Review Exercises and Sample Exam

6.8 Review Exercises and Sample Exam

Review Exercises

Introduction to Factoring

Determine the missing factor.

1. $12 x^{3} - 24 x^{2} + 4 x = 4 x (?)$

2. $10 y^{4} - 35 y^{3} - 5 y^{2} = 5 y^{2} (?)$

3. $- 18 a^{5} + 9 a^{4} - 27 a^{3} = - 9 a^{3} (?)$

4. $- 21 x^{2} y + 7 x y^{2} - 49 x y = - 7 x y (?)$

Factor out the GCF.

5. $22 x^{2} + 11 x$

6. $15 y^{4} - 5 y^{3}$

7. $18 a^{3} - 12 a^{2} + 30 a$

8. $12 a^{5} + 20 a^{3} - 4 a$

9. $9 x^{3} y^{2} - 18 x^{2} y^{2} + 27 x y^{2}$

10. $16 a^{5} b^{5} c - 8 a^{3} b^{6} + 24 a^{3} b^{2} c$

Factor by grouping.

11. $x^{2} + 2 x - 5 x - 10$

12. $2 x^{2} - 2 x - 3 x + 3$

13. $x^{3} + 5 x^{2} - 3 x - 15$

14. $x^{3} - 6 x^{2} + x - 6$

15. $x^{3} - x^{2} y - 2 x + 2 y$

16. $a^{2} b^{2} - 2 a^{3} + 6 a b - 3 b^{3}$

Factoring Trinomials of the Form x² + bx + c

Are the following factored correctly? Check by multiplying.

17. $x^{2} + 5 x + 6 = (x + 6) (x - 1)$

18. $x^{2} + 3 x - 10 = (x + 5) (x - 2)$

19. $x^{2} + 6 x + 9 = {(x + 3)}^{2}$

20. $x^{2} - 6 x - 9 = (x - 3) (x + 3)$

Factor.

21. $x^{2} - 13 x - 14$

22. $x^{2} + 13 x + 12$

23. $y^{2} + 10 y + 25$

24. $y^{2} - 20 y + 100$

25. $a^{2} - 8 a - 48$

26. $b^{2} - 18 b + 45$

27. $x^{2} + 2 x + 24$

28. $x^{2} - 10 x - 16$

29. $a^{2} + a b - 2 b^{2}$

30. $a^{2} b^{2} + 5 a b - 50$

Factoring Trinomials of the Form ax² + bx + c

Factor.

31. $5 x^{2} - 27 x - 18$

32. $3 x^{2} - 14 x + 8$

33. $4 x^{2} - 28 x + 49$

34. $9 x^{2} + 48 x + 64$

35. $6 x^{2} - 29 x - 9$

36. $8 x^{2} + 6 x + 9$

37. $60 x^{2} - 65 x + 15$

38. $16 x^{2} - 40 x + 16$

39. $6 x^{3} - 10 x^{2} y + 4 x y^{2}$

40. $10 x^{3} y - 82 x^{2} y^{2} + 16 x y^{3}$

41. $- y^{2} + 9 y + 36$

42. $- a^{2} - 7 a + 98$

43. $16 + 142 x - 18 x^{2}$

44. $45 - 132 x - 60 x^{2}$

Factoring Special Binomials

Factor completely.

45. $x^{2} - 81$

46. $25 x^{2} - 36$

47. $4 x^{2} - 49$

48. $81 x^{2} - 1$

49. $x^{2} - 64 y^{2}$

50. $100 x^{2} y^{2} - 1$

51. $16 x^{4} - y^{4}$

52. $x^{4} - 81 y^{4}$

53. $8 x^{3} - 125$

54. $27 + y^{3}$

55. $54 x^{4} y - 2 x y^{4}$

56. $3 x^{4} y^{2} + 24 x y^{5}$

57. $64 x^{6} - y^{6}$

58. $x^{6} + 1$

General Guidelines for Factoring Polynomials

Factor completely.

59. $8 x^{3} - 4 x^{2} + 20 x$

60. $50 a^{4} b^{4} c + 5 a^{3} b^{5} c^{2}$

61. $x^{3} - 12 x^{2} - x + 12$

62. $a^{3} - 2 a^{2} - 3 a b + 6 b$

63. $- y^{2} - 15 y + 16$

64. $x^{2} - 18 x + 72$

65. $144 x^{2} - 25$

66. $3 x^{4} - 48$

67. $20 x^{2} - 41 x - 9$

68. $24 x^{2} + 14 x - 20$

69. $a^{4} b - 343 a b^{4}$

70. $32 x^{7} y^{2} + 4 x y^{8}$

Solving Equations by Factoring

Solve.

