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6.3E: Exercises

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Practice Makes Perfect

Factor Trinomials of the Form $$x^2+bx+c$$

In the following exercises, factor each trinomial of the form $$x^2+bx+c$$.

1. $$p^2+11p+30$$

$$(p+5)(p+6)$$

2. $$w^2+10w+21$$

3. $$n^2+19n+48$$

$$(n+3)(n+16)$$

4. $$b^2+14b+48$$

5. $$a^2+25a+100$$

$$(a+5)(a+20)$$

6. $$u^2+101u+100$$

7. $$x^2−8x+12$$

$$(x−2)(x−6)$$

8. $$q^2−13q+36$$

9. $$y^2−18y+45$$

$$(y−3)(y−15)$$

10. $$m^2−13m+30$$

11. $$x^2−8x+7$$

$$(x−1)(x−7)$$

12. $$y^2−5y+6$$

13. $$5p−6+p^2$$

$$(p−1)(p+6)$$

14. $$6n−7+n^2$$

15. $$8−6x+x^2$$

$$(x−4)(x−2)$$

16. $$7x+x^2+6$$

17. $$x^2−12−11x$$

$$(x−12)(x+1)$$

18. $$−11−10x+x^2$$

In the following exercises, factor each trinomial of the form $$x^2+bxy+cy^2$$.

19. $$x^2−2xy−80y^2$$

$$(x+8y)(x−10y)$$

20. $$p^2−8pq−65q^2$$

21. $$m^2−64mn−65n^2$$

$$(m+n)(m−65n)$$

22. $$p^2−2pq−35q^2$$

23. $$a^2+5ab−24b^2$$

$$(a+8b)(a−3b)$$

24. $$r^2+3rs−28s^2$$

25. $$x^2−3xy−14y^2$$

Prime

26. $$u^2−8uv−24v^2$$

27. $$m^2−5mn+30n^2$$

Prime

28. $$c^2−7cd+18d^2$$

Factor Trinomials of the Form $$ax^2+bx+c$$ Using Trial and Error

In the following exercises, factor completely using trial and error.

29. $$p^3−8p^2−20p$$

$$p(p−10)(p+2)$$

30. $$q^3−5q^2−24q$$

31. $$3m^3−21m^2+30m$$

$$3m(m−5)(m−2)$$

32. $$11n^3−55n^2+44n$$

33. $$5x^4+10x^3−75x^2$$

$$5x^2(x−3)(x+5)$$

34. $$6y^4+12y^3−48y^2$$

35. $$2t^2+7t+5$$

$$(2t+5)(t+1)$$

36. $$5y^2+16y+11$$

37. $$11x^2+34x+3$$

$$(11x+1)(x+3)$$

38. $$7b^2+50b+7$$

39. $$4w^2−5w+1$$

$$(4w−1)(w−1)$$

40. $$5x^2−17x+6$$

41. $$4q^2−7q−2$$

$$(4q+1)(q−2)$$

42. $$10y^2−53y−111$$

43. $$6p^2−19pq+10q^2$$

$$(2p−5q)(3p−2q)$$

44. $$21m^2−29mn+10n^2$$

45. $$4a^2+17ab−15b^2$$

$$(4a−3b)(a+5b)$$

46. $$6u^2+5uv−14v^2$$

47. $$−16x^2−32x−16$$

$$−16(x+1)(x+1)$$

48. $$−81a^2+153a+18$$

49. $$−30q^3−140q^2−80q$$

$$- 10q(3q+2)(q+4)$$

50. $$−5y^3−30y^2+35y$$

Factor Trinomials of the Form $$ax^2+bx+c$$ using the ‘ac’ Method

In the following exercises, factor using the ‘ac’ method.

51. $$5n^2+21n+4$$

$$(5n+1)(n+4)$$

52. $$8w^2+25w+3$$

53. $$4k^2−16k+15$$

$$(2k−3)(2k−5)$$

54. $$5s^2−9s+4$$

55. $$6y^2+y−15$$

$$(3y+5)(2y−3)$$

56. $$6p^2+p−22$$

57. $$2n^2−27n−45$$

$$(2n+3)(n−15)$$

58. $$12z^2−41z−11$$

59. $$60y^2+290y−50$$

$$10(6y−1)(y+5)$$

60. $$6u^2−46u−16$$

61. $$48z^3−102z^2−45z$$

$$3z(8z+3)(2z−5)$$

62. $$90n^3+42n^2−216n$$

63. $$16s^2+40s+24$$

$$8(2s+3)(s+1)$$

64. $$24p^2+160p+96$$

65. $$48y^2+12y−36$$

$$12(4y−3)(y+1)$$

66. $$30x^2+105x−60$$

Factor Using Substitution

In the following exercises, factor using substitution.

67. $$x^4−x^2−12$$

$$(x^2+3)(x^2−4)$$

68. $$x^4+2x^2−8$$

69. $$x^4−3x^2−28$$

$$(x^2−7)(x^2+4)$$

70. $$x^4−13x^2−30$$

71. $$(x−3)^2−5(x−3)−36$$

$$(x−12)(x+1)$$

72. $$(x−2)^2−3(x−2)−54$$

73. $$(3y−2)^2−(3y−2)−2$$

$$(3y−4)(3y−1)$$

74. $$(5y−1)^2−3(5y−1)−18$$

Mixed Practice

In the following exercises, factor each expression using any method.

75. $$u^2−12u+36$$

$$(u−6)(u−6)$$

76. $$x^2−14x−32$$

77. $$r^2−20rs+64s^2$$

$$(r−4s)(r−16s)$$

78. $$q^2−29qr−96r^2$$

79. $$12y^2−29y+14$$

$$(4y−7)(3y−2)$$

80. $$12x^2+36y−24z$$

81. $$6n^2+5n−4$$

$$(2n−1)(3n+4)$$

82. $$3q^2+6q+2$$

83. $$13z^2+39z−26$$

$$13(z^2+3z−2)$$

84. $$5r^2+25r+30$$

85. $$3p^2+21p$$

$$3p(p+7)$$

86. $$7x^2−21x$$

87. $$6r^2+30r+36$$

$$6(r+2)(r+3)$$

88. $$18m^2+15m+3$$

89. $$24n^2+20n+4$$

$$4(2n+1)(3n+1)$$

90. $$4a^2+5a+2$$

91. $$x^4−4x^2−12$$

$$(x^2+2)(x^2−6)$$

92. $$x^4−7x^2−8$$

93. $$(x+3)^2−9(x+3)−36$$

$$(x−9)(x+6)$$

94. $$(x+2)^2−25(x+2)−54$$

Writing Exercises

95. Many trinomials of the form $$x^2+bx+c$$ factor into the product of two binomials $$(x+m)(x+n)$$. Explain how you find the values of $$m$$ and $$n$$.

96. Tommy factored $$x^2−x−20$$ as $$(x+5)(x−4)$$. Sara factored it as $$(x+4)(x−5)$$. Ernesto factored it as $$(x−5)(x−4)$$. Who is correct? Explain why the other two are wrong.

97. List, in order, all the steps you take when using the “$$ac$$” method to factor a trinomial of the form $$ax^2+bx+c$$.

98. How is the “$$ac$$” method similar to the “undo FOIL” method? How is it different?