 + 8x(20 - 15x) + (8x - 6 x 2 )5(4 - 3x) + 2x(4 - 3x)(5 + 2x)(4 - 3x)

55. Given GCF of 4v:16v + 12 v 2 4(4v) + 3v(4v)4v(4 + 3v)

56. Given GCF of 5x:15x - 25 x 2 5x(3) - 5x(5x)5x(3 - 5x)

57. Given GCF of -8 k 2 :-16 k 3 - 24 k 2 -8 k 2 (2k) - 8 k 2 (3)-8 k 2 (2k + 3)

58. Given GCF of 1:-x - 10-1(x) - 1(10)-1(x + 10)

59. x 2 + 5x; polynomial has 2 terms, so it is a binomial. x 2 + 5xx(x) + 5(x)x(x + 5)

60. 28 c 2 - 49c; polynomial has 2 terms, so it is a binomial.28 c 2 - 49c 4c(7c) - 7(7c)7c(4c - 7)

61. a 4 + a 3 + a 2 ; polynomial has 3 terms, so it is a trinomial. a 4 + a 3 + a 2 a 2 ( a 2 ) + a( a 2 ) + 1( a 2 ) a 2 ( a 2 + a + 1)

62. 36 + 99r - 40 r 2 - 110 r 3 ; polynomial has 4 terms, so it is a polynomial.36 + 99r - 40 r 2 - 110 r 3 (36 + 99r) - (40 r 2 + 110 r 3 )9(4 + 11r) - 10 r 2 (4 + 11r)(11r + 4)(-10 r 2 + 9)

63a. Let x be the interest rate of the CDs that Justin’s aunt purchased for him.

For CDs purchased in 2004, n = 3 and P = 100. The value of the CDs purchased in 2004 is 100x 3 . For CDs purchased in 2005, n = 2 and P = 200. The value of the CDs purchased in 2005 is 200x 2 . For CDs purchased in 2005, n = 2 and P = 400. The value of the CDs purchased in 2006 is 400x.


CS10_A1_MESK710372_C07.indd 228 3/30/11 11:23:29 PM

b. The total value is 100 x 3 + 200 x 2 + 400x + 800.

c. 100 x 3 + 200 x 2 + 400x + 800 (100 x 3 + 200 x 2 ) + (400x + 800) 100 x 2 (x + 2) + 400(x + 2) (100 x 2 + 400)(x + 2) 100( x 2 + 4)(x + 2) 100( 1.09 2 + 4)(1.09 + 2) = 1603.1229; $1603.12

64. The area of the figure is the sum of the areas of the rectangle and the triangle. The area of the rectangle is 2x(2x + 6), and the area of the triangle is

1 __ 2 · 2x(x + 8). The sum is

2x(2x + 6) + 1 __ 2 · 2x(x + 8).

2x(2x + 6) + 1 __ 2 · 2x(x + 8)

4 x 2 + 12x + x 2 + 8x5 x 2 + 20x5x(x + 4)

65. Method 1(3a - 3b) - (4a - 4b)3(a - b) - 4(a - b)(3 - 4)(a - b)-1(a - b)(b - a)

Method 2(3a - 4a) - (3b - 4b)a(3 - 4) - b(3 - 4)(a - b)(3 - 4)(a - b)(-1)(b - a)

66. 1 __ 2 ( x 3 - 2x + 2 x 2 - 4)

1 __ 2 [( x 3 - 2x) + (2 x 2 - 4)]

1 __ 2 [x( x 2 - 2) + 2( x 2 - 2)]

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

The base of the triangle is x 2 - 2.

67. If the sum of two binomials is 0, they areopposite binomials.

68a. Either a or b, or both must equal 0.

b. The product of t and (3 - t) is 0, so at least one of the factors must be 0.

c. The two times are t = 0 and t = 3.

69a. Commutative Property of Addition

b. Association Property of Addition

c. Distribution Property

d. Distribution Property

70. A is incorrect because n 2 ≠ n 2 (0).

teSt PreP

71. D;

24 x 3 - 12 x 2 12 x 2 (2x) - 12 x 2 (1)12 x 2 (2x - 1)

72. G;

18 x 2 + 36x18x(x) + 18x(2)18x(x + 2)

73. C;

x 2 + 3x - 6x - 18( x 2 + 3x) - (6x + 18)x(x + 3) - 6(x + 3)(x - 6)(x + 3)


74. 6a b 2 - 24 a 2 6a( b 2 ) - 6a(4a)6a( b 2 - 4a)

75. -72 a 2 b 2 - 45ab-9ab(8ab) - 9ab(5)-9ab(8ab + 5)

76. -18 a 2 b 2 + 21ab-3ab(6ab) - 3ab(-7)-3ab(6ab - 7)

77. ab + bc + ad + cd(ab + bc) + (ad + cd)b(a + c) + d(a + c)(b + d)(a + c)

78. 4 y 2 + 8ay - y -2a(4 y 2 + 8ay) - (y + 2a)4y(y + 2a) - 1(y + 2a)(4y - 1)(y + 2a)

79. x 3 - 4 x 2 + 3x -12( x 3 - 4 x 2 ) + (3x - 12) x 2 (x - 4) + 3(x - 4)( x 2 + 3)(x - 4)

80a. A = π R 2 - π r 2 A = π( R 2 ) - π( r 2 ) A = π( R 2 - r 2 )

b. A = π( 12 2 - 5 2 ) = π(119) ≈ 374 cm 2

FActorinG x 2 + bx + c

CheCk it OUt!

1a. x 2 + 10x + 24 (x + 1)(x + 24) = x 2 + 25x + 24 7 (x + 2)(x + 12) = x 2 + 14x + 24 7 (x + 3)(x + 8) = x 2 + 11x + 24 7 (x + 4)(x + 6) = x 2 + 10x + 24 3 The factors of x 2 + 10x + 24 are (x + 4) and (x + 6). x 2 + 10x + 24 = (x + 4)(x + 6)

b. x 2 + 7x + 12 (x + 1)(x + 12) = x 2 + 13x + 127

(x + 2)(x + 6) = x 2 + 8x + 127

(x + 3)(x + 4) = x 2 + 7x + 12 3 The factors of x 2 + 7x + 12 are (x + 3) and (x + 4). x 2 + 7x + 12 = (x + 3)(x + 4)

2a. x 2 + 8x + 12 ________________ Factors of 12 Sum 1 and 12 137

2 and 6 83

(x + 2)(x + 6)

b. x 2 - 5x + 6 _______________ Factors of 6 Sum -1 and -6 -77

-2 and -3 -53

(x - 2)(x - 3)

c. x 2 + 13x + 42 ________________ ________________ Factors of 42 Sum 1 and 42 437

2 and 21 237

3 and 14 17 7 6 and 7 133

(x + 6)(x + 7)


7-3

CS10_A1_MESK710372_C07.indd 229 3/30/11 11:23:30 PM

d. x 2 - 13x + 40 ________________ Factors of 40 Sum -1 and -40 -417

-2 and -20 -227

-4 and -10 -147

-5 and -8 -133

(x - 5)(x - 8)

3a. x 2 + 2x - 15 __________________ __________________ Factors of -15 Sum -1 and 15 147 -3 and 5 23

(x - 3)(x + 5)

b. x 2 - 6x + 8 _______________ Factors of 8 Sum -1 and -8 -97

-2 and -4 -63

(x - 2)(x - 4)

c. x 2 - 8x - 20 __________________ __________________ Factors of -20 Sum 1 and -20 -197

2 and -10 -83

(x -10)(x + 2)

4. n 2 - 7n + 10 ________________ Factors of 10 Sum -1 and -10 -117

-2 and -5 -73

(n - 5)(n - 2)

n n 2 - 7n + 10

0 0 2 - 7(0) + 10 = 10

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

2 2 2 - 7(2) + 10 = 0

3 3 2 - 7(3) + 10 = -2

4 4 2 - 7(4) + 10 = -2

n (n - 5)(n - 2)

0 (0 - 5)(0 - 2) = 10

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

2 (2 - 5)(2 - 2) = 0

3 (3 - 5)(3 - 2) = -2

4 (4 - 5)(4 - 2) = -2

think and disCUss

1. Find the 2 factors of 14 that have a sum of 9: 2 and 7. Then use these numbers as the constants in the factors: (x + 2)(x + 7).

check (x + 2)(x + 7) = x 2 + 2x + 7x + 14 = x 2 + 9x + 143

2. For x 2 + bx + c = (x + m)(x + n), if c > 0 and b > 0, m > 0 and n > 0. If c > 0 and b < 0, m < 0 and n < 0. If c < 0 and b > 0, the greater of m and n is positive and the lesser is negative. If c < 0 and b < 0, the greater of m and n is negative and the lesser is positive.

3.

c is positive, and b is positive.

x2 + 5x + 6 (x + 2)(x + 3)

c is positive, and b is negative.

x2 - 5x + 6 (x - 2)(x - 3)

c is negative, and b is negative.

x2 - x - 6 (x + 2)(x - 3)

c is negative, and b is positive.

x2 + x - 6 (x - 2)(x + 3)

Factoringx2 + bx + c


1. x 2 + 13x + 36(x + 1)(x + 36) = x 2 + 37x + 36 7(x + 2)(x + 18) = x 2 + 20x + 367

(x + 3)(x + 12) = x 2 + 15x + 367

(x + 4)(x + 9) = x 2 + 13x + 363

The factors of x 2 + 13x + 36 are (x + 4) and (x + 9). x 2 + 13x + 36 = (x + 4)(x + 9)

2. x 2 + 11x + 24(x + 1)(x + 24) = x 2 + 25x + 247

(x + 2)(x + 12) = x 2 + 14x + 247

(x + 3)(x + 8) = x 2 + 11x + 243


3. x 2 + 14x + 40(x + 1)(x + 40) = x 2 + 41x + 407

(x + 2)(x + 20) = x 2 + 22x + 407

(x + 4)(x + 10) = x 2 + 14x + 403


4. x 2 + 4x + 3 _______________ Factors of 3 Sum 1 and 3 43

(x + 1)(x + 3)

5. x 2 + 10x + 16 ________________ Factors of 16 Sum 1 and 16 177

2 and 8 103

(x + 2)(x + 8)

6. x 2 + 15x + 44 ________________ Factors of 44 Sum 1 and 44 45 72 and 22 247

4 and 11 153

(x + 4)(x + 11)

7. x 2 - 7x + 6 _______________ Factors of 6 Sum -1 and -6 -73

(x - 1)(x - 6)

8. x 2 - 9x + 14 ________________ Factors of 14 Sum -1 and -14 -157

-2 and -7 -93

(x - 2)(x - 7)


CS10_A1_MESK710372_C07.indd 230 3/30/11 11:23:33 PM


-2 and -12 -147

-3 and -8 -113

-4 and -6 -103

(x - 3)(x - 8)

10. x 2 + 6x -7 _________________ _________________ Factors of -7 Sum -1 and 7 63

(x - 7)(x + 1)

11. x 2 + 6x - 27 __________________ Factors of -27 Sum -1 and 27 267

-3 and 9 63

(x - 3)(x + 9)

12. x 2 + x - 30 __________________ Factors of -30 Sum -1 and 30 297

-2 and 15 137

-3 and 10 77

-5 and 6 13

(x - 5)(x + 6)

13. x 2 - x - 2 _________________ Factors of -2 Sum 1 and -2 -13

(x + 1)(x - 2)

14. x 2 - 3x - 18 __________________ __________________ Factors of -18 Sum 1 and -17 -167

2 and -9 -77

3 and -6 -33

(x + 3)(x - 6)

15. x 2 - 4x - 45 __________________ __________________ Factors of -45 Sum 1 and -45 -447

3 and -15 -127

5 and -9 -43

(x + 5)(x - 9)

16. n 2 + 6n - 7 _________________ Factors of -7 Sum -1 and 7 63

(n - 1)(n + 7)

n n 2 + 6n - 7

0 0 2 + 6(0) - 7 = -7

1 1 2 + 6(1) - 7 = 0

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

3 3 2 + 6(3) - 7 = 20

4 4 2 + 6(4) - 7 = 33

n (n - 1)(n + 7)

0 (0 - 1)(0 + 7) = -7

1 (1 - 1)(1 + 7) = 0

2 (2 - 1)(2 + 7) = 9

3 (3 - 1)(3 + 7) = 20

4 (4 - 1)(4 + 7) = 33


17. x 2 + 13x + 30(x + 1)(x + 30) = x 2 + 31x + 307

(x + 2)(x + 15) = x 2 + 17x + 307

(x + 3)(x + 10) = x 2 + 13x + 303


18. x 2 + 11x + 28(x + 1)(x + 28) = x 2 + 29x + 287

(x + 2)(x + 14) = x 2 + 16x + 287

(x + 4)(x + 7) = x 2 + 11x + 283


19. x 2 + 16x + 48(x + 1)(x + 48) = x 2 + 49x + 487

(x + 2)(x + 24) = x 2 + 26x + 487

(x + 3)(x + 16) = x 2 + 19x + 487

(x + 4)(x + 12) = x 2 + 16x + 483



(x + 1)(x + 11)


2 and 14 163

(x + 2)(x + 14)


2 and 18 207

3 and 12 153

(x + 3)(x + 12)

23. x 2 - 6x + 5 _______________ Factors of 5 Sum -1 and -5 -63

(x - 1)(x - 5)


-2 and -9 -117

-3 and -6 -93

(x - 3)(x - 6)


-2 and -16 -187

-4 and -8 -123

(x - 4)(x - 8)


CS10_A1_MESK710372_C07.indd 231 3/30/11 11:23:34 PM


-2 and 6 47

-3 and 4 13

(x - 3)(x + 4)

27. x 2 + 4x - 21 __________________ Factors of -21 Sum -1 and 21 20 7-3 and 7 4 3(x - 3)(x + 7)

28. x 2 + 9x - 36 __________________ Factors of -36 Sum -1 and 36 35 7-2 and 18 16 7-3 and 12 9 3(x - 3)(x + 12)

29. x 2 - 12x - 13 __________________ Factors of -13 Sum 1 and -13 -12 3(x + 1)(x - 13)

30. x 2 - 10x - 24 __________________ Factors of -24 Sum 1 and -24 -23 72 and -12 -10 3(x + 2)(x - 12)


32. n 2 - 12n - 45 __________________ Factors of -45 Sum 1 and -45 -44 73 and -15 -12 3(n + 3)(n - 15)

n n 2 - 12n - 45

0 0 2 - 12(0) - 45 = -45

1 1 2 - 12(1) - 45 = -56

2 2 2 - 12(2) - 45 = -65

3 3 2 - 12(3) - 45 = -72

4 4 2 - 12(4) - 45 = -77

n (n + 3)(n - 15)

0 (0 + 3)(0 - 15) = -45

1 (1 + 3)(1 - 15) = -56

2 (2 + 3)(2 - 15) = -65

3 (3 + 3)(3 - 15) = -72

4 (4 + 3)(4 - 15) = -77

33. C; x 2 + 3x - 10 __________________ Factors of -10 Sum -1 and 10 97

-2 and 5 33

(x - 2)(x + 5)

34. A; x 2 - 7x + 10 ________________ Factors of 10 Sum -1 and -10 -11 7-2 and -5 -7 3(x - 2)(x - 5)

35. D; x 2 - 9x - 10 __________________ Factors of -10 Sum 1 and -10 -9 3(x + 1)(x - 10)

36. B; x 2 + 11x + 10 ________________ Factors of 10 Sum 1 and 10 11 3(x + 1)(x + 10)

37. They are inverse operations.


-2 and 10 87

-4 and 5 1 3(x - 4)(x + 5)

39. x 2 - 11x + 18 ________________ Factors of 18 Sum -1 and -18 -19 7-2 and -9 -11 3(x - 2)(x - 9)


41. x 2 + 10x + 9 _______________ Factors of 9 Sum 1 and 9 10 3(x + 1)(x + 9)

42. x 2 - 12x + 32 __________________ Factors of +32 Sum -1 and -32 -33 7-2 and -16 -18 7-4 and -8 -12 3

x 2 - 12x + 32 = (x - 4)(x - 8).