71. $(x - 9) (x + 10) = 0$

72. $- 3 x (x + 8) = 0$

73. $6 (x + 1) (x - 1) = 0$

74. $(x - 12) (x + 4) (2 x - 1) = 0$

75. $x^{2} + 5 x - 50 = 0$

76. $3 x^{2} - 13 x + 4 = 0$

77. $3 x^{2} - 12 = 0$

78. $16 x^{2} - 9 = 0$

79. $(x - 2) (x + 6) = 20$

80. $2 (x - 2) (x + 3) = 7 x - 9$

81. $\frac{5}{2} x^{2} - \frac{20}{3} x = 0$

82. $\frac{2}{3} x^{2} - \frac{5}{12} x + \frac{1}{24} = 0$

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 $h (t) = - 16 t^{2} + 196$ , 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 $25 a^{2} b^{2} c$ , $50 a b^{4}$ , and $35 a^{3} b^{3} c^{2}$ .

2. Determine the missing factor: $24 x^{2} y^{3} - 16 x^{3} y^{2} + 8 x^{2} y = 8 x^{2} y (?)$ .

Factor.

3. $12 x^{5} - 15 x^{4} + 3 x^{2}$

4. $x^{3} - 4 x^{2} - 2 x + 8$

5. $x^{2} - 7 x + 12$

6. $9 x^{2} - 12 x + 4$

7. $x^{2} - 81$

8. $x^{3} + 27 y^{3}$

Factor completely.

9. $x^{3} + 2 x^{2} - 4 x - 8$

10. $x^{4} - 1$

11. $- 6 x^{3} + 20 x^{2} - 6 x$

12. $x^{6} - 1$

Solve.

13. $(2 x + 1) (x - 7) = 0$

14. $3 x (4 x - 3) (x + 1) = 0$

15. $x^{2} - 64 = 0$

16. $x^{2} + 4 x - 12 = 0$

17. $\frac{2}{3} x^{2} + \frac{8}{9} x - \frac{1}{6} = 0$

18. $(x - 5) (x - 3) = - 1$

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

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

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 $h (t) = - 16 t^{2} + 64 t + 36$ . How long will it take the projectile to hit the ground?

Review Exercises Answers

1: $(3 x^{2} - 6 x + 1)$

3: $(2 a^{2} - a + 3)$

5: $11 x (2 x + 1)$

7: $6 a (3 a^{2} - 2 a + 5)$

9: $9 x y^{2} (x^{2} - 2 x + 3)$

11: $(x + 2) (x - 5)$

13: $(x + 5) (x^{2} - 3)$

15: $(x - y) (x^{2} - 2)$

17: No

19: Yes

21: $(x - 14) (x + 1)$

23: ${(y + 5)}^{2}$

25: $(a - 12) (a + 4)$

27: Prime

29: $(a - b) (a + 2 b)$

31: $(5 x + 3) (x - 6)$

33: ${(2 x - 7)}^{2}$

35: Prime

37: $5 (3 x - 1) (4 x - 3)$

39: $2 x (3 x - 2 y) (x - y)$

41: $- 1 (y - 12) (y + 3)$

43: $- 2 (9 x + 1) (x - 8)$

45: $(x + 9) (x - 9)$

47: $(2 x + 7) (2 x - 7)$

49: $(x + 8 y) (x - 8 y)$

51: $(4 x^{2} + y^{2}) (2 x + y) (2 x - y)$

53: $(2 x - 5) (4 x^{2} + 10 x + 25)$

55: $2 x y (3 x - y) (9 x^{2} + 3 x y + y^{2})$

57: $(2 x + y) (4 x^{2} - 2 x y + y^{2}) (2 x - y) (4 x^{2} + 2 x y + y^{2})$

59: $4 x (2 x^{2} - x + 5)$

61: $(x - 12) (x + 1) (x - 1)$

63: $- 1 (y + 16) (y - 1)$

65: $(12 x + 5) (12 x - 5)$

67: $(4 x - 9) (5 x + 1)$

69: $a b (a - 7 b) (a^{2} + 7 a b + 49 b^{2})$

71: 9, −10

73: −1, 1

75: −10, 5

77: ±2

79: −8, 4

81: 0, 8/3

83: $x^{2} + x - 42 = 0$

85: $18 x^{2} - 7 x - 1 = 0$

87: {8, 12} or {−6, −16}

89: 16 feet

91: Length: 14 centimeters; width: 5 centimeters

Sample Exam Answers

1: $5 a b^{2}$

3: $3 x^{2} (4 x^{3} - 5 x^{2} + 1)$

5: $(x - 4) (x - 3)$

7: $(x + 9) (x - 9)$

9: ${(x + 2)}^{2} (x - 2)$

11: $- 2 x (3 x - 1) (x - 3)$

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