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6.8 Review Exercises and Sample Exam

Review Exercises

Introduction to Factoring

Determine the missing factor.

1. 12x324x2+4x=4x(?)

2. 10y435y35y2=5y2(?)

3. 18a5+9a427a3=9a3(?)

4. 21x2y+7xy249xy=7xy(?)

Factor out the GCF.

5. 22x2+11x

6. 15y45y3

7. 18a312a2+30a

8. 12a5+20a34a

9. 9x3y218x2y2+27xy2

10. 16a5b5c8a3b6+24a3b2c

Factor by grouping.

11. x2+2x5x10

12. 2x22x3x+3

13. x3+5x23x15

14. x36x2+x6

15. x3x2y2x+2y

16. a2b22a3+6ab3b3

Factoring Trinomials of the Form x2 + bx + c

Are the following factored correctly? Check by multiplying.

17. x2+5x+6=(x+6)(x1)

18. x2+3x10=(x+5)(x2)

19. x2+6x+9=(x+3)2

20. x26x9=(x3)(x+3)

Factor.

21. x213x14

22. x2+13x+12

23. y2+10y+25

24. y220y+100

25. a28a48

26. b218b+45

27. x2+2x+24

28. x210x16

29. a2+ab2b2

30. a2b2+5ab50

Factoring Trinomials of the Form ax2 + bx + c

Factor.

31. 5x227x18

32. 3x214x+8

33. 4x228x+49

34. 9x2+48x+64

35. 6x229x9

36. 8x2+6x+9

37. 60x265x+15

38. 16x240x+16

39. 6x310x2y+4xy2

40. 10x3y82x2y2+16xy3

41. y2+9y+36

42. a27a+98

43. 16+142x18x2

44. 45132x60x2

Factoring Special Binomials

Factor completely.

45. x281

46. 25x236

47. 4x249

48. 81x21

49. x264y2

50. 100x2y21

51. 16x4y4

52. x481y4

53. 8x3125

54. 27+y3

55. 54x4y2xy4

56. 3x4y2+24xy5

57. 64x6y6

58. x6+1

General Guidelines for Factoring Polynomials

Factor completely.

59. 8x34x2+20x

60. 50a4b4c+5a3b5c2

61. x312x2x+12

62. a32a23ab+6b

63. y215y+16

64. x218x+72

65. 144x225

66. 3x448

67. 20x241x9

68. 24x2+14x20

69. a4b343ab4

70. 32x7y2+4xy8

Solving Equations by Factoring

Solve.

71. (x9)(x+10)=0

72. 3x(x+8)=0

73. 6(x+1)(x1)=0

74. (x12)(x+4)(2x1)=0

75. x2+5x50=0

76. 3x213x+4=0

77. 3x212=0

78. 16x29=0

79. (x2)(x+6)=20

80. 2(x2)(x+3)=7x9

81. 52x2203x=0

82. 23x2512x+124=0

Find a quadratic equation with integer coefficients, given the following solutions.

83. −7, 6

84. 0, −10

85. −1/9, 1/2

86. ±3/2

Applications Involving Quadratic Equations

Set up an algebraic equation and then solve the following.

87. An integer is 4 less than twice another. If the product of the two integers is 96, then find the integers.

88. The sum of the squares of two consecutive positive even integers is 52. Find the integers.

89. A 20-foot ladder leaning against a wall reaches a height that is 4 feet more than the distance from the wall to the base of the ladder. How high does the ladder reach?

90. The height of an object dropped from the top of a 196-foot building is given by h(t)=16t2+196, where t represents the number of seconds after the object has been released. How long will it take the object to hit the ground?

91. The length of a rectangle is 1 centimeter less than three times the width. If the area is 70 square centimeters, then find the dimensions of the rectangle.

92. The base of a triangle is 4 centimeters more than twice the height. If the area of the triangle is 80 square centimeters, then find the measure of the base.

Sample Exam

1. Determine the GCF of the terms 25a2b2c, 50ab4, and 35a3b3c2.

2. Determine the missing factor: 24x2y316x3y2+8x2y=8x2y(?).

Factor.

3. 12x515x4+3x2

4. x34x22x+8

5. x27x+12

6. 9x212x+4

7. x281

8. x3+27y3

Factor completely.

9. x3+2x24x8

10. x41

11. 6x3+20x26x

12. x61

Solve.

13. (2x+1)(x7)=0

14. 3x(4x3)(x+1)=0

15. x264=0

16. x2+4x12=0

17. 23x2+89x16=0

18. (x5)(x3)=1

19. 3x(x+3)=14x+2

20. (3x+1)(3x+2)=9x+3

For each problem, set up an algebraic equation and then solve.

21. An integer is 4 less than twice another. If the product of the two integers is 70, then find the integers.

22. The sum of the squares of two consecutive positive odd integers is 130. Find the integers.

23. The length of a rectangle is 4 feet more than twice its width. If the area is 160 square feet, then find the dimensions of the rectangle.

24. The height of a triangle is 6 centimeters less than four times the length of its base. If the area measures 27 square centimeters, then what is the height of the triangle?

25. The height of a projectile launched upward at a speed of 64 feet/second from a height of 36 feet is given by the function h(t)=16t2+64t+36. How long will it take the projectile to hit the ground?

Review Exercises Answers

1: (3x26x+1)

3: (2a2a+3)

5: 11x(2x+1)

7: 6a(3a22a+5)

9: 9xy2(x22x+3)

11: (x+2)(x5)

13: (x+5)(x23)

15: (xy)(x22)

17: No

19: Yes

21: (x14)(x+1)

23: (y+5)2

25: (a12)(a+4)

27: Prime

29: (ab)(a+2b)

31: (5x+3)(x6)

33: (2x7)2

35: Prime

37: 5(3x1)(4x3)

39: 2x(3x2y)(xy)

41: 1(y12)(y+3)

43: 2(9x+1)(x8)

45: (x+9)(x9)

47: (2x+7)(2x7)

49: (x+8y)(x8y)

51: (4x2+y2)(2x+y)(2xy)

53: (2x5)(4x2+10x+25)

55: 2xy(3xy)(9x2+3xy+y2)

57: (2x+y)(4x22xy+y2)(2xy)(4x2+2xy+y2)

59: 4x(2x2x+5)

61: (x12)(x+1)(x1)

63: 1(y+16)(y1)

65: (12x+5)(12x5)

67: (4x9)(5x+1)

69: ab(a7b)(a2+7ab+49b2)

71: 9, −10

73: −1, 1

75: −10, 5

77: ±2

79: −8, 4

81: 0, 8/3

83: x2+x42=0

85: 18x27x1=0

87: {8, 12} or {−6, −16}

89: 16 feet

91: Length: 14 centimeters; width: 5 centimeters

Sample Exam Answers

1: 5ab2

3: 3x2(4x35x2+1)

5: (x4)(x3)

7: (x+9)(x9)

9: (x+2)2(x2)

11: 2x(3x1)(x3)

13: −1/2, 7

15: ±8

17: −3/2, 1/6

19: −1/3, 2

21: {7, 10} or {−14, −5}

23: Width: 8 feet; length: 20 feet

25: 4½ sec