43. x 2 + 13x + 42 ________________ Factors of 42 Sum 1 and 42 43 72 and 21 23 73 and 14 17 76 and 7 13 3(x + 6)(x + 7)

44. x 2 - 7x + 12 ________________ Factors of 12 Sum -1 and -12 -13 7-2 and -6 -8 7-3 and -4 -7 3(x - 3)(x - 4)

45. x 2 + 11x + 18 ________________ Factors of 18 Sum 1 and 18 19 72 and 9 11 3(x + 2)(x + 9)


CS10_A1_MESK710372_C07.indd 232 3/30/11 11:23:35 PM

46. x 2 - 6x - 27 __________________ Factors of -27 Sum 1 and -27 -267

3 and -9 -63

(x + 3)(x - 9)

47. x 2 + 5x - 24 __________________ Factors of -24 Sum -1 and 24 23 7-2 and 12 10 7-3 and 8 53

(x - 3)(x + 8)

48. x 2 - 10x + 21 ________________ Factors of 21 Sum -1 and -21 -22 7-3 and -7 -10 3(x - 3)(x - 7)

49. x 2 + 4x - 45 __________________ Factors of -45 Sum -1 and 45 44 7-3 and 15 12 7-5 and 9 4 3(x - 5)(x + 9)

50. n 2 + 11n + 28 ________________ Factors of 28 Sum 1 and 28 29 72 and 14 16 74 and 7 11 3(n + 4)(n + 7)

n n 2 + 11n + 28

0 0 2 + 11(0) + 28 = 28

1 1 2 + 11(1) + 28 = 40

2 2 2 + 11(2) + 28 = 54

3 3 2 + 11(3) + 28 = 70

4 4 2 + 11(4) + 28 = 88

n (n + 4)(n + 7)

0 (0 + 4)(0 + 7) = 28

1 (1 + 4)(1 + 7) = 40

2 (2 + 4)(2 + 7) = 54

3 (3 + 4)(3 + 7) = 70

4 (4 + 4)(4 + 7) = 88

51. Approximately 1.5 yards

52. x 2 + 8x + 12 = x 2 + 2x + 6x + 2 · 6 = (x + 2)(x + 6) The width of the rectangle is (x + 2) ft.

53a. x 2 + 3x + 2 = x 2 + x + 2x + 1 · 2 = (x + 1)(x + 2) Length: (x + 2) ft Width: (x + 1) ft

b. x 2 + 8x + 15 = x 2 + 3x + 5x + 3 · 5 = (x + 3)(x + 5) Length: (x + 3) ft Width: (x + 5) ft

c. The length will increase by 1 ft. The width will increase by 4 ft.

54. x 2 + 5x + 6 x 2 + 2x + 3x + 2 · 3(x + 2)(x + 3)

55. x 2 + 6x + 8 x 2 + 2x + 4x + 2 · 4(x + 2)(x + 4)

56. x 2 + 2x - 8(x + 4)(x - 2)

57. Sign of c: positiveBinomial Factors: (x - 1)(x - 3)Signs of Numbers in Binomials: both negative

58. Sign of c: negativeBinomial Factors: (x - 1)(x + 3)Signs of Numbers in Binomials: negative; positive

59. Sign of c: negativeBinomial Factors: (x + 1)(x - 3)Signs of Numbers in Binomials: positive; negative

60. x 2 + 6x + 8 = x 2 + 2x + 4x + 2 · 4 = (x + 2)(x + 4)

The width of the rectangle is (x + 2).2 ≠ 4 so (x + 2) ≠ (x + 4)The rectangle cannot be a square.

61a. v = 0, a = 2

d = vt + 1 __ 2 a t 2

= (0)t + 1 __ 2 (2) t 2

= t 2

b. v = 4, a = 0

d = vt + 1 __ 2

a t 2

= (4)t + 1 __ 2

(0) t 2

= 4t c. t 2 - 4t t(t) - 4(t) t(t - 4)

62. x 2 + 9x + 14 = x 2 + 2x + 7x + 2 · 7 = (x + 2)(x + 7)

The width of the platform is (x + 2) ft.

63. True

64. False; the correct factorization is (x - 1)(x + 2).

65. False; the correct factorization is (x - 4)(x + 1).

66. False; many trinomials cannot be factored, e.g. x 2 - 12x - 32.

67. x 2 - 6x + 8 x 2 - 2x - 4x + 2 · 4(x - 2)(x - 4)

68. x 2 - 2x - 8 x 2 + 2x - 4x - 2 · 4(x + 2)(x - 4)

69. x 2 + 2x - 8 x 2 - 2x + 4x - 2 · 4(x - 2)(x + 4)

70. x 2 + 6x + 8 x 2 + 2x + 4x + 2 · 4(x + 2)(x + 4)

71a. x 2 + 12x + 20 = x 2 + 2x + 10x + 2 · 10 = (x + 2)(x + 10) The length of the fountain is (x + 10) ft.

b. Length: (x + 10) + 2 · 2 = (x + 14) ft Width: (x + 2) + 2 · 2 = (x + 6) ft

c. (x + 14)(x + 6) = x 2 + 14x + 6x + 84 = x 2 + 20x + 84 The total area covered is ( x 2 + 20x + 84) ft 2 .


CS10_A1_MESK710372_C07.indd 233 3/30/11 11:23:37 PM

72. x 2 + bx + 6 _______________ Factors of 6 Sum

1 and 6 7 2 and 3 5-2 and -3 -5-1 and -6 -7Possible values of b are 7, 5, -5, and -7.

teSt PreP

73. D; (x + 2)(x - 12) = x 2 + 2x - 12x - 24 = x 2 - 10x - 24

74. H; x 2 + bx - 20 __________________ Factors of -20 Sum -1 and 20 19-2 and 10 8-4 and 5 1Possible values of b are 19, 8, and 1.

75. C; x 2 + bx - 36 __________________ Factors of -36 Sum -1 and 36 35-2 and 18 16-3 and 12 9-4 and 9 5-6 and 6 0Possible values of b are 35, 16, 9, 5, and 0.

76. x 2 + 2x - 24 b = 2 and c = -24; look for factors of -24 whose sum is 2. The factor with the greater absolute value is positive.

__________________ Factors of -24 Sum -1 and 24 237

-2 and 12 107

-3 and 8 5 7-4 and 6 2 3

The factors are -4 and 6.(x - 4)(x + 6)


77. x 4 + 18 x 2 + 81 ________________ Factors of 81 Sum 1 and 81 823 and 27 309 and 9 18( x 2 + 9)( x 2 + 9)

78. y 4 - 5 y 2 - 24 __________________ Factors of -24 Sum 1 and -24 -232 and -12 -103 and -8 -5( y 2 + 3)( y 2 - 8)

79. d 4 + 22 d 2 + 21 ________________ Factors of 21 Sum 1 and 21 22( d 2 + 1)( d 2 + 21)

80. (u + v) 2 + 2(u + v) - 3 _________________ Factors of -3 Sum -1 and 3 2(u + v - 1)(u + v + 3)

81. (de) 2 - (de) - 20 __________________ Factors of -20 Sum 1 and -20 -192 and -10 -84 and -5 -1(de + 4)(de - 5)

82. (m - n) 2 - 4(m - n) - 45 __________________ Factors of -45 Sum 1 and -45 -443 and -15 -125 and -9 -4(m - n + 5)(m - n - 9)

83. x 2 + bx + 28 ________________ Factors of 28 Sum 1 and 28 292 and 14 164 and 7 11Possible values of b are 29, 16 and 11.

84. x 2 + bx + 32 ________________ Factors of 32 Sum -1 and -32 -33-2 and -16 -18-4 and -8 -12Possible values of b are -33, -18 and -12.

85a. x 2 + 13x + 42 = x 2 + 6x + 7x + 6 · 7 = (x + 6)(x + 7) The length of the garden is (x + 7) ft.

b. 2 · [(x + 6) + (x + 7)] = 2 · (2x + 13) = 4x + 26 The perimeter is (4x + 26) ft.

c. 2 · [4(5) + 26] = 2 · 46 = 92.00 The cost to fence the garden is $92.00

d. 0.28 · [ (5) 2 + 13(5) + 42] = 0.28 · 132 = 36.96 The cost of fertilizer is $36.96.

e. 92+ 36.96 = 128.96 The total cost to fence and fertilize is $128.96.

FActorinG ax 2 + bx + c

CheCk it OUt!

1a. 6 x 2 + 11x + 3(1x + 3)(6x + 1) = 6 x 2 + 19x + 37

(1x + 1)(6x + 3) = 6 x 2 + 9x + 3 7(2x + 3)(3x + 1) = 6 x 2 + 11x + 3 3The factors of 6 x 2 + 11x + 3 are (2x + 3) and (3x + 1).6 x 2 + 11x + 3 = (2x + 3)(3x + 1)

b. 3 x 2 - 2x - 8(1x - 8)(3x + 1) = 3 x 2 - 23x - 87

(1x - 4)(3x + 2) = 3 x 2 -10x - 87

(1x + 4)(3x - 2) = 3 x 2 + 10x - 87

(1x + 8)(3x - 1) = 3 x 2 + 23x -87

(3x - 8)(1x + 1) = 3 x 2 - 5x - 87

(3x - 4)(1x + 2) = 3 x 2 + 2x - 87

(3x + 4)(1x - 2) = 3 x 2 - 2x - 83

The factors of 3 x 2 - 2x - 8 are (3x + 4) and (x - 2).3 x 2 - 2x - 8 = (3x + 4)(x - 2)

2a. 6 x 2 + 17x + 5 ___________________________________ ___________________________________ Factors of 6 Factors of 5 Outer + Inner 1 and 6 1 and 5 1(5) + 6(1) = 117

2 and 3 5 and 1 2(1) + 3(5) = 173

(2x + 5)(3x + 1)

b. 9 x 2 - 15x + 4 ___________________________________ Factors of 9 Factors of 4 Outer + Inner 1 and 9 -1 and -4 1(-4) + 9(-1) = -137

1 and 9 -4 and -1 1(-1) + 9(-4) = -377

1 and 9 -2 and -2 1(-2) + 9(-2) = -207

3 and 3 -1 and -4 3(-4) + 3(-1) = -153

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


7-4

CS10_A1_MESK710372_C07.indd 234 3/30/11 11:23:38 PM

c. 3 x 2 + 13x + 12 ____________________________________ Factors of 3 Factors of 12 Outer + Inner 1 and 3 1 and 12 1(12) + 3(1) = 15 71 and 3 12 and 1 1(1) + 3(12) = 37 71 and 3 2 and 6 1(6) + 3(2) = 12 71 and 3 6 and 2 1(2) + 3(6) = 20 71 and 3 3 and 4 1(4) + 3(3) = 13 3(x + 3)(3x + 4)

3a. 6 x 2 + 7x - 3 ____________________________________ Factors of 6 Factors of -3 Outer + Inner 1 and 6 1 and -3 1(-3) + 6(1) = 3 7

1 and 6 -1 and 3 1(3) + 6(-1) = -3 7

1 and 6 3 and -1 1(-1) + 6(3) = 17 7

1 and 6 -3 and 1 1(1) + 6(-3) = -177

2 and 3 1 and -3 2(-3) + 3(1) = -3 7

2 and 3 -1 and 3 2(3) + 3(-1) = 3 7

2 and 3 3 and -1 2(-1) + 3(3) = 7 3

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

b. 4 n 2 - n - 3 ____________________________________ Factors of 4 Factors of -3 Outer + Inner 1 and 4 1 and -3 1(-3) + 4(1) = 1 7

1 and 4 -1 and 3 1(3) + 4(-1) = -13

(n - 1)(4n + 3)

4a. -6 x 2 - 17x - 12-1(6 x 2 + 17x + 12) ____________________________________ Factors of 6 Factors of 12 Outer + Inner 1 and 6 1 and 12 1(12) + 6(1) = 18 71 and 6 12 and 1 1(1) + 6(12) = 737

1 and 6 2 and 6 1(6) + 6(2) = 187

1 and 6 6 and 2 1(2) + 6(6) = 387

1 and 6 3 and 4 1(4) + 6(3) = 227

1 and 6 4 and 3 1(3) + 6(4) = 277

2 and 3 1 and 12 2(12) + 3(1) = 277

2 and 3 12 and 1 2(1) + 3(12) = 387

2 and 3 2 and 6 2(6) + 3(2) = 187

2 and 3 6 and 2 2(2) + 3(6) = 227

2 and 3 3 and 4 2(4) + 3(3) = 173

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

b. -3 x 2 - 17x - 10-1(3 x 2 + 17x + 10) ____________________________________ Factors of 3 Factors of 10 Outer + Inner 1 and 3 1 and 10 1(10) + 3(1) = 127

1 and 3 10 and 1 1(1) + 3(10) = 317

1 and 3 2 and 5 1(5) + 3(2) = 117

1 and 3 5 and 2 1(2) + 3(5) = 173

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

think and disCUss

1. The signs of the numbers are all positive.

2.

b > 0 b < 0

b < 0 b > 0

Factoring ax2 + bx + c

3x2 + 10x + 8 = (3x + 4)(x + 2) 3x2 - 10x + 8 = (3x - 4)(x - 2)

3x2 - 10x - 8 = (3x + 2)(x - 4)

c > 0

c < 0

3x2 + 10x - 8 = (3x - 2)(x + 4)


1. 2 x 2 + 9x + 10(1x + 10)(2x + 1) = 2 x 2 + 21x + 107

(1x + 5)(2x + 2) = 2 x 2 + 12x + 107

(1x + 2)(2x + 5) = 2 x 2 + 9x + 10 3

The factors of 2 x 2 + 9x + 10 are (x + 2) and (2x + 5).2 x 2 + 9x + 10 = (x + 2)(2x + 5)

2. 5 x 2 + 31x + 6(1x + 6)(5x + 1) = 5 x 2 + 31x + 63


3. 5 x 2 + 7x - 6(1x - 6)(5x + 1) = 5 x 2 - 29x - 67

(1x - 3)(5x + 2) = 5 x 2 - 13x - 67

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

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

The factors of 5 x 2 + 7x - 6 are (x + 2) and (5x - 3).5 x 2 + 7x - 6 = (x + 2)(5x - 3)

4. 6 x 2 + 37x + 6(1x + 6)(6x + 1) = 6 x 2 + 37x + 63


5. 3 x 2 - 14x - 24(1x - 24)(3x + 1) = 3 x 2 - 71x - 247

(1x - 12)(3x + 2) = 3 x 2 - 34x - 247

(1x - 8)(3x + 3) = 3 x 2 - 21x - 247

(1x - 6)(3x + 4) = 3 x 2 - 14x - 243

The factors of 3 x 2 - 14x - 24 are (x - 6) and (3x + 4).3 x 2 - 14x - 24 = (x - 6)(3x + 4)

6. 6x 2 + x - 2(1x - 2)(6x + 1) = 6 x 2 - 11x - 27

(1x - 1)(6x + 2) = 6 x 2 - 4x - 2 7

(1x + 1)(6x - 2) = 6 x 2 + 4x -2 7

(1x + 2)(6x - 1) = 6 x 2 + 11x - 27

(2x - 2)(3x + 1) = 6 x 2 - 4x -2 7

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

The factors of 6 x 2 + x - 2 are (2x - 1) and (3x + 2).6 x 2 + x - 2 = (2x - 1)(3x + 2).

7. 5 x 2 + 11x + 2 ___________________________________ Factors of 5 Factors of 2 Outer + Inner 1 and 5 1 and 2 1(2) + 5(1) = 7 7

1 and 5 2 and 1 1(1) + 5(2) = 113

(x + 2)(5x + 1)


1 and 2 5 and 1 1(1) + 2(5) = 11 3(x + 5)(2x + 1)


CS10_A1_MESK710372_C07.indd 235 3/30/11 11:23:40 PM

9. 4 x 2 - 9x + 5 ___________________________________ Factors of 4 Factors of 5 Outer + Inner 1 and 4 -1 and -5 1(-5) + 4(-1) = -9 3(x - 1)(4x - 5)

10. 2 y 2 - 11y + 14 ___________________________________ Factors of 2 Factors of 14 Outer + Inner 1 and 2 -1 and -14 1(-14) + 2(-1) = -167

1 and 2 -14 and -1 1(-1) + 2(-14) = -297

1 and 2 -2 and -7 1(-7) + 2(-2) = -113

(2y - 7)(y - 2)

11. 5 x 2 + 9x + 4 ___________________________________ Factors of 5 Factors of 4 Outer + Inner 1 and 5 1 and 4 1(4) + 5(1) = 93

(x + 1)(5x + 4)

12. 3 x 2 + 7x + 2 ___________________________________ Factors of 3 Factors of 2 Outer + Inner 1 and 3 1 and 2 1(2) + 3(1) = 57

1 and 3 2 and 1 1(1) + 3(2) = 73

(x + 2)(3x + 1)

13. 4 a 2 + 8a - 5 ____________________________________ Factors of 4 Factors of -5 Outer + Inner 1 and 4 1 and -5 1(-5) + 4(1) = -1 7

1 and 4 -1 and 5 1(5) + 4(-1) = 1 7

1 and 4 5 and -1 1(-1) + 4(5) = 19 7

1 and 4 -5 and 1 1(1) + 4(-5) = -197

2 and 2 1 and -5 2(-5) + 2(1) = - 87

2 and 2 -1 and 5 2(5) + 2(-1) = 8 3

(2a - 1)(2a + 5)

14. 15 x 2 + 4x - 3 ____________________________________ Factors of 15 Factors of -3 Outer + Inner 1 and 15 1 and -3 1(-3) + 15(1) = 12 7

1 and 15 -1 and 3 1(3) + 15(-1) = -127

1 and 15 3 and -1 1(-1) + 15(3) = 447

1 and 15 -3 and 1 1(1) + 15(-3) = -447

3 and 5 1 and -3 3(-3) + 5(1) = -47

3 and 5 -1 and 3 3(3) + 5(-1) = 4 3

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

15. 2 x 2 + x - 6 ____________________________________ Factors of 2 Factors of -6 Outer + Inner 1 and 2 1 and -6 1(-6) + 2(1) = -4 71 and 2 -1 and 6 1(6) + 2(-1) = 4 7

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

(x + 2)(2x - 3)

16. 6 n 2 - 11n - 10 ____________________________________ Factors of 6 Factors of -10 Outer + Inner 1 and 6 1 and -10 1(-10) + 6(1) = -4 71 and 6 -1 and 10 1(10) + 6(-1) = 4 71 and 6 2 and -5 1(-5) + 6(2) = 7 7

1 and 6 -2 and 5 1(5) + 6(-2) = -7 71 and 6 5 and -2 1(-2) + 6(5) = 28 71 and 6 -5 and 2 1(2) + 6(-5) = -287

1 and 6 10 and -1 1(-1) + 6(10) = 59 71 and 6 -10 and 1 1(1) + 6(-10) = -597

2 and 3 2 and -5 2(-5) + 3(2) = -4 72 and 3 -2 and 5 2(5) + 3(-2) = 4 72 and 3 5 and -2 2(-2) + 3(5) = 11 72 and 3 -5 and 2 2(2) + 3(-5) = -113

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

17. 10 x 2 - 9x - 1 _____________________________________ Factors of 10 Factors of -1 Outer + Inner 1 and 10 1 and -1 1(-1) + 10(1) = 9 7

1 and 10 -1 and 1 1(1) + 10(-1) = -93

(x - 1)(10x + 1)

18. 7 x 2 - 3x - 10 _____________________________________ Factors of 7 Factors of -10 Outer + Inner 1 and 7 1 and -10 1(-10) + 7(1) = -33

(x + 1)(7x - 10)

19. -2 x 2 + 5x + 12-1(2 x 2 - 5x - 12) ____________________________________ Factors of 2 Factors of -12 Outer + Inner 1 and 2 1 and -12 1(-12) + 2(1) = -107

1 and 2 -1 and 12 1(12) + 2(-1) = 10 7

1 and 2 2 and -6 1(-6) + 2(2) = -2 7

1 and 2 -2 and 6 1(6) + 2(-2) = 2 7

1 and 2 3 and -4 1(-4) + 2(3) = 2 7

1 and 2 -3 and 4 1(4) + 2(-3) = -2 7

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

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

20. -4 n 2 - 16n + 9-1(4 n 2 + 16n - 9) ____________________________________ Factors of 4 Factors of -9 Outer + Inner 1 and 4 1 and -9 1(-9) + 4(1) = -5 71 and 4 -1 and 9 1(9) + 4(-1) = 5 71 and 4 3 and -3 1(-3) + 4(3) = 9 71 and 4 -3 and 3 1(3) + 4(-3) = -9 71 and 4 9 and -1 1(-1) + 4(9) = 35 71 and 4 -9 and 1 1(1) + 4(-9) = -35 72 and 2 1 and -9 2(-9) + 2(1) = -16 72 and 2 -1 and 9 2(9) + 2(-1) = 16 3-1(2n - 1)(2n + 9)

21. -5 x 2 + 7x + 6-1(5 x 2 - 7x - 6) ____________________________________ Factors of 5 Factors of -6 Outer + Inner 1 and 5 1 and -6 1(-6) + 5(1) = -1 71 and 5 -1 and 6 1(6) + 5(-1) = 1 7

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

1 and 5 -2 and 3 1(3) + 5(-2) = -7 3-1(x - 2)(5x + 3)

22. -6 x 2 + 13x - 2-1(6 x 2 - 13x + 2) ___________________________________ Factors of 6 Factors of 2 Outer + Inner 1 and 6 -1 and -2 1(-2) + 6(-1) = -8 7

1 and 6 -2 and -1 1(-1) + 6(-2) = -133

-1(x - 2)(6x - 1)

23. -4 x 2 - 8x + 5-1(4 x 2 + 8x - 5) ____________________________________ Factors of 4 Factors of -5 Outer + Inner 1 and 4 1 and -5 1(-5) + 4(1) = -1 7

1 and 4 -1 and 5 1(5) + 4(-1) = 1 7

1 and 4 5 and -1 1(-1) + 4(5) = 19 7

1 and 4 -5 and 1 1(1) + 4(-5) = -19 72 and 2 1 and -5 2(-5) + 2(1) = -87

2 and 2 -1 and 5 2(5) + 2(-1) = 8 3

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


CS10_A1_MESK710372_C07.indd 236 3/30/11 11:23:42 PM

24. -5 x 2 + x + 18-1(5 x 2 - x - 18) ____________________________________ Factors of 5 Factors of -18 Outer + Inner 1 and 5 1 and -18 1(-18) + 5(1) = -137

1 and 5 -1 and 18 1(18) + 5(-1) = 13 7

1 and 5 2 and -9 1(-9) + 5(2) = 1 7

1 and 5 -2 and 9 1(9) + 5(-2) = -1 3

-1(x - 2)(5x + 9)


25. 9 x 2 + 9x + 2(1x + 2)(9x + 1) = 9 x 2 + 19x + 27

(1x + 1)(9x + 2) = 9 x 2 + 11x + 27

(3x + 2)(3x + 1) = 9 x 2 + 9x + 2 3

The factors of 9 x 2 + 9x + 2 are (3x + 2) and (3x + 1).9 x 2 + 9x + 2 = (3x + 2)(3x + 1)

26. 2 x 2 + 7x + 5(1x + 5)(2x + 1) = 2 x 2 + 11x + 57

(1x + 1)(2x + 5) = 2 x 2 + 7x + 5 3


27. 3 n 2 + 8n + 4(1n + 4)(3n + 1) = 3 n 2 + 13n + 47

(1n + 2)(3n + 2) = 3 n 2 + 8n + 4 3

The factors of 3 n 2 + 8n + 4 are (n + 2) and (3n + 2).3 n 2 + 8n + 4 = (n + 2)(3n + 2)

28. 10 d 2 + 17d + 7(1d + 7)(10d + 1) = 10 d 2 + 71d + 77

(1d + 1)(10d + 7) = 10 d 2 + 17d + 73

The factors of 10 d 2 + 17d + 7 are (d + 1) and (10d + 7).10 d 2 + 17d + 7 = (d + 1)(10d + 7)

29. 4 c 2 - 17c + 15(1c - 15)(4c - 1) = 4 c 2 - 61c + 157

(1c - 5)(4c - 3) = 4 c 2 - 23c + 157

(1c - 3)(4c - 5) = 4 c 2 - 17c + 153

The factors of 4 c 2 - 17c + 15 are (c - 3) and (4c - 5).4 c 2 - 17c + 15 = (c - 3)(4c - 5)

30. 6 x 2 + 14x + 4(1x + 4)(6x + 1) = 6 x 2 + 25x + 47

(1x + 2)(6x + 2) = 6 x 2 + 14x + 43

The factors of 6 x 2 + 14x + 4 are (x + 2) and (6x + 2).6 x 2 + 14x + 4 = 2(3x + 1)(x + 2)

31. 8 x 2 + 22x + 5(1x + 5)(8x + 1) = 8 x 2 + 41x + 57

(1x + 1)(8x + 5) = 8 x 2 + 13x + 57

(2x + 5)(4x + 1) = 8 x 2 + 22x + 53

The factors of 8 x 2 + 22x + 5 are (2x + 5) and (4x + 1).8 x 2 + 22x + 5 = (2x + 5)(4x + 1)

32. 6 x 2 - 13x + 6 (1x - 6)(6x - 1) = 6 x 2 - 37x + 67

(1x - 3)(6x - 2) = 6 x 2 - 20x + 67

(1x - 2)(6x - 3) = 6 x 2 - 15x + 67

(1x - 1)(6x - 6) = 6 x 2 - 12x + 67

(2x - 6)(3x - 1) = 6 x 2 - 20x + 67

(2x - 3)(3x - 2) = 6 x 2 - 13x + 63

The factors of 6 x 2 - 13x + 6 are (2x - 3) and (3x - 2).6 x 2 - 13x + 6 = (2x - 3)(3x - 2)

33. 5 x 2 + 9x - 18(1x + 18)(5x - 1) = 5 x 2 + 89x - 187

(1x + 9)(5x - 2) = 5 x 2 + 43x - 187

(1x + 6)(5x - 3) = 5 x 2 + 27x - 187

(1x + 3)(5x - 6) = 5 x 2 + 9x - 18 3

The factors of 5 x 2 + 9x - 18 are (x + 3) and (5x - 6).5 x 2 + 9x - 18 = (x + 3)(5x - 6)

34. 6 x 2 + 23x + 7 ___________________________________ Factors of 6 Factors of 7 Outer + Inner 1 and 6 1 and 7 1(7) + 6(1) = 13 71 and 6 7 and 1 1(1) + 6(7) = 43 72 and 3 1 and 7 2(7) + 3(1) = 17 72 and 3 7 and 1 2(1) + 3(7) = 23 3(2x + 7)(3x + 1)

35. 10 n 2 - 17n + 7 ___________________________________ Factors of 10 Factors of 7 Outer + Inner 1 and 10 -1 and -7 1(-7) + 10(-1) = -173

(n - 1)(10n - 7)

36. 3 x 2 + 11x + 6 ___________________________________ Factors of 3 Factors of 6 Outer + Inner 1 and 3 1 and 6 1(6) + 3(1) = 9 71 and 3 6 and 1 1(1) + 3(6) = 19 71 and 3 2 and 3 1(3) + 3(2) = 9 71 and 3 3 and 2 1(2) + 3(3) = 11 3(x + 3)(3x + 2)

37. 7 x 2 + 15x + 2 ___________________________________ Factors of 7 Factors of 2 Outer + Inner 1 and 7 1 and 2 1(2) + 7(1) = 9 71 and 7 2 and 1 1(1) + 7(2) = 15 3

(x + 2)(7x + 1)

38. 3 n 2 + 4n + 1 ___________________________________ Factors of 3 Factors of 1 Outer + Inner 1 and 3 1 and 1 1(1) + 3(1) = 4 3(n + 1)(3n + 1)

39. 3 x 2 - 19x + 20 ___________________________________ Factors of 3 Factors of 20 Outer + Inner 1 and 3 -1 and -20 1(-20) + 3(-1) = -237

1 and 3 -20 and -1 1(-1) + 3(-20) = -617

1 and 3 -2 and -10 1(-10) + 3(-2) = -167

1 and 3 -10 and -2 1(-2) + 3(-10) = -327

1 and 3 -4 and -5 1(-5) + 3(-4) = -177

1 and 3 -5 and -4 1(-4) + 3(-5) = -193

(x - 5)(3x - 4)


CS10_A1_MESK710372_C07.indd 237 3/30/11 11:23:44 PM

40. 6 x 2 + 11x + 4 ___________________________________ Factors of 6 Factors of 4 Outer + Inner 1 and 6 1 and 4 1(4) + 6(1) = 10 71 and 6 4 and 1 1(1) + 6(4) = 25 71 and 6 2 and 2 1(2) + 6(2) = 14 72 and 3 1 and 4 2(4) + 3(1) = 11 3(2x + 1)(3x + 4)


1 and 4 -21 and -1 1(-1) + 4(-21) = -857

1 and 4 -3 and -7 1(-7) + 4(-3) = -197

1 and 4 -7 and -3 1(-3) + 4(-7) = -313

(x - 7)(4x - 3)

42. 10 x 2 + 31x + 15 _____________________________________ Factors of 10 Factors of 15 Outer + Inner 1 and 10 1 and 15 1(15) + 10(1) = 25 7

1 and 10 15 and 1 1(1) + 10(15) = 1517

1 and 10 3 and 5 1(5) + 10(3) = 35 7

1 and 10 5 and 3 1(3) + 10(5) = 53 7

2 and 5 1 and 15 2(15) + 5(1) = 35 7

2 and 5 15 and 1 2(1) + 5(15) = 77 7

2 and 5 3 and 5 2(5) + 5(3) = 25 7

2 and 5 5 and 3 2(3) + 5(5) = 31 3(2x + 5)(5x + 3)

43. 12 y 2 + 17y - 5 ____________________________________ Factors of 12 Factors of -5 Outer + Inner 1 and 12 1 and -5 1(-5) + 12(1) = 7 7

1 and 12 -1 and 5 1(5) + 12(-1) = -7 7

1 and 12 5 and -1 1(-1) + 12(5) = 59 7

1 and 12 -5 and 1 1(1) + 12(-5) = -597

2 and 6 1 and -5 2(-5) + 6(1) = -4 7

2 and 6 -1 and 5 2(5) + 6(-1) = 4 7

2 and 6 5 and -1 2(-1) + 6(5) = 28 7

2 and 6 -5 and 1 2(1) + 6(-5) = -287

3 and 4 1 and -5 3(-5) + 4(1) = -117

3 and 4 -1 and 5 3(5) + 4(-1) = 11 7

3 and 4 5 and -1 3(-1) + 4(5) = 17 3

(3y + 5)(4y - 1)

44. 3 x 2 + 10x - 8 ____________________________________ Factors of 3 Factors of -8 Outer + Inner 1 and 3 1 and -8 1(-8) + 3(1) = -5 71 and 3 -1 and 8 1(8) + 3(-1) = 5 7

1 and 3 2 and -4 1(-4) + 3(2) = 2 7

1 and 3 -2 and 4 1(4) + 3(-2) = -2 71 and 3 4 and -2 1(-2) + 3(4) = 10 3

(x + 4)(3x - 2)

45. 4 x 2 + 4x - 3 ____________________________________ Factors of 4 Factors of -3 Outer + Inner 1 and 4 1 and -3 1(-3) + 4(1) = 1 7

1 and 4 -1 and 3 1(3) + 4(-1) = -1 7

1 and 4 3 and -1 1(-1) + 4(3) = 11 7

1 and 4 -3 and 1 1(1) + 4(-3) = -11 72 and 2 1 and -3 2(-3) + 2(1) = -4 7

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

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

46. 2 n 2 - 7n - 4 ____________________________________ Factors of 2 Factors of -4 Outer + Inner 1 and 2 1 and -4 1(-4) + 2(1) = -2 71 and 2 -1 and 4 1(4) + 2(-1) = 2 71 and 2 2 and -2 1(-2) + 2(2) = 2 71 and 2 -2 and 2 1(2) + 2(-2) = -2 71 and 2 4 and -1 1(-1) + 2(4) = 7 71 and 2 -4 and 1 1(1) + 2(-4) = -7 3(n - 4)(2n + 1)

47. 3 x 2 - 4x - 15 ____________________________________ Factors of 3 Factors of -15 Outer + Inner 1 and 3 1 and -15 1(-15) + 3(1) = -127

1 and 3 -1 and 15 1(15) + 3(-1) = 12 7

1 and 3 3 and -5 1(-5) + 3(3) = 4 7

1 and 3 -3 and 5 1(5) + 3(-3) = -4 3

(x - 3)(3x + 5)

48. 3 n 2 - n - 4 ____________________________________ Factors of 3 Factors of -4 Outer + Inner 1 and 3 1 and -4 1(-4) + 3(1) = -1 3(n + 1)(3n - 4)

49. -4 x 2 - 4x + 15-1(4 x 2 + 4x - 15) ____________________________________ Factors of 4 Factors of -15 Outer + Inner 1 and 4 1 and -15 1(-15) + 4(1) = -117

1 and 4 -1 and 15 1(15) + 4(-1) = 11 7

1 and 4 3 and -5 1(-5) + 4(3) = 7 7

1 and 4 -3 and 5 1(5) + 4(-3) = -7 7

1 and 4 5 and -3 1(-3) + 4(5) = 17 7

1 and 4 -5 and 3 1(3) + 4(-5) = -177

1 and 4 15 and -1 1(-1) + 4(15) = 59 7

1 and 4 -15 and 1 1(1) + 4(-15) = -597

2 and 2 1 and -15 2(-15) + 2(1) = -287

2 and 2 -1 and 15 2(15) + 2(-1) = 28 7

2 and 2 3 and -5 2(-5) + 2(3) = -4 7

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

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

50. -3 x 2 + 16x - 16-1(3 x 2 - 16x + 16) ___________________________________ Factors of 3 Factors of 16 Outer + Inner 1 and 3 -1 and -16 1(-16) + 3(-1) = -197

1 and 3 -16 and -1 1(-1) + 3(-16) = -497

1 and 3 -2 and -8 1(-8) + 3(-2) = -147

1 and 3 -8 and -2 1(-2) + 2(-8) = -187

1 and 3 -4 and -4 1(-4) + 3(-4) = -163

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

51. -3 x 2 - x + 2-1(3 x 2 + x - 2) ____________________________________ Factors of 3 Factors of -2 Outer + Inner 1 and 3 1 and -2 1(-2) + 3(1) = 1 3-1(x + 1)(3x - 2)

52. 12 x 2 + 24x + 3x + 6(12 x 2 + 24x) + (3x + 6)12x(x + 2) + 3(x + 2)12 x 2 + 27 + 6; 3(4x + 1)(x + 2)

53. 2 x 2 - 4x - x + 2(2 x 2 - 4x) - (x - 2)2x(x - 2) - (x - 2)(2x - 1)(x - 2)

54. 5 x 2 + 35x - 4x - 28(5 x 2 + 35x) - (4x + 28)5x(x + 7) - 4(x + 7)(5x - 4)(x + 7)


CS10_A1_MESK710372_C07.indd 238 3/30/11 11:23:46 PM

55. 9 n 2 + 17n + 8(1n + 8)(9n + 1) = 9 n 2 + 73n + 87

(1n + 4)(9n + 2) = 9 n 2 + 38n + 87

(1n + 2)(9n + 4) = 9 n 2 + 22n + 87

(1n + 1)(9n + 8) = 9 n 2 + 17n + 83

The factors of 9 n 2 + 17n + 8 are (n + 1) and(9n + 8).9 n 2 + 17n + 8 = (n + 1)(9n + 8)

56. 2 x 2 - 7x - 4(1x - 4)(2x + 1) = 2 x 2 - 7x - 43

The factors of 2 x 2 - 7x - 4 are (x - 4) and(2x + 1).2 x 2 - 7x - 4 = (x - 4)(2x + 1)

57. 4 x 2 - 12x + 5(1x - 5)(4x - 1) = 4 x 2 - 21x + 57

(1x - 1)(4x - 5) = 4 x 2 - 9x + 5 7

(2x - 5)(2x - 1) = 4 x 2 - 12x + 53

The factors of 4 x 2 - 12x + 5 are (2x - 5) and (2x - 1).4 x 2 - 12x + 5 = (2x - 5)(2x - 1)

58. 5 x 2 - 4x + 12(1x - 12)(5x - 1) = 5 x 2 - 61x + 127

(1x - 6)(5x - 2) = 5 x 2 - 32x + 127

(1x - 4)(5x - 3) = 5 x 2 - 23x + 127

(1x - 3)(5x - 4) = 5 x 2 - 19x + 127

(1x - 2)(5x - 6) = 5 x 2 - 16x + 127

(1x - 1)(5x - 12) = 5 x 2 - 17x + 127

(5 x 2 - 4x + 12) cannot be factored.

59. 3 x 2 + 14x + 16(1x + 16)(3x + 1) = 3 x 2 + 49x + 167

(1x + 8)(3x + 2) = 3 x 2 + 26x + 167

(1x + 4)(3x + 4) = 3 x 2 + 16x + 167

(1x + 2)(3x + 8) = 3 x 2 + 14x + 163


60. -3 x 2 - 11x + 4-1(3 x 2 + 11x - 4)(1x - 4)(3x + 1) = 3 x 2 - 11x - 47

(1x + 4)(3x - 1) = 3 x 2 + 11x - 43

The factors of 3 x 2 + 11x - 4 are (x + 4) and (3x - 1).-3 x 2 - 11x + 4 = -1(x + 4)(3x - 1)

61. 6 x 2 - x - 12(1x - 12)(6x + 1) = 6 x 2 - 71x - 127

(1x + 12)(6x - 1) = 6 x 2 + 71x - 127

(1x - 6)(6x + 2) = 6 x 2 - 34x - 127

(1x + 6)(6x - 2) = 6 x 2 + 34x - 127

(1x - 4)(6x + 3) = 6 x 2 - 21x - 127

(1x + 4)(6x - 3) = 6 x 2 + 21x - 127

(1x - 3)(6x + 4) = 6 x 2 - 14x - 127

(1x + 3)(6x - 4) = 6 x 2 + 14x - 127

(1x - 2)(6x + 6) = 6 x 2 - 6x - 12 7

(1x + 2)(6x - 6) = 6 x 2 + 6x - 12 7

(1x - 1)(6x + 12) = 6 x 2 + 6x - 12 7

(1x + 1)(6x - 12) = 6 x 2 - 6x - 12 7

(2x - 12)(3x + 1) = 6 x 2 - 34x - 127

(2x + 12)(3x - 1) = 6 x 2 + 34x - 127

(2x - 6)(3x + 2) = 6 x 2 - 14x - 127

(2x + 6)(3x - 2) = 6 x 2 + 14x - 127

(2x - 4)(3x + 3) = 6 x 2 - 6x - 12 7

(2x + 4)(3x - 3) = 6 x 2 + 6x - 12 7

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

The factors of 6 x 2 - x - 12 are (2x - 3) and (3x + 4).6 x 2 - x - 12 = (2x - 3)(3x + 4)

62. 10 a 2 + 11a + 3(1a + 3)(10a + 1) = 10 a 2 + 31a + 37

(1a + 1)(10a + 3) = 10 a 2 + 13a + 37

(2a + 3)(5a + 1) = 10 a 2 + 17a + 37

(2a + 1)(5a + 3) = 10 a 2 + 11a + 33

The factors of 10 a 2 + 11a + 3 are (2a + 1) and(5a + 3).10 a 2 + 11a + 3 = (2a + 1)(5a + 3)

63. 4 x 2 - 12x + 9(1x - 9)(4x - 1) = 4 x 2 - 37x + 97

(1x - 3)(4x - 3) = 4 x 2 - 15x + 97

(1x - 1)(4x - 9) = 4 x 2 - 13x + 97

(2x - 9)(2x - 2) = 4 x 2 - 22x + 97

(2x - 3)(2x - 3) = 4 x 2 - 12x + 93

The factors of 4 x 2 - 12x + 9 are (2x - 3) and(2x - 3).4 x 2 - 12x + 9 = (2x - 3)(2x - 3)

64. 6 x 2 + 11x + 5 (1x + 6)(6x + 1) = 6 x 2 + 37x + 5(1x + 1)(6x + 5) = 6 x 2 + 11x + 5The factors of 6 x 2 + 11x + 5 are (x + 1) and(6x + 5).The length of the rectangle is (6x + 5) cm.

65. 6 x 2 + 13x + 6a = 6 and c = 6; Outer + Inner = 13 ___________________________________ Factors of 6 Factors of 6 Outer + Inner 1 and 6 1 and 6 1(6) + 6(1) = 12 71 and 6 6 and 1 1(1) + 6(6) = 37 71 and 6 2 and 3 1(3) + 6(2) = 15 71 and 6 3 and 2 1(2) + 6(3) = 20 72 and 3 1 and 6 2(6) + 3(1) = 15 72 and 3 6 and 1 2(1) + 3(6) = 20 72 and 3 2 and 3 2(3) + 3(2) = 12 72 and 3 3 and 2 2(2) + 3(3) = 13 3(2x + 3)(3x + 2)


CS10_A1_MESK710372_C07.indd 239 3/30/11 11:23:48 PM

66. 8 x 2 + 18x - 58 x 2 + 20x - 2x - 5(8 x 2 + 20x) + (-2x - 5)4x(2x + 5) - 1(2x + 5)(4x - 1)(2x + 5)

67. 4 x 2 + 9x + 24 x 2 + 8x + x + 2(4 x 2 + 8x) + (x + 2)4x(x + 2) + 1(x + 2)(4x + 1)(x + 2)

68. 2 w 2 + 7w + 6(1w + 6)(2w + 1) = 2 w 2 + 13w + 6(1w + 3)(2w + 2) = 2 w 2 + 8w + 6(1w + 2)(2w + 3) = 2 w 2 + 7w + 6The length of Rebecca’s old garden is (2w) yd, and the width is (w)yd.The length of Rebecca’s new garden is (2w + 3) yd, and the width is (w + 2) yd.Length increased by 3 yd, and width increased by 2 yd.

69a. v = 20, h = 6 -16 t 2 + vt + h = -16 t 2 + 20t + 6

b. -16 t 2 + 20t + 6 -2(8 t 2 - 10t - 3) (1t + 1)(8t - 3) = 8 t 2 + 5t - 37

(1t - 1)(8t + 3) = 8 t 2 - 5t - 37

(1t + 3)(8t - 1) = 8 t 2 + 23t - 37

(1t - 3)(8t + 1) = 8 t 2 - 23t - 37

(2t + 1)(4t - 3) = 8 t 2 - 2t - 37

(2t - 1)(4t + 3) = 8 t 2 + 2t - 37

(2t + 3)(4t - 1) = 8 t 2 + 10t - 37

(2t - 3)(4t + 1) = 8 t 2 - 10t - 33

-2(4t + 1)(2t - 3)

c. -1(2t - 3)(8t + 2) = -1(2(1) - 3)(8(1) + 2) = -1(-1)(10) = 10 The height of the football after 1 second is 10 ft.

70. Possible answer: The student tried factors of 12 instead of factors of 2 · 12.

71a. 2 t 2 = 10t - 8 2 t 2 - 10t + 8 = 0

b. 2 t 2 - 10t + 8 2( t 2 - 5t + 4) (t - 4)(t - 1) = t 2 - 5t + 43

The factors of t 2 - 5t + 4 are (t - 4) and (t - 1). 2 t 2 - 10t + 8 = 2(t - 4)(t - 1)

c. The boats are the same distance from the start point when 2 t 2 - 10t + 8 = 0. From factorization, 2(t - 4)(t - 1) = 0, so (t - 4) = 0 or (t - 1) = 0. Therefore the boats are the same distance from the

start point when t = 1 and t = 4.

72. D; (x - 5)(6x + 1)6 x 2 + x - 30x - 56 x 2 - 29x - 5

73. B;(x - 5)(6x - 1)6 x 2 - x - 30x + 5 6 x 2 - 31x + 5

74. A;(x + 5)(6x - 1)6 x 2 - x + 30x - 56 x 2 + 29x - 5

75. C;(x + 5)(6x - 1)6 x 2 - x + 30x - 56 x 2 + 29x - 5

76a. Both signs are positive, or both signs are negative.

b. One sign is positive, and the other is negative.

teSt PreP

77. B;3 x 2 + bx - 8 ____________________________________ Factors of 3 Factors of -8 Outer + Inner 1 and 3 1 and -8 1(-8) + 3(1) = -5 1 and 3 -1 and 8 1(8) + 3(-1) = 5 1 and 3 2 and -4 1(-4) + 3(2) = 2 1 and 3 -2 and 4 1(4) + 3(-2) = -2 1 and 3 4 and -2 1(-2) + 3(4) = 10 1 and 3 -4 and 2 1(2) + 3(-4) = -10 1 and 3 8 and -1 1(-1) + 3(8) = 23 1 and 3 -8 and 1 1(1) + 3(-8) = -23 Possible values of b are -23, -10, -5, -2, 2, 5, 10, and 23.

78. H;5 x 2 + 15x + 4x + 12(5 x 2 + 15x) + (4x + 12)5x(x + 3) + 4(x + 3)(5x + 4)(x + 3)

79. A;24 x 2 - 49x + 2(1x - 2)(24x -1) = 24 x 2 - 49x + 2 3The factors of 24 x 2 - 49x + 2 are (x - 2) and(24x - 1).

80. G;c = -15; 2 x 2 + x - 15(1x + 1)(2x - 15) = 2 x 2 - 13x - 15 7(1x - 1)(2x + 15) = 2 x 2 + 13x - 15 7(1x + 3)(2x - 5) = 2 x 2 + x - 15 3

The factors of 2 x 2 + x - 15 are (x + 3) and (2x - 5).c = -9; 2 x 2 + x - 9(1x + 1)(2x - 9) = 2 x 2 - 7x - 9 7

(1x - 1)(2x + 9) = 2 x 2 + 7x - 9 7

(1x + 3)(2x - 3) = 2 x 2 + 3x - 9 7

(1x - 3)(2x + 3) = 2 x 2 - 3x - 9 7

(1x + 9)(2x - 1) = 2 x 2 + 17x - 9 7(1x - 9)(2x + 1) = 2 x 2 - 17x -9 7(2 x 2 + x - 9) cannot be factored.c = -6; 2 x 2 + x - 6(1x + 1)(2x - 6) = 2 x 2 - 4x - 6 7(1x - 1)(2x + 6) = 2 x 2 + 4x - 6 7(1x + 2)(2x - 3) = 2 x 2 + x -6 3

The factors of 2 x 2 + x - 6 are (x + 2) and (2x - 3).c = -1; 2 x 2 + x - 1(1x + 1)(2x - 1) = 2 x 2 + x -1 3The factors of 2 x 2 + x - 1 are (x + 1) and (2x - 1).


81. 1 + 4x + 4 x 2 ___________________________________ Factors of 4 Factors of 1 Outer + Inner 1 and 4 1 and 1 1(1) + 4(1) = 57

2 and 2 1 and 1 1(2) + 2(1) = 43

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

82. 1 - 14x + 49 x 2 ___________________________________ Factors of 49 Factors of 1 Outer + Inner 1 and 49 -1 and -1 1(-1) + 49(-1) = -507

7 and 7 -1 and -1 7(-1) + 7(-1) = -143

(7x - 1)(7x - 1)


CS10_A1_MESK710372_C07.indd 240 3/30/11 11:23:50 PM

83. 1 + 18x + 81 x 2 ____________________________________ Factors of 81 Factors of 1 Outer + Inner 1 and 81 1 and 1 1(1) + 81(1) = 82 79 and 9 1 and 1 9(1) + 9(1) = 18 3(9x + 1)(9x + 1)

84. 25 + 30x + 9 x 2 ____________________________________ Factors of 9 Factors of 25 Outer + Inner 1 and 9 1 and 25 1(25) + 9(1) = 34 7

1 and 9 25 and 1 1(1) + 9(25) = 226 71 and 9 5 and 5 1(5) + 9(5) = 50 7

3 and 3 1 and 25 3(25) + 3(1) = 78 7

3 and 3 5 and 5 3(5) + 3(5) = 30 3

(3x + 5)(3x + 5)

85. 4 + 20x + 25 x 2 ____________________________________ Factors of 25 Factors of 4 Outer + Inner 1 and 25 1 and 4 1(4) + 25(1) = 29 7

1 and 25 4 and 1 1(1) + 25(4) = 101 71 and 25 2 and 2 1(2) + 25(2) = 52 7

5 and 5 1 and 4 5(4) + 5(1) = 25 7

5 and 5 2 and 2 5(2) + 5(2) = 20 3

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

86. 4 - 12x + 9 x 2 ___________________________________ Factors of 9 Factors of 4 Outer + Inner 1 and 9 -1 and -4 1(-4) + 9(-1) = -13 71 and 9 -4 and -1 1(-1) + 9(-4) = -37 71 and 9 -2 and -2 1(-2) + 9(-2) = -20 73 and 3 -1 and -4 3(-4) + 3(-1) = -15 73 and 3 -2 and -2 3(-2) + 3(-2) = -12 3(3x - 2)(3x - 2)

87. 3 x 2 + bx + 2 ___________________________________ Factors of 3 Factors of 2 Outer + Inner 1 and 3 1 and 2 1(2) + 3(1) = 51 and 3 2 and 1 1(1) + 3(2) = 71 and 3 -1 and -2 1(-2) + 3(-1) = -51 and 3 -2 and -1 1(-1) + 3(-2) = -7Possible values of b are -7, -5, 5, and 7.

88. 3 x 2 + bx - 2 ____________________________________ Factors of 3 Factors of -2 Outer + Inner 1 and 3 1 and -2 1(-2) + 3(1) = 11 and 3 -1 and 2 1(2) + 3(-1) = -11 and 3 2 and -1 1(-1) + 3(2) = 51 and 3 -2 and 1 1(1) + 3(-2) = -5Possible values of b are -5, -1, 1, and 5.

89. 5 x 2 + bx + 1 ___________________________________ Factors of 5 Factors of 1 Outer + Inner 1 and 5 1 and 1 1(1) + 5(1) = 61 and 5 -1 and -1 1(-1) + 5(-1) = -6Possible values of b are -6 and 6.

reADy to Go on? section A Quiz

1. 54 = 2 · 3 · 3 · 3 = 2 · 3 3

2. 42 = 2 · 3 · 7

3. 50 = 2 · 5 · 5 = 2 · 5 2

4. 120 = 2 · 2 · 2 · 3 · 5 = 2 3 · 3 · 5

5. 44 = 2 · 2 · 11 = 2 2 · 11

6. 78 = 2 · 3 · 13

7. 6 p 3 = 2 · 3 · p · p · p2p = 2 · pThe GCF of 6p 3 and 2p is 2p.

8. 12 x 3 = 2 · 2 · 3 · x · x · x18 x 4 = 2 · 3 · 3 · x · x · x · xThe GCF of 12 x 3 and 18 x 4 is 6 x 3 .

9. -15 = -1 · 3 · 520 s 4 = 2 · 2 · 5 · s · s · s · sThe GCF of -15 and 20 s 4 is 5.

10. 3a = 3 · a4 b 2 = 2 · 2 · b · bThe GCF of 3a and 4 b 2 is 1.

11. The 24 American League games’ balls and 30 National League games’ balls must be divided into groups of equal size. The number of balls in each row must be a common factor of 24 and 30.factors of 24: 1, 2, 3, 4, 6, 8, 12, 24factors of 30: 1, 2, 3, 5, 6, 10, 15, 30The GCF of 24 and 30 is 6.The greatest possible number of balls in each row is 6. Find the number of rows.

24 balls from American League games

_______________________________ 6 balls per row

= 4 rows

30 balls from National League games

______________________________ 6 balls per row

= 5 rows

When the greatest possible number of balls is in each row, there are 9 rows in total.

12. 2 d 3 = 2 · d · d · d4d = 2 · 2 · dThe GCF of 2d 3 and 4d is 2d.2 d 3 + 4d d 2 (2d) + 2(2d)2d( d 2 + 2)

13. m 2 = m · m8 m 5 = 2 · 2 · m · m · m · m · mThe GCF of m 2 and 8 m 5 is m 2 . m 2 - 8 m 5 1( m 2 ) - 8 m 3 ( m 2 ) m 2 (1 - 8 m 3 )

14. 12 x 4 = 2 · 2 · 3 · x · x · x · x 8 x 3 = 2 · 2 · 2 · x · x · x 4 x 2 = 2 · 2 · x · xThe GCF of 12x 4 , 8 x 3 and 4 x 2 is 4 x 2 .12 x 4 - 8 x 3 - 4 x 2 4 x 2 (3 x 2 - 2x - 1) ____________________________________ Factors of 3 Factors of -1 Outer + Inner 1 and 3 1 and -1 1(-1) + 3(1) = 2 7

1 and 3 -1 and 1 1(1) + 3(-1) = -23

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

15. 3 k 2 = 3 · k · k6k = 2 · 3 · k3 = 3The GCF of 3k 2 , 6k, and 3 is 3.3 k 2 + 6k - 3 k 2 (3) + 2k(3) - 1(3)3( k 2 + 2k - 1)


CS10_A1_MESK710372_C07.indd 241 3/30/11 11:23:52 PM

16. sπr = s · π · rπ r 2 = π · r · rThe GCF of sπr and πr 2 is πr.sπr + π r 2 s(πr) + r(πr)πr(s + r)

17. w 3 - 4 w 2 + w - 4( w 3 - 4 w 2 ) + (w - 4) w 2 (w - 4) + 1(w - 4)( w 2 + 1)(w - 4)

18. 3 x 3 + 6 x 2 - 4x - 8(3 x 3 + 6 x 2 ) - (4x + 8)3 x 2 (x + 2) - 4(x + 2)(3 x 2 - 4)(x + 2)

19. 2 p 3 - 6 p 2 + 15 - 5p(2 p 3 - 6 p 2 ) - (5p - 15)2 p 2 (p - 3) - 5(p - 3)(2 p 2 - 5)(p - 3)

20. n 3 - 6 n 2 + 5n - 30( n 3 - 6 n 2 ) + (5n - 30) n 2 (n - 6) + 5(n - 6) = ( n 2 + 5)(n - 6)

21. n 2 + 9n + 20 ________________ Factors of 20 Sum 1 and 20 21 72 and 10 12 74 and 5 93

(n + 4)(n + 5)

22. d 2 - 6d - 7 _________________ Factors of -7 Sum 1 and -7 -63

(d + 1)(d - 7)

23. x 2 - 6x + 8 _______________ Factors of 8 Sum -1 and -8 -9 7-2 and -4 -63

(x - 2)(x - 4)

24. y 2 + 7y - 30 __________________ Factors of -30 Sum -1 and 30 297

-2 and 15 137

-3 and 10 7 3(y - 3)(y + 10)

25. k 2 - 6k + 5 _______________ Factors of 5 Sum -1 and -5 -63

(k - 1)(k - 5)

26. c 2 - 10c + 24 ________________ Factors of 24 Sum -1 and -24 -257

-2 and -12 -147

-3 and -8 -117

-4 and -6 -103

(c - 4)(c - 6)

27. n(n+ 3) - 4 n 2 + 3n - 4 _________________ Factors of -4 Sum -1 and 4 33

(n - 1)(n + 4)

n n 2 + 3n - 4

0 0 2 + 3(0) - 4 = -4

1 1 2 + 3(1) - 4 = 0

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

3 3 2 + 3(3) - 4 = 14

4 4 2 + 3(4) - 4 = 24

n (n - 1)(n + 4)

0 (0 + 4)(0 - 1) = -4

1 (1 + 4)(1 - 1) = 0

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

3 (3 + 4)(3 - 1) = 14

4 (4 + 4)(4 - 1) = 24


1 and 2 5 and 1 1(1) + 2(5) = 113

(x + 5)(2x + 1)

29. 3 n 2 + 16n + 21 ____________________________________ Factors of 3 Factors of 21 Outer + Inner 1 and 3 1 and 21 1(21) + 3(1) = 247

1 and 3 21 and 1 1(1) + 3(21) = 647

1 and 3 3 and 7 1(7) + 3(3) = 163

(n + 3)(3n + 7)

30. 5 y 2 - 7y - 6 ____________________________________ Factors of 5 Factors of -6 Outer + Inner 1 and 5 1 and -6 1(-6) + 5(1) = -17

1 and 5 -1 and 6 1(6) + 5(-1) = 1 7

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

1 and 5 -2 and 3 1(3) + 5(-2) = -73

(y - 2)(5y + 3)

31. 4 g 2 - 10g + 6 ___________________________________ Factors of 4 Factors of 6 Outer + Inner 1 and 4 -1 and -6 1(-6) + 4(-1) = -103

(g - 1)(4g - 6)2(g - 1)(2g - 3)

32. 6 p 2 - 18p - 24 ____________________________________ Factors of 6 Factors of -24 Outer + Inner 1 and 6 1 and -24 1(-24) + 6(1) = -183

(p + 1)(6p - 24)6(p + 1)(p - 4)


CS10_A1_MESK710372_C07.indd 242 3/30/11 11:23:54 PM

33. 12 d 2 + 7d - 12 _____________________________________ Factors of 12 Factors of -12 Outer + Inner 1 and 12 1 and -12 1(-12) + 12(1) = 01 and 12 -1 and 12 1(12) + 12(-1) = 01 and 12 2 and -6 1(-6) + 12(2) = 181 and 12 -2 and 6 1(6) + 12(-2) = -182 and 6 1 and -12 2(-12) + 6(1) = -182 and 6 -1 and 12 2(12) + 6(-1) = 182 and 6 2 and -6 2(-6) + 6(2) = 02 and 6 -2 and 6 2(6) + 6(-2) = 02 and 6 3 and -4 2(-4) + 6(3) = 102 and 6 -3 and 4 2(4) + 6(-3) = -103 and 4 1 and -12 3(-12) + 4(1) = -323 and 4 -1 and 12 3(12) + 4(-1) = 323 and 4 2 and -6 3(-6) + 4(2) = -103 and 4 -2 and 6 3(6) + 4(-2) = 103 and 4 3 and -4 3(-4) + 4(3) = 03 and 4 -3 and 4 3(4) + 4(-3) = 03 and 4 4 and -3 3(-3) + 4(4) = 7(3d + 4)(4d - 3)

34. (4x + 2) cm

FActorinG sPeciAL ProDucts

CheCk it OUt!

1a. Yes, the trinomial is a perfect square. x 2 + 4x + 4 x 2 + 2(x)(2) + 2 2 (x + 2) 2

b. Yes, the trinomial is a perfect square. x 2 - 14x + 49 x 2 - 2(x)(7) + 7 2 (x - 7) 2

c. No, the trinomial is not a perfect square. 9x 2 = (3x) 2 , 4 = 2 2 , but -6x ≠ 2(3x)(2)

2. 9 x 2 + 6x + 1 (3x) 2 + 2(3x)(1) + 1 2 (3x + 1) 2 The perimeter of each sheet is 4(3x + 1) m.x = 3, 4(3x + 1) = 4(3(3) + 1) = 40The perimeter is 40 m when x = 3 m.

3a. Yes, the binomial is a difference of two squares.1 - 4 x 2 1 2 - (2x) 2 (1 + 2x)(1 - 2x)1 - 4 x 2 = (1 + 2x)(1 - 2x)

b. Yes, the binomial is a difference of two squares.p 8 - 49 q 6 ( p 4 ) 2 - (7 q 3 ) 2 ( p 4 + 7 q 3 )( p 4 - 7 q 3 ) p 8 - 49 q 6 = ( p 4 + 7 q 3 )( p 4 - 7 q 3 )

c. No, the binomial is not a difference of two squares because 4 y 5 is not a perfect square.

think and disCUss

1. 1 - x 4 1 2 - ( x 2 ) 2

(1 + x 2 )(1 - x 2 )a = 1, b = x 2

2. x 2 + 8x + 16 x 2 + 2(x)(4) + 4 2 (x + 4) 2 a = x, b = 4

3.

Special Product Factored Form

Perfect-square trinomial with positive coefficient of middle term: x2 + 2x + 1

Perfect-square trinomial with negative coefficient of middle term: x2 - 2x + 1

Difference of two squares: x2 - 1

(x + 1)2

(x - 1)2

(x - 1)(x + 1)

exerCises

Guided Practice

1. Yes x 2 - 4x + 4 x 2 - 2(x)(2) + 2 2 (x - 2) 2

2. No, the trinomial is not a perfect square because the last term of the trinomial is not positive.

3. Yes9 x 2 - 12x + 4 (3x) 2 - 2(3x)(2) + 2 2 (3x - 2) 2

4. Yes x 2 + 2x + 1 x 2 + 2(x)(1) + 1 2 (x + 1 )

2

5. Yes x 2 - 6x + 9 x 2 - 2(x)(3) + 3 2 (x - 3) 2


7. x 2 + 24x + 144 x 2 + 2(x)(12) + 12 2 (x + 12) 2 The length and width are both (x + 12) yd.The perimeter of the park is 4(x + 12) yd.x = 10; 4(x + 12) = 4(10 + 12) = 88The perimeter of the park is 88 yd when x = 10 yd.

8. Yes1 - 4 x 2 1 2 - (2x) 2 (1 + 2x)(1 - 2x)

9. Yes s 2 - 4 2 (s + 4)(s - 4)

10. Yes81 x 2 - 1 (9x) 2 - 1 2 (9x + 1)(9x - 1)

11. Yes4 x 4 - 9 y 2

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

12. No, the binomial is not a difference of two squares because 50 is not a perfect square.

13. Yes x 6 - 9

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

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

3


7-5

CS10_A1_MESK710372_C07.indd 243 3/30/11 11:23:56 PM


14. Yes4 x 2 - 4x + 1 (2x) 2 - 2(2x)(1) + 1 2 (2x - 1) 2


16. Yes36 x 2 - 12x + 1 (6x) 2 - 2(6x)(1) + 1 2 (6x - 1) 2

17. No, the trinomial is not a perfect square.25 x 2 = (5x) 2 , 4 = 2 2 , but 10x ≠ 2(5x)(2)

18. Yes 9x 2 + 18x + 9 (3x) 2 + 2(3x) + 3 2 (3x + 3) 2 9(x + 1 ) 2

19. Yes 16x 2 - 40x + 25(4x ) 2 - 2(4x)(5) + 5 2 (4x - 5) 2

20. 4 x 2 - 44x + 121 (2x) 2 - 2(2x)(11) + 11 2 (2x - 11) 2

The length and width are both (2x - 11) mm.The perimeter of the rectangle is 4(2x - 11) mm.x = 41, 4(2x - 11) = 4(2(41) - 11) = 4(71) = 284The perimeter of the rectangle is 284 mm when x = 41mm.

21. Yes 1 2 - 4 x 2 1 2 - (2x) 2 (1 + 2x)(1 - 2x)

22. Yes25 m 2 - 16 n 2 (5m) 2 - (4n) 2 (5m + 4n)(5m - 4n)

23. No, the binomial is not a difference of two squares because 4x and 9y are not perfect squares.

24. Yes49 p 12 - 9 q 6 (7 p 6 ) 2 - (3 q 3 ) 2 (7 p 6 + 3 q 3 )(7 p 6 - 3 q 3 )

25. Yes9 2 - 100 x 4 9 2 - (10 x 2 ) 2 (9 + 10 x 2 )(9 - 10 x 2 )

26. No, the binomial is not a difference of two squares because x 3 and y 3 are not perfect squares.

27. 14x = 2(x)(7) b = 7 and b 2 = 7 2 = 49 x 2 + 14x + 49

28. 9 x 2 = (3x) 2 , 25 = 5 2 a = 3x, b = 5, and 2ab = 2(3x)(5) = 30x9 x 2 + 30x + 25

29. 36y = 2(2y)(9)a = 2y and a 2 = (2y) 2 = 4 y 2 4 y 2 - 36y + 81

30. Perfect-square trinomial x 2 - 8x + 16 x 2 - 2(x)(4) + 4 2 (x - 4) 2

31. Difference of 2 squares100 x 2 - 81 y 2 (10x) 2 - (9y) 2 (10x + 9y)(10x - 9y)

32. Perfect-square trinomial36 x 2 + 24x + 4 (6x) 2 + 2(6x)(2) + 2 2 (6x + 2) 2

33. Difference of 2 squares4 r 6 - 25 s 6 (2 r 3 ) 2 - (5 s 3 ) 2 (2 r 3 + 5 s 3 )(2 r 3 - 5 s 3 )

34. Perfect-square trinomial 49x 2 - 70x + 25 (7x) 2 - 2(7x)(5) + 5 2 (7x - 5) 2

35. Difference of 2 squares ( x 7 ) 2 - (12) 2 ( x 7 + 12)( x 7 - 12)

36. Possible answer: they are similar in that the first and last terms of each are perfect squares. They are different in that a perfect-square trinomial has 3 terms and a difference of 2 squares has 2 terms.

37. Possible answer: multiply a binomial by itself. Choose 2 perfect squares, find 2 times the product of their square roots, and then write these 3 expressions as a sum.

38. x 2 - 22x + 121 x 2 - 2(x)(11) + 11 2 (x - 11) 2 b = -11

39. 256 = 16 2 a = x, b = 16 2ab = 2(x)(16) = 32xc = 32

40a. x 2 - 25 x 2 - 5 2 (x + 5)(x - 5)

b. ℓ = (x + 5) ft w = (x - 5) ft 5 feet were added to the length and subtracted

from the width.

c. ℓ = (x + 5) = (8 + 5) = 13 ft w = (x - 5) = (8 - 5) = 3 ft

41a. 25 z 2 - 40z + 16 (5z) 2 - 2(5z)(4) + 4 2 (5z - 4) 2 The length of a side of the square is 5z - 4.

b. The perimeter of the square is 4(5z - 4) = 20z - 16.

c. z = 3 5z - 4 = 5(3) - 4 = 11 4(5z - 4) = 4(11) = 44 (5z - 4) 2 = (11) 2 = 121 When z = 3, the length of a side is 11, the

perimeter is 44, and the area is 121.

42a. The area of the larger rectangle is 3x(x) = 3 x 2 . The area of the smaller rectangle is 3y(y) = 3 y 2 .

b. The area of the green region is 3 x 2 - 3 y 2 .

c. 3 x 2 - 3 y 2 3( x 2 - y 2 ) 3(x + y)(x - y)

43a. x = -5 x 2 + 10x + 25 = (-5) 2 + 10(-5) + 25 = 0 (x + 5) 2 = (-5 + 5) 2 = 0 (x - 5) 2 = (-5 - 5) 2 = 100 x 2 - 10x + 25 = (-5) 2 - 10(-5) + 25 = 100 x 2 - 25 = (-5) 2 - 25 = 0

b. x = -1 x 2 + 10x + 25 = (-1) 2 + 10(-1) + 25 = 16 (x + 5) 2 = (-1 + 5) 2 = 16 (x - 5) 2 = (-1 - 5) 2 = 36 x 2 - 10x + 25 = (-1) 2 - 10(-1) + 25 = 36 x 2 - 25 = (-1) 2 - 25 = -24


CS10_A1_MESK710372_C07.indd 244 3/30/11 11:23:58 PM

c. x = 0 x 2 + 10x + 25 = (0) 2 + 10(0) + 25 = 25 (x + 5) 2 = (0 + 5) 2 = 25 (x - 5) 2 = (0 - 5) 2 = 25 x 2 - 10x + 25 = (0) 2 - 10(0) + 25 = 25 x 2 - 25 = (0) 2 - 25 = -25

d. x = 1 x 2 + 10x + 25 = (1) 2 + 10(1) + 25 = 36 (x + 5) 2 = (1 + 5) 2 = 36 (x - 5) 2 = (1 - 5) 2 = 16 x 2 - 10x + 25 = (1) 2 - 10(1) + 25 = 16 x 2 - 25 = (1) 2 - 25 = -24

e. x = 5 x 2 + 10x + 25 = (5) 2 + 10(5) + 25 = 100 (x + 5) 2 = (5 + 5) 2 = 100 (x - 5) 2 = (5 - 5) 2 = 0 x 2 - 10x + 25 = (5) 2 - 10(5) + 25 = 0 x 2 - 25 = (5) 2 - 25 = 0

44. Columns 1 and 2 have equivalent values because x 2 + 10x + 25 = (x + 5) 2 .Columns 3 and 4 have equivalent values because (x - 5) 2 = x 2 - 10x + 25.

45. The missing labels are (a + b) and (a - b).

46. Student A is incorrect because (5x)(5x) ≠ 25 x 4 , and (-3)(3) ≠ 9 y 2 .

teSt PreP

47. C; x 2 - 2xy + y 2 x 2 - 2(x)(y) + y 2 (x - y) 2 x = 0, y = 0; (x - y) 2 = (0 - 0) 2 = 0x = -1, y = -1; (x - y) 2 = (-1 + 1) 2 = 0x = 1, y = 1; (x - y) 2 = (1 - 1) 2 = 0x = 1, y = -1; (x - y) 2 = (1 + 1) 2 = 4

48. J;4 x 2 + 20x + 25 (2x) 2 + 2(2x)(5) + 5 2 (2x + 5) 2

49. x 2 - 18x + 81 x 2 - 2(x)(9) + 9 2 (x - 9) 2 x = 10, x 2 - 18x + 81 = (x - 9) 2 = (10 - 9) 2 = 1


50a. 81 x 4 - 16 (9 x 2 ) 2 - 4 2 (9 x 2 + 4)(9 x 2 - 4)

b. 9 x 2 - 4 (3x) 2 - 2 2 (3x + 2)(3x - 2)

c. Possible answer: x 4 - 1 ( x 2 ) 2 - 1 2 ( x 2 + 1)( x 2 - 1) ( x 2 + 1)( x 2 - 1 2 ) ( x 2 + 1)(x + 1)(x - 1)

51a. a = 2, b = (v + 2)

b. 4 - (v + 2) 2 (2 + (v + 2))(2 - (v - 2)) (v + 4)(-v) - v 2 - 4v 52. x 3 - 1

x 3 - 1 3 a = x, b = 1(x - 1)( x 2 + (x)(1) + 1 2 )(x - 1)( x 2 + x + 1)

53. 27 y 3 - 64 (3y) 3 - 4 3 a = 3y, b = 4(3y - 4)( (3y) 2 + (3y)(4) + 4 2 )(3y - 4)(9 y 2 + 12y + 16)

54. n 6 - 8 ( n 2 ) 3 - 2 3 a = n 2 , b = 2

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

choosinG A FActorinG methoD

CheCk it OUt!

1a. Yes, 5 x 2 (x - 1) is completely factored.

b. No, (4x + 4)(x + 1) is not completely factored.(4x + 4)(x + 1)4(x + 1)(x + 1) 4(x + 1) 2

2a. 4 x 3 + 16 x 2 + 16x4x( x 2 + 4x + 4)4x (x + 2) 2

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

3a. 3 x 2 + 7x + 4 __________________________________ Factors of 3 Factors of 4 Outer + Inner 1 and 3 1 and 4 1(4) + 3(1) = 73

(x + 1)(3x + 4)

b. 2 p 5 + 10 p 4 - 12 p 3 2 p 3 ( p 2 + 5p - 6) _________________ Factors of -6 Sum -1 and 6 53

2 p 3 (p - 1)(p + 6)

c. 9 q 6 + 30 q 5 + 24 q 4 3 q 4 (3 q 2 + 10q + 8) ___________________________________ Factors of 3 Factors of 8 Outer + Inner 1 and 3 1 and 8 1(8) + 3(1) = 117

1 and 3 8 and 1 1(1) + 3(8) = 257

1 and 3 2 and 4 1(4) + 3(2) = 103

3 q 4 (q + 2)(3q + 4)

d. 2 x 4 + 182( x 4 ) + 2(9)2( x 4 + 9)


7-6

CS10_A1_MESK710372_C07.indd 245 3/30/11 11:24:00 PM

think and disCUss

1. Possible answer: ( x 2 + 1)( x 2 - 1)

2. Possible answer: x 2 + 1; x 2 + x + 1 3.

Factoring Methods

Polynomial Method

1. 16x4 - 25y8

2. x2 + 10x + 25

3. 9t2 + 27t + 18t 4

4. a2 + 3a - 7a - 21

5. 100b2 + 81

A. Factoring out the GCF

B. Factoring by grouping

C. Unfactorable

D. Difference of two squares

E. Perfect-Square trinomial


1. Yes, 3x(9 x 2 + 1) is completely factored.

2. No2(4 x 3 - 3 x 2 - 8x)2x(4 x 2 - 3x - 8)

3. Yes, 2 k 2 (4 - k 3 ) is completely factored.

4. Yes, (2x + 3)(3x - 5) is completely factored.

5. No4(4 p 2 - 1)4[ (2p) 2 - 1 2 ]4(2 p 2 + 1)(2 p 2 - 1)

6. Yes, a( a 3 + 2ab + b 2 ) is completely factored.

7. 3 x 5 - 12 x 3 3 x 3 ( x 2 - 4)3 x 3 (x + 2)(x - 2)

8. 4 x 3 + 8 x 2 + 4x4x( x 2 + 2x + 1)4x (x + 1) 2

9. 8p q 2 + 8pq + 2p2p(4 q 2 + 4q + 1)2p (2q + 1) 2

10. 18r s 2 - 2r2r(9 s 2 - 1)2r(3s + 1)(3s - 1)

11. m n 5 - m 3 nmn( n 4 - m 2 )mn( n 2 + m)( n 2 - m)

12. 2 x 2 y - 20xy + 50y2y( x 2 - 10x + 25)2y (x - 5) 2

13. 6 x 4 - 3 x 3 - 9 x 2 3 x 2 (2 x 2 - x - 3) ____________________________________ Factors of 2 Factors of -3 Outer + Inner 1 and 2 1 and -3 1(-3) + 2(1) = -13

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

14. 3 y 2 + 14y + 4 ___________________________________ Factors of 3 Factors of 4 Outer + Inner 1 and 3 1 and 4 1(4) + 3(1) = 7 7

1 and 3 4 and 1 1(1) + 3(4) = 13 71 and 3 2 and 2 1(2) + 3(2) = 8 7

3 y 2 + 14y + 4 cannot be factored.

15. p 5 + 3 p 3 + p 2 + 3( p 5 + 3 p 3 ) + ( p 2 + 3) p 3 ( p 2 + 3) + 1( p 2 + 3)( p 3 + 1)( p 2 + 3)

16. 7 x 5 + 21 x 4 - 28 x 3 7 x 3 ( x 2 + 3x - 4) _________________ Factors of -4 Sum -1 and 4 3 37 x 3 (x - 1)(x + 4)

17. 2 z 2 + 11z + 6 ___________________________________ Factors of 2 Factors of 6 Outer + Inner 1 and 2 1 and 6 1(6) + 2(1) = 8 7

1 and 2 6 and 1 1(1) + 2(6) = 13 71 and 2 2 and 3 1(3) + 2(2) = 7 7

1 and 2 3 and 2 1(2) + 2(3) = 8 7

2 z 2 + 11z + 6 cannot be factored.

18. 9 p 2 - q 2 + 3p + q(9 p 2 - q 2 ) + (3p + q)(3p - q)(3p + q) + 1(3p + q)(3p - q + 1)(3p + q)


19. No2x( y 3 - 4 y 2 + 5y)2xy( y 2 - 4y + 5)

20. No2r(25 r 6 - 36)

2r[ (5 r 3 ) 2 - 6 2 ]2r(5 r 3 + 6)(5 r 3 - 6)

21. No3 n 2 ( n 2 - 25)3 n 2 ( n 2 - 5 2 )3 n 2 (n + 5)(n - 5)

22. Yes, 2m(m + 1)(m + 4) is completely factored.

23. Yes, 2 y 2 (4 x 2 + 9) is completely factored.

24. Yes, 4(7g + 9 h 2 ) is completely factored.

25. -4 x 3 + 24 x 2 - 36x-4x( x 2 - 6x + 9)-4x (x - 3) 2

26. 24 r 2 - 6 r 4 6 r 2 (4 - r 2 )6 r 2 (2 + r)(2 - r)

27. 5 d 2 - 60d + 1355( d 2 - 12d + 27) __________________ Factors of 27 Sum -1 and -27 -287

-3 and -9 -12 35(d - 3)(d - 9)

28. 4 y 8 + 36 y 7 + 81 y 6 y 6 (4 y 2 + 36y + 81) y 6 (2y + 9) 2


CS10_A1_MESK710372_C07.indd 246 3/30/11 11:24:02 PM

29. 98 x 3 - 50x y 2 2x(49 x 2 - 25 y 2 )2x(7x + 5y)(7x - 5y)

30. 4 x 3 y - 4 x 2 y - 8xy4xy( x 2 - x - 2) ___________________ Factors of -2 Sum 1 and -2 -1 34xy(x + 1)(x - 2)


1 and 5 -14 and -1 1(-1) + 5(-14) = -717

1 and 5 -2 and -7 1(-7) + 5(-2) = -177

1 and 5 -7 and -2 1(-2) + 5(-7) = -377

5 x 2 - 10x + 14 cannot be factored.

32. 121 x 2 + 36 y 2 cannot be factored.

33. p 4 - 16( p 2 + 4)( p 2 - 4)( p 2 + 4)(p + 2)(p - 2)

34. 4 m 6 - 30 m 5 + 36 m 4 2 m 4 (2 m 2 - 15m + 18) ___________________________________ Factors of 2 Factors of 18 Outer + Inner 1 and 2 -1 and -18 1(-18) + 2(-1) = -207

1 and 2 -18 and -1 1(-1) + 2(-18) = -377

1 and 2 -2 and -9 1(-9) + 2(-2) = -137

1 and 2 -9 and -2 1(-2) + 2(-9) = -207

1 and 2 -3 and -6 1(-6) + 2(-3) = -127

1 and 2 -6 and -3 1(-3) + 2(-6) = -153

2 m 4 (m - 6)(2m - 3)

35. 2 k 3 + 3 k 2 + 6k + 9(2 k 3 + 3 k 2 ) + (6k + 9) k 2 (2k + 3) + 3(2k + 3)( k 2 + 3)(2k + 3)( k 2 + 3)(2k + 3)

36. a b 4 - 16aa( b 4 - 16)a( b 2 + 4)( b 2 - 4)a( b 2 + 4)(b + 2)(b - 2)

37. Let x be Ella’s age. x 2 + 12x + 36 x 2 + 2(x)(6) + 6 2 (x + 6) 2

38. Let d be the distance from point A to point B. d 2 - 81 d 2 - 9 2 (d + 9)(d - 9)

39. Let s be the number of seconds Bob can hold. s 2 - 16s + 28 ________________ Factors of 28 Sum -1 and -28 -29 7-2 and -14 -163

(s - 2)(s - 14)

40. Let a be the number of apples on the tree.3 a 2 - 22a + 35 ___________________________________ Factors of 3 Factors of 35 Outer + Inner 1 and 3 -1 and -35 1(-35) + 3(-1) = -38 7

1 and 3 -35 and -1 1(-1) + 3(-35) = -1067

1 and 3 -5 and -7 1(-7) + 3(-5) = -22 3

(a - 5)(3a - 7)

41. Let b be Beth’s score. b 2 - 49 b 2 - 7 2 (b + 7)(b - 7)

42. -5 t 2 + 30t + 1-1(5 t 2 - 30t - 1) ____________________________________ Factors of 5 Factors of -1 Outer + Inner 1 and 5 1 and -1 1(-1) + 5(1) = 4 7

1 and 5 -1 and 1 1(1) + 5(-1) = -47

-5 t 2 + 30t + 1 is fully factored.

43. The next step is to check for a pattern, such as a perfect-square trinomial, or a difference of 2 squares.

44. 12 (x + 1) 2 + 60(x + 1) + 753[ 4(x + 1) 2 + 20(x + 1) + 25]3[( 2(x + 1)) 2 + 2(2(x + 1))(5) + 5 2 ]3 [2(x + 1) + 5] 2 3 (2x + 7) 2

45. (2x + 3) 2 - (x - 4) 2

[(2x + 3) + (x - 4)][(2x + 3) - (x - 4)](3x - 1)(x + 7)

46. 45x (x - 2) 2 + 60x(x - 2) + 20x5x[9 (x - 2) 2 + 12(x - 2) + 4]5x [3(x - 2)) 2 + 2(3(x - 2))(2) + 2 2 ]5x [3(x - 2) + 2] 2 5x (3x - 4) 2

47. (3x - 5) 2 - (y + 2) 2 [(3x - 5) + (y + 2)][(3x - 5) - (y + 2)](3x + y - 3)(3x - y - 7)

48a. x 2 + 2x - 15 __________________ Factors of -15 Sum -1 and 15 147

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

b. x + 5

x - 3

c. x = 7 ℓ = x + 5 = 7 + 5 = 12 ft w = x - 3 = 7 - 3 = 4 ft

49. Method 1: 4 x 2 - 100 = 4( x 2 - 25) = 4(x + 5)(x - 5)

Method 2: 4 x 2 - 100 = (2x + 10)(2x - 10) = 2(x + 5) · 2(x - 5) = 4(x + 5)(x - 5)


CS10_A1_MESK710372_C07.indd 247 3/30/11 11:24:04 PM

50. 2 x 2 + 5xy + 3 y 2 ___________________________________ Factors of 2 Factors of 3 Outer + Inner 1 and 2 1 and 3 1(3) + 2(1) = 53

(x + y)(2x + 3y)x = -10.1, y = 10.052 x 2 + 5xy + 3 y 2 = (x + y)(2x + 3y)

= approx. 0

51. Possible answer: (2x - 1) 2 ≠ 4 x 2 - 4x - 1

52. (a + b) 6 53. (a + b) 8

54. (a + b) 7

teSt PreP

55. C;6 x 2 + 7x - 10 _____________________________________ Factors of 6 Factors of -10 Outer + Inner 1 and 6 1 and -10 1(-10) + 6(1) = -47

1 and 6 -1 and 10 1(10) + 6(-1) = 4 7

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

(x + 2)(6x - 5)

56. H;16 x 12 - 25616( x 12 - 16)16( x 6 + 4)( x 6 - 4)16( x 6 + 4)( x 3 + 2)( x 3 - 2)

57. C

58a. 8 x 3 + 24 x 2 + 18x 2x(4 x 2 + 12x + 9) 2x[ (2x) 2 + 2(2x)(3) + 3 2 ] 2x (2x + 3) 2 First factor out the GCF of 8 x 3 , 24 x 2 and 18x,

which is 2x; then use the pattern for a perfect-square trinomial.

b. The polynomial could be factored by finding factors of 8 and factors of 18 that would result in 24 as the sum of the outer and inner products. Then one binomial would need to be factored again.


59a. 72π p 3 + 48π p 2 + 8π

8p(9π p 2 + 6πp + π) 8p[π(9 p 2 + 6p + 1)] 8p[π (3p + 1) 2 ]

b. The radius of the cylinder is (3p + 1) cm.

c. 3p + 1 = 4 3p = 3 p = 1 h = 8p = 8(1) = 8 cm V = 8p[π (3p + 1) 2 ] = 8[π( 4) 2 ] = 128π cm 3

60. g 7 + g 3 + g 5 + g 4 g 3 ( g 4 ) + g 3 (1) + g 3 ( g 2 ) + g 3 (g) g 3 ( g 4 + g 2 + g + 1)

61. h 2 + h 8 + h 6 + h 4 h 2 (1) + h 2 ( h 6 ) + h 2 ( h 4 ) + h 2 ( h 2 ) h 2 ( h 6 + h 4 + h 2 + 1) h 2 [( h 6 + h 4 ) + ( h 2 + 1)] h 2 [ h 4 ( h 2 + 1) + ( h 2 + 1)] h 2 ( h 4 + 1)( h 2 + 1)

62. x n + 2 + x n + 1 + x n x n ( x 2 ) + x n (x) + x n (1) x n ( x 2 + x + 1)

63. x n + 5 + x n + 4 + x n + 3 x n + 3 ( x 2 ) + x n + 3 (x) + x n + 3 (1) x n + 3 ( x 2 + x + 1)

64a. h = w + 5 ℓ = w + 9

b. V = w(w + 5)(w + 9) = ( w 2 + 5w)(w + 9) = w 3 + 9 w 2 + 5 w 2 + 45w = w 3 + 14 w 2 + 45w

reADy to Go on? section b Quiz

1. Yes x 2 + 8x + 16 x 2 + 2(x)(4) + 4 2 (x + 4) 2

2. Yes4 x 2 - 20x + 25( 2x) 2 - 2(2x)(5) + 5 2 ( 2x - 5) 2

3. No, ( x 2 + 3x + 9) is not a perfect square because 3x ≠ 2(x)(3).

4. No, - 4x ≠ 2(x)(2)

5. Yes9 x 2 - 12x + 4 (3x) 2 - 2(3x)(2) + 2 2 (3x - 2) 2

6. No, ( x 2 - 12x - 36) is not a perfect square because the last term is not positive.

7. x 2 + 20x + 100 x 2 + 2(x)(10) + 1 0 2 (x + 10) 2 ℓ = w = (x + 10) ftThe perimeter of a window is 4(x + 10) ft.x = 4, 4(x + 10) = 4(4 + 10) = 56The perimeter of a window is 56 ft when x = 4 ft.

8. Yes x 2 - 121 x 2 - 1 1 2 (x + 11)(x - 11)

9. No, (4 t 2 - 20) is not a difference of 2 squares because 20 is not a perfect square.

10. Yes1 - 9 y 4 1 2 - (3 y 2 ) 2 (1 + 3 y 2 )(1 - 3 y 2 )

11. Yes25 m 2 - 4 m 6 (5m) 2 - (2 m 3 ) 2 m 2 (5 + 2 m 2 )(5 - 2 m 2 )

12. No, (16 x 2 + 49) is not a difference of 2 squares because the last term is not negative.


CS10_A1_MESK710372_C07.indd 248 3/30/11 11:24:06 PM

13. Yes r 4 - t 2 ( r 2 ) 2 - t 2 ( r 2 + t)( r 2 - t)

14a. 36 d 2 - 36d + 9 (6d) 2 - 2(6d)(3) + 3 2 (6d - 3) 2 ℓ = (6d - 3) in

b. The perimeter of the square is 4(6d - 3) in.

c. d = 2 6d - 3 = 6(2) - 3 = 9 4(6d - 3) = 4(9) = 36 (6d - 3) 2 = (9) 2 = 81 When d = 2 in, the length of a side is 9 in, the

perimeter is 36 in, and the area is 81 in 2 .

15. Yes, 5( x 2 + 3x + 1) is completely factored.

16. No6x(5 x 2 - x) 6x[5x(x) - 1(x)]6 x 2 (5x - 1)

17. No3t( t 4 - 9) 3t[ ( t 2 ) 2 - 3 2 ]3t( t 2 + 3)( t 2 - 3)

18. No.2( m 2 - 10m + 25)2[ m 2 - 2(m)(5) + 5 2 ]2 (m - 5) 2

19. Yes, 3(2 y 2 - 5)(y + 1) is completely factored.

20. No(2n + 6)(n - 4)2(n + 3)(n - 4)

21. 3 x 3 - 12 x 2 + 12x3x( x 2 - 4x + 4)3x[( x 2 - 2(x)(2) + 2 2 ]3x(x - 2) 2

22. 16 m 3 - 4m4m(4 m 2 - 1)4m[(2m ) 2 - 1 2 ]4m(2m + 1)(2m - 1)

23. 5 x 3 y - 45xy 5xy( x 2 - 9)5xy( x 2 - 3 2 )5xy(x + 3)(x - 3)

24. 3 t 2 + 5t - 1 ____________________________________ Factors of 3 Factors of -1 Outer + Inner 1 and 3 1 and -1 1(-1) + 3(1) = -2 71 and 3 -1 and 1 1(1) + 3(-1) = 2 7

3 t 2 + 5t - 1 cannot be factored.

25. 3c 2 + 12c - 633( c 2 + 4c - 21) __________________ Factors of -21 Sum -1 and 21 207

-3 and 7 43

3(c - 3)(c + 7)

26. x 5 - 81xx( x 4 - 81)x[ ( x 2 ) 2 - 9 2 ]x( x 2 + 9)( x 2 - 9)x( x 2 + 9)(x + 3)(x - 3)

27. Let ℓ be the length. ℓ 2 - 36 ℓ 2 - 6 2 (ℓ + 6)(ℓ - 6)

28. Let a be Michael’s age. a 2 - 8a + 16 a 2 - 2(a)(4) + 4 2 (a - 4) 2

29. Let v be the speed.2 v 2 + 2v - 122( v 2 + v - 6)2(v + 3)(v - 2)

30. Let h be Jessie’s height.3 h 3 + 3 h 2 - 6h3h( h 2 + h - 2)3h(h + 2)(h - 1)

31. A = (9x)(8x) - (8y)(4y) = 72 x 2 - 32 y 2 = 8(9 x 2 - 4 y 2 ) = 8(3x + 2y)(3x - 2y)

stuDy GuiDe: review

1. Prime factorization 2. Greatest common factor

FaCtOrs and Greatest COMMOn FaCtOrs

3. 12 = 2 · 2 · 3 = 2 2 · 3 4. 20 = 2 · 2 · 5 = 2 2 · 5

5. 32 = 2 · 2 · 2 · 2 · 2 = 2 5 6. 23 is a prime number

7. 40 = 2 · 2 · 2 · 5 = 2 3 · 5

8. 64 = 2 · 2 · 2 · 2 · 2 · 2 = 2 6

9. 66 = 2 · 3 · 11 10. 114 = 2 · 3 · 19

11. 15 = 3 · 550 = 2 · 2 · 5The GCF of 15 and 50 is 5.

12. 36 = 2 · 2 · 3 · 3132 = 2 · 2 · 3 · 11The GCF of 36 and 132 is 12.

13. 29 is a prime number.30 = 2 · 3 · 5The GCF of 29 and 30 is 1.

14. 54 = 2 · 3 · 3 · 381 = 3 · 3 · 3 · 3The GCF of 54 and 81 is 27.

15. 20 = 2 · 2 · 548 = 2 · 2 · 2 · 2 · 3The GCF of 20 and 48 is 4.

16. 9m = 3 · 3 · m3 is a prime number.The GCF of 9m and 3 is 3.

17. 4x = 2 · 2 · x2 x 2 = 2 · x · xThe GCF of 4x and 2 x 2 is 2x.

18. -18 b 4 = -1 · 2 · 3 · 3 · b · b · b · b 27 b 2 = 3 · 3 · 3 · b · bThe GCF of -18 b 4 and 27 b 2 is 9 b 2 .

19. 100r = 2 · 2 · 5 · 5 · r25 r 5 = 5 · 5 · r · r · r · r · rThe GCF of 100r and 25r 5 is 25r.


7-1

CS10_A1_MESK710372_C07.indd 249 3/30/11 11:24:08 PM

20. The 42 types of boxed nails and 36 types of boxed screws must be divided into groups of equal size. The number of boxes in each row must be a common factor of 42 and 36.factors of 42: 1, 2, 3, 6, 7, 14, 21, 42factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36The GCF of 42 and 36 is 6.The greatest possible number of boxes in each row is 6. Find the number of rows.

42 types of boxed nails

___________________ 6 boxes per row

= 7 rows

36 types of boxed screws

_____________________ 6 boxes per row

= 6 rows

When the greatest possible number of boxes is in each row, there are 13 rows in total.

FaCtOrinG by GCF

21. 5x = 5 · x15 x 3 = 3 · 5 · x · x · xGCF: 5 · x = 5x5x - 15 x 3 = 5x(1) - 5x(3 x 2 )

= 5x(1 - 3 x 2 ) 22. 16b = 2 · 2 · 2 · 2 · b

32 = 2 · 2 · 2 · 2 · 2GCF: 2 · 2 · 2 · 2 = 16-16b + 32 = 16(-b) + 16(2)

= 16(-b + 2) 23. -1(14v + 21)

14v = 2 · 7 · v21 = 3 · 7GCF: 7-1(14v + 21)-1[7(2v) + 7(3)]-7(2v + 3)

24. 4 a 2 = 2 · 2 · a · a12a = 2 · 2 · 3 · a8 = 2 · 2 · 2GCF: 2 · 2 = 44 a 2 - 12a - 8 = 4( a 2 ) - 4(3a) - 4(2)

= 4( a 2 - 3a - 2)

25. 5 g 5 = 5 · g · g · g · g · g10 g 3 = 2 · 5 · g · g · g15g = 3 · 5 · gGCF: 5 · g = 5g5 g 5 - 10 g 3 - 15g = 5g( g 4 ) - 5g(2 g 2 ) - 5g(3)

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

26. 40 p 2 = 2 · 2 · 2 · 5 · p · p10p = 2 · 5 · p30 = 2 · 3 · 5GCF: 2 · 5 = 1040 p 2 - 10p + 30 = 10(4 p 2 ) - 10(p) + 10(3)

= 10(4 p 2 - p + 3)

27. 6 x 2 = 2 · 3 · x · x5x = 5 · xGCF: x6 x 2 + 5x = x(6x) + x(5)

= x(6x + 5)The dimension of the lot is (6x + 5) ft by x ft.

28. 2x(x - 4) + 9(x - 4)(2x + 9)(x - 4)

29. t(3t + 5) - 6(3t + 5)(t - 6)(3t + 5)

30. 5(6 - n) - 3n(6 - n)(5 - 3n)(6 - n)

31. b(b + 4) + 2(b + 4) (b + 2)(b + 4)

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

33. n 3 + n - 4 n 2 - 4( n 3 - 4 n 2 ) + (n - 4) n 2 (n - 4) + 1(n - 4)( n 2 + 1)(n - 4)

34. 6 b 2 - 8b + 15b - 20(6 b 2 - 8b) + (15b - 20)2b(3b - 4) + 5(3b - 4)(2b + 5)(3b - 4)

35. 2 h 3 - 7h + 14 h 2 - 49(2 h 3 + 14 h 2 ) - (7h + 49)2 h 2 (h + 7) - 7(h + 7)(2 h 2 - 7)(h + 7)

36. 3 t 2 + 18t + t + 6(3 t 2 + 18t) + (t + 6)3t(t + 6) + 1(t + 6)(3t + 1)(t + 6)

37. 10 m 3 + 15 m 2 - 2m - 3(10 m 3 + 15 m 2 ) - (2m + 3)5 m 2 (2m + 3) - 1(2m + 3)(5 m 2 - 1)(2m + 3)

38. 8 p 3 + 4p - 6 p 2 - 3(8 p 3 - 6 p 2 ) + (4p - 3)2 p 2 (4p - 3) + 1(4p - 3)(2 p 2 + 1)(4p - 3)

39. 5r - 10 + 2r - r 2 (5r - 10) - ( r 2 - 2r)5(r - 2) - r(r - 2)-1(r - 5)(r - 2)

40. b 3 - 5b + 15 - 3 b 2 ( b 3 - 3 b 2 ) - (5b - 15) b 2 (b - 3) - 5(b - 3)( b 2 - 5)(b - 3)

41. 6t - t 3 - 4 t 2 + 24-( t 3 + 4 t 2 ) + (6t + 24)- t 2 (t + 4) + 6(t + 4)-1( t 2 - 6)(t + 4)

42. 12h - 3 h 2 + h - 4-1(3 h 2 - 12h) + (h - 4)-3h(h - 4) + 1(h - 4)-1(3h - 1)(h - 4)

43. d - d 2 + d - 1-( d 2 - d) + (d - 1)-d(d - 1) + 1(d - 1)-1(d - 1) 2

44. 6b - 5 b 2 + 10b - 12-(5 b 2 - 6b) + (10b - 12)-b(5b - 6) + 2(5b - 6)-1(b - 2)(5b - 6)

45. 5t - t 2 - t + 5-( t 2 - 5t) - (t - 5)-t(t - 5) - 1(t - 5)-1(t + 1)(t - 5)

46. 8 b 2 - 2 b 3 - 5b + 20-(2 b 3 - 8 b 2 ) - (5b - 20)-2 b 2 (b - 4) - 5(b - 4)-1(2 b 2 + 5)(b - 4)

47. 3r - 3 r 2 - 1 + r-(3 r 2 - 3r) + (r - 1)-3r(r - 1) + 1(r - 1)-1(3r - 1)(r - 1)

48. Left rectangle: x(2x + 3) = 2 x 2 + 3xRight rectangle: 2(4x + 6) = 8x + 12Combined: (2 x 2 + 3x) + (8x + 12) = (2 x 2 + 8x) + (3x + 12)

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

FaCtOrinG x 2 + bx + c

49. x 2 + 6x + 5(x + 1)(x + 5)

50. x 2 + 6x + 8(x + 2)(x + 4)

51. x 2 + 8x + 15(x + 3)(x + 5)


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52. x 2 - 8x + 12(x - 2)(x - 6)

53. x 2 + 10x + 25 x 2 + 2(x)(5) + 5 2 (x + 5) 2

54. x 2 - 13x + 22(x - 2)(x - 11)

55. x 2 + 24x + 80(x + 4)(x + 20)

56. x 2 - 26x + 120(x - 6)(x - 20)

57. x 2 + 5x - 84(x + 12)(x - 7)

58. x 2 - 5x - 24(x + 3)(x - 8)

59. x 2 - 3x - 28(x + 4)(x - 7)

60. x 2 + 4x - 5(x + 5)(x - 1)

61. x 2 + x - 6(x + 3)(x - 2)

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

63. x 2 - 2x - 48(x + 6)(x - 8)

64. x 2 - 5x - 36(x + 4)(x - 9)

65. x 2 - 6x - 72(x + 6)(x - 12)

66. x 2 - 3x - 70(x + 7)(x - 10)

67. x 2 + 14x - 120(x + 20)(x - 6)

68. x 2 + 6x - 7(x + 7)(x - 1)

69. y 2 + 8y + 15(y + 3)(y + 5)ℓ= (y + 5) mw = (y + 3) m

FaCtOrinG ax 2 + bx + c

70. 2 x 2 + 11x + 5 ___________________________________ Factors of 2 Factors of 5 Outer + Inner 1 and 2 1 and 5 1(5) + 2(1) = 71 and 2 5 and 1 1(1) + 2(5) = 11(x + 5)(2x + 1)

71. 3 x 2 + 10x + 7 ___________________________________ Factors of 3 Factors of 7 Outer + Inner 1 and 3 1 and 7 1(7) + 3(1) = 10(x + 1)(3x + 7)

72. 2 x 2 - 3x + 1 ___________________________________ Factors of 2 Factors of 1 Outer + Inner 1 and 2 -1 and -1 1(-1) + 2(-1) = -3(x - 1)(2x - 1)

73. 3 x 2 + 8x + 4 ___________________________________ Factors of 3 Factors of 4 Outer + Inner 1 and 3 1 and 4 1(4) + 3(1) = 71 and 3 4 and 1 1(1) + 3(4) = 131 and 3 2 and 2 1(2) + 3(2) = 8(x + 2)(3x + 2)

74. 5 x 2 + 28x + 15 ____________________________________ Factors of 5 Factors of 15 Outer + Inner 1 and 5 1 and 15 1(15) + 5(1) = 201 and 5 15 and 1 1(1) + 5(15) = 761 and 5 3 and 5 1(5) + 5(3) = 201 and 5 5 and 3 1(3) + 5(5) = 28(x + 5)(5x + 3)

75. 6 x 2 - 19x + 15 ____________________________________ Factors of 6 Factors of 15 Outer + Inner 1 and 6 -1 and -15 1(-15) + 6(-1) = -211 and 6 -15 and -1 1(-1) + 6(-15) = -911 and 6 -3 and -5 1(-5) + 6(-3) = -231 and 6 -5 and -3 1(-3) + 6(-5) = -332 and 3 -1 and -15 2(-15) + 3(-1) = -332 and 3 -15 and -1 2(-1) + 3(-15) = -472 and 3 -3 and -5 2(-5) + 3(-3) = -19(2x - 3)(3x - 5)

76. 4 x 2 + 13x + 10 ____________________________________ Factors of 4 Factors of 10 Outer + Inner 1 and 4 1 and 10 1(10) + 4(1) = 141 and 4 10 and 1 1(1) + 4(10) = 411 and 4 2 and 5 1(5) + 4(2) = 13(x + 2)(4x + 5)

77. 3 x 2 + 10x + 8 ___________________________________ Factors of 3 Factors of 8 Outer + Inner 1 and 3 1 and 8 1(8) + 3(1) = 111 and 3 8 and 1 1(1) + 3(8) = 251 and 3 2 and 4 1(4) + 3(2) = 10(x + 2)(3x + 4)

78. 7 x 2 - 37x + 10 ____________________________________ Factors of 7 Factors of 10 Outer + Inner 1 and 7 -1 and -10 1(-10) + 7(-1) = -171 and 7 -10 and -1 1(-1) + 7(-10) = -711 and 7 -2 and -5 1(-5) + 7(-2) = -191 and 7 -5 and -2 1(-2) + 7(-5) = -37(x - 5)(7x - 2)

79. 9 x 2 + 18x + 8 ___________________________________ Factors of 9 Factors of 8 Outer + Inner 1 and 9 1 and 8 1(8) + 9(1) = 171 and 9 8 and 1 1(1) + 9(8) = 731 and 9 2 and 4 1(4) + 9(2) = 221 and 9 4 and 2 1(2) + 9(4) = 383 and 3 1 and 8 3(8) + 3(1) = 273 and 3 2 and 4 3(4) + 3(2) = 18(3x + 2)(3x + 4)

80. 2 x 2 - x - 1 ____________________________________ Factors of 2 Factors of -1 Outer + Inner 1 and 2 1 and -1 1(-1) + 2(1) = 11 and 2 -1 and 1 1(1) + 2(-1) = -1(x - 1)(2x + 1)


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81. 3 x 2 - 11x - 4 ____________________________________ Factors of 3 Factors of -4 Outer + Inner 1 and 3 1 and -4 1(-4) + 3(1) = -11 and 3 -1 and 4 1(4) + 3(-1) = 11 and 3 2 and -2 1(-2) + 3(2) = 41 and 3 -2 and 2 1(2) + 3(-2) = -41 and 3 4 and -1 1(-1) + 3(4) = 111 and 3 -4 and 1 1(1) + 3(-4) = -11(x - 4)(3x + 1)

82. 2 x 2 - 11x + 5 ___________________________________ Factors of 2 Factors of 5 Outer + Inner 1 and 2 -1 and -5 1(-5) + 2(-1) = -71 and 2 -5 and -1 1(-1) + 2(-5) = -11(x - 5)(2x - 1)

83. 7 x 2 - 19x - 6 ____________________________________ Factors of 7 Factors of -6 Outer + Inner 1 and 7 1 and -6 1(-6) + 7(1) = 11 and 7 -1 and 6 1(6) + 7(-1) = -11 and 7 2 and -3 1(-3) + 7(2) = 111 and 7 -2 and 3 1(3) + 7(-2) = -111 and 7 3 and -2 1(-2) + 7(3) = 191 and 7 -3 and 2 1(2) + 7(-3) = -19(x - 3)(7x + 2)

84. 5 x 2 - 9x - 2 ____________________________________ Factors of 5 Factors of -2 Outer + Inner 1 and 5 1 and -2 1(-2) + 5(1) = 31 and 5 -1 and 2 1(2) + 5(-1) = -31 and 5 2 and -1 1(-1) + 5(2) = 91 and 5 -2 and 1 1(1) + 5(-2) = -9(x - 2)(5x + 1)

85. -6 x 2 - x + 2-1(6 x 2 + x - 2) ____________________________________ Factors of 6 Factors of -2 Outer + Inner 1 and 6 1 and -2 1(-2) + 6(1) = 41 and 6 -1 and 2 1(2) + 6(-1) = -41 and 6 2 and -1 1(-1) + 6(2) = 111 and 6 -2 and 1 1(1) + 6(-2) = -112 and 3 1 and -2 2(-2) + 3(1) = -12 and 3 -1 and 2 2(2) + 3(-1) = 1-1(2x - 1)(3x + 2)

86. 6 x 2 - x - 5 ____________________________________ Factors of 6 Factors of -5 Outer + Inner 1 and 6 1 and -5 1(-5) + 6(1) = 11 and 6 -1 and 5 1(5) + 6(-1) = -1(x - 1)(6x + 5)

87. 6 x 2 + 17x - 14 _____________________________________ Factors of 6 Factors of -14 Outer + Inner 1 and 6 1 and -14 1(-14) + 6(1) = -81 and 6 -1 and 14 1(14) + 6(-1) = 81 and 6 2 and -7 1(-7) + 6(2) = 51 and 6 -2 and 7 1(7) + 6(-2) = -51 and 6 7 and -2 1(-2) + 6(7) = 401 and 6 -7 and 2 1(2) + 6(-7) = -401 and 6 14 and -1 1(-1) + 6(14) = 831 and 6 -14 and 1 1(1) + 6(-14) = -832 and 3 1 and -14 2(-14) + 3(1) = -252 and 3 -1 and 14 2(14) + 3(-1) = 252 and 3 2 and -7 2(-7) + 3(2) = -82 and 3 -2 and 7 2(7) + 3(-2) = 82 and 3 7 and -2 2(-2) + 3(7) = 17(2x + 7)(3x - 2)

88. -4 x 2 + 8x + 5-1(4 x 2 - 8x - 5) ____________________________________ Factors of 4 Factors of -5 Outer + Inner 1 and 4 1 and -5 1(-5) + 4(1) = -11 and 4 -1 and 5 1(5) + 4(-1) = 11 and 4 5 and -1 1(-1) + 4(5) = 191 and 4 -5 and 1 1(1) + 4(-5) = -192 and 2 1 and -5 2(-5) + 2(1) = -8-1(2x + 1)(2x - 5)

89. -10 x 2 + 11x + 6-1(10 x 2 - 11x - 6) _____________________________________ Factors of 10 Factors of -6 Outer + Inner 1 and 10 1 and -6 1(-6) + 10(1) = 41 and 10 -1 and 6 1(6) + 10(-1) = -41 and 10 2 and -3 1(-3) + 10(2) = 171 and 10 -2 and 3 1(3) + 10(-2) = -171 and 10 3 and -2 1(-2) + 10(3) = 281 and 10 -3 and 2 1(2) + 10(-3) = -281 and 10 6 and -1 1(-1) + 10(6) = 591 and 10 -6 and 1 1(1) + 10(-6) = -592 and 5 1 and -6 2(-6) + 5(1) = -72 and 5 -1 and 6 2(6) + 5(-1) = 72 and 5 2 and -3 2(-3) + 5(2) = 42 and 5 -2 and 3 2(3) + 5(-2) = -42 and 5 3 and -2 2(-2) + 5(3) = 112 and 5 -3 and 2 2(2) + 5(-3) = -11-1(2x - 3)(5x + 2)

90. 12 x 2 + 4x - 15x - 5(12 x 2 + 4x) - (15x + 5)4x(3x + 1) - 5(3x + 1)(4x - 5)(3x + 1)

FaCtOrinG sPeCiaL PrOdUCts

91. Yes x 2 + 12x + 36 x 2 + 2(x)(6) + 6 2 (x + 6) 2

92. No, ( x 2 + 5x + 25) is not a perfect-square trinomial because 5x ≠ 2(x)(5).

93. No, (4 x 2 - 2x + 1) is not a perfect-square trinomial because -2x ≠ 2(2x)(1).

94. Yes9 x 2 + 12x + 4(3 x) 2 + 2(3x)(2) + 2 2 (3x + 2) 2

95. No, (16 x 2 + 8x + 4) is not a perfect-square trinomial because 8x ≠ 2(4x)(2).

96. Yes x 2 + 14x + 49 x 2 + 2(x)(7) + 7 2 (x + 7) 2

97. Yes100 x 2 - 81(10 x) 2 - 9 2 (10x + 9)(10x - 9)

98. No, ( x 2 - 2) is not a difference of 2 squares because 2 is not a perfect square.

99. No, (5 x 4 - 10 y 6 ) is not a difference of 2 squares because 5 x 4 - 10 y 6 = 5( x 4 - 2 y 6 ) and 2 y 6 is not a

perfect square.


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100. Yes (-1 2) 2 - ( x 3 ) 2

(-12 + x 3 )(-12 - x 3 ) -1( x 3 + 12)( x 3 - 12)

101. No, (121 b 2 + 9 c 8 ) is not a difference of 2 squares because the operation between the terms is addition, and not subtracting.

102. Yes 100 p 2 - 25 q 2

25(4 p 2 - q 2 ) 25[(2p ) 2 - q 2 ] 25(2p + q)(2p - q)

103. difference of 2 squares x 2 - 25 x 2 - 5 2 (x + 5)(x - 5)

104. Perfect-square trinomial x 2 + 20x + 100 x 2 + 2(x)(10) + 10 2 (x + 10 ) 2

105. Difference of 2 squares j 2 - k 4 j 2 - ( k 2 ) 2 (j + k 2 )(j - k 2 )

106. Perfect-square trinomial 9 x 2 - 42x + 49 (3x ) 2 - 2(3x)(7) + 7 2 (3x - 7 ) 2

107. Perfect-square trinomial 81 x 2 + 144x + 64 (9x ) 2 + 2(9x)(8) + 8 2 (9x + 8) 2

108. Difference of 2 squares 16 b 4 - 121 c 6 (4 b 2 ) 2 - (11 c 3 ) 2 (4 b 2 + 11 c 3 )(4 b 2 - 11 c 3 )

ChOOsinG a FaCtOrinG MethOd

109. No 4 x 2 + 10x + 6 (4x + 6)(x + 1) 2(2x + 3)(x + 1)

110. Yes, 3( y 2 + 25) is completely factored.

111. No b 4 - 81 ( b 2 + 9)( b 2 - 9) ( b 2 + 9)(b + 3)(b - 3)

112. Yes, (x - 3 ) 2 is completely factored.

113. 4 x 2 - 64 4( x 2 - 16) 4(x + 4)(x - 4)

114. 3 b 5 - 6 b 4 - 24 b 3 3 b 3 ( b 2 - 2b - 8) 3 b 2 (b + 2)(b - 4)

115. a 4 b 3 - a 2 b 5 a 2 b 3 ( a 2 - b 2 ) a 2 b 3 (a + b)(a - b)

116. t 20 - t 4 t 4 ( t 16 - 1) t 4 ( t 8 + 1)( t 8 - 1) t 4 ( t 8 + 1)( t 4 + 1)( t 4 - 1) t 4 ( t 8 + 1)( t 4 + 1)( t 2 + 1)( t 2 - 1) t 4 ( t 8 + 1)( t 4 + 1)( t 2 + 1)(t + 1)(t - 1)

117. 5 x 2 + 20x + 15 5( x 2 + 4x + 3) 5(x + 1)(x + 3)

118. 2 x 4 - 50 x 2 2 x 2 ( x 2 - 25) 2 x 2 (x + 5)(x - 5)

119. 8t + 32 + 2st + 8s 2(st + 4s + 4t + 16 2[(st + 4s) + (4t + 16)] 2[s(t + 4) + 4(t + 4)] 2(s + 4)(t + 4)

120. 25 m 3 - 90 m 2 - 40m 5m(5 m 2 - 18m - 8)

____________________________________ Factors of 5 Factors of -8 Outer + Inner 1 and 5 1 and -8 1(-8) + 5(1) = -3 1 and 5 -1 and 8 1(8) + 5(-1) = 3 1 and 5 2 and -4 1(-4) + 5(2) = 6 1 and 5 -2 and 4 1(4) + 5(-2) = -6 1 and 5 4 and -2 1(-2) + 5(4) = 18 1 and 5 -4 and 2 1(2) + 5(-4) = -18 5m(m - 4)(5m + 2)

121. 32 x 4 - 48 x 3 + 8 x 2 - 12x 4x(8 x 3 - 12 x 2 + 2x - 3) 4x[(8 x 3 - 12 x 2 ) + (2x - 3)] 4x[4 x 2 (2x - 3) + 1(2x - 3)] 4x(4 x 2 + 1)(2x - 3)

122. 6 s 4 t + 12 s 3 t 2 + 6 s 2 t 3 6 s 2 t( s 2 + 2st + t 2 ) 6 s 2 t(s + t) 2

123. 10 m 3 + 4 m 2 - 90m - 36 2(5 m 3 + 2 m 2 - 45m - 18) 2[(5 m 3 + 2 m 2 ) - (45m + 18)] 2[ m 2 (5m + 2) - 9(5m + 2)] 2( m 2 - 9)(5m + 2) 2(m + 3)(m - 3)(5m + 2)

chAPter test

1. 3 t 4 = 3 · t · t · t · t8 t 2 = 2 · 2 · 2 · t · tThe GCF of 3 t 2 and 8t 2 is t 2 .

2. 2 y 3 = 2 · y · y · y-12y = -1 · 2 · 2 · 3 · yThe GCF of 2y 3 and -12y is 2y.

3. 15 n 5 = 3 · 5 · n · n · n · n · n9 n 4 = 3 · 3 · n · n · n · nThe GCF of 15 n 5 and 9 n 4 is 3 n 4 .

4. 360 = 2 · 2 · 2 · 3 · 3 · 5 = 2 3 · 3 2 · 5


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5. The 16 Liberty nickels, 24 Buffalo nuckels, and 40 Jefferson nickels must be divided into groups of equal size. The number of nickels in each row must be a common factor of 16, 24 and 40.factors of 16: 1, 2, 4, 8, 16factors of 24: 1, 2, 3, 4, 6, 8, 12, 24factors of 40: 1, 2, 4, 5, 8, 10, 20, 40The GCF of 16, 24 and 40 is 8.The greatest possible number of nickels in each row is 8. Find the number of rows.

16 Liberty nickels

_______________ 8 per row

= 2 rows

24 Buffalo nickels _______________ 8 per row

= 3 rows

40 Jefferson nickels _________________ 8 per row

= 5 rows

When the greatest possible number of nickels is in each row, there are 10 rows in total.

6. 24 m 2 + 4 m 3 4 m 2 (6) + 4 m 2 (m)4 m 2 (6 + m)

7. 9 x 5 - 12x 3x(3 x 4 ) - 3x(4)3x(3 x 4 - 4)

8. -2 r 4 - 6-2( r 4 ) - 2(3)-2( r 4 + 3)

9. 3(c - 5) + 4c(c - 5)(3 + 4c)(c - 5)

10. 10 x 3 + 4x - 25 x 2 - 10(10 x 3 - 25 x 2 ) + (4x - 10)5 x 2 (2x - 5) + 2(2x - 5)(5 x 2 + 2)(2x - 5)

11. 4 y 3 - 4 y 2 - 3 + 3y(4 y 3 - 4 y 2 ) + (3y - 3)4 y 2 (y - 1) + 3(y - 1)(4 y 2 + 3)(y - 1)

12. -5 t 2 + 50t + 5-1(5 t 2 - 50t - 5)-1[5( t 2 ) - 5(10t) - 5(1)]-5( t 2 - 10t - 1)

13. x 2 + 6x + 5(x + 1)(x + 5)

14. x 2 - 4x - 21(x + 3)(x - 7)

15. x 2 - 8x + 15(x - 3)(x - 5)

16. 2 x 2 + 9x + 7 ___________________________________ Factors of 2 Factors of 7 Outer + Inner 1 and 2 1 and 7 1(7) + 2(1) = 9(x + 1)(2x + 7)

17. 2 x 2 + 9x - 18 _____________________________________ Factors of 2 Factors of -18 Outer + Inner 1 and 2 1 and -18 1(-18) + 2(1) = -161 and 2 -1 and 18 1(18) + 2(-1) = 161 and 2 2 and -9 1(-9) + 2(2) = -51 and 2 -2 and 9 1(9) + 2(-2) = 51 and 2 3 and -6 1(-6) + 2(3) = 01 and 2 -3 and 6 1(6) + 2(-3) = 01 and 2 6 and -3 1(-3) + 2(6) = 9(x + 6)(2x - 3)

18. -3 x 2 - 2x + 8-1(3 x 2 + 2x - 8) ____________________________________ Factors of 3 Factors of -8 Outer + Inner 1 and 3 1 and -8 1(-8) + 3(1) = -51 and 3 -1 and 8 1(8) + 3(-1) = 51 and 3 2 and -4 1(-4) + 3(2) = 2-1(x + 2)(3x - 4)

19. Yes a 2 + 14a + 49 a 2 + 2(a)(7) + 7 2 (a + 7) 2

20. No, (2 x 2 + 10x + 25) is not a perfect-square trinomial because 2 x 2 is not a perfect square.

21. Yes9 t 2 - 6t + 1(3t ) 2 - 3(3t)(1) + 1 2 (3t - 1) 2

22. Yes b 2 - 16 b 2 - 4 2 (b + 4)(b - 4)

23. No, (25 y 2 - 10) is not a difference of 2 squares because 10 is not a perfect square.

24. Yes9 a 2 - b 10 (3a ) 2 - ( b 5 ) 2 (3a + b 5 )(3a - b 5 )

25. 9 x 2 + 30x + 25(3x ) 2 + 2(3x)(5) + 5 2 (3x + 5) 2 P = 4(3x + 5) ftx = 4, P = 4(3x + 5) = 4[3(4) + 5] = 68The perimeter is 68 ft when x = 4 ft.

26. No(6x - 3)(x + 5)3(2x - 1)(x + 5)

27. Yes, ( v 5 + 10)( v 5 - 10) is completely factored.

28. Yes, (2b + 3)(3b - 2) is completely factored.

29. 8 x 3 + 72 x 2 + 160x8x( x 2 + 9x + 20)8x(x + 4)(x + 5)

30. 3 x 5 - 27 x 3 3 x 3 ( x 2 - 9)3 x 3 (x + 3)(x - 3)

31. 8 x 3 + 64 x 2 - 20x - 1604(2 x 3 + 16 x 2 - 5x - 40)4[(2 x 3 + 16 x 2 ) - (5x + 40)]4[2 x 2 (x + 8) - 5(x + 8)]4(2 x 2 - 5)(x + 8)

32. c d 4 - c 7 d 6 c d 4 (1 - c 6 d 2 )c d 4 (1 + c 3 d)(1 - c 3 d)

33. 100 x 2 - 80x + 164(25 x 2 - 20x + 4)4[(5x ) 2 - 2(5x)(2) + 2 2 ]4(5x - 2) 2

34. 7 m 8 - 77( m 8 - 1)7( m 4 + 1)( m 4 - 1)7( m 4 + 1)( m 2 + 1)( m 2 - 1)7( m 4 + 1)( m 2 + 1)(m + 1)(m - 1)


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CHAPTER Factoring Polynomials 7 Solutions Key · 2 ˜ 2 5 5 Variable term as a product x ˜ x...

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