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# 6.3E: Exercises

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## Practice Makes Perfect

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

1. $$16y^2+24y+9$$

$$(4y+3)^2$$

2. $$25v^2+20v+4$$

3. $$36s^2+84s+49$$

$$(6s+7)^2$$

4. $$49s^2+154s+121$$

5. $$100x^2−20x+1$$

$$(10x−1)^2$$

6. $$64z^2−16z+1$$

7. $$25n^2−120n+144$$

$$(5n−12)^2$$

8. $$4p^2−52p+169$$

9. $$49x^2+28xy+4y^2$$

$$(7x+2y)^2$$

10. $$25r^2+60rs+36s^2$$

11. $$100y^2−52y+1$$

$$(50y−1)(2y−1)$$

12. $$64m^2−34m+1$$

13. $$10jk^2+80jk+160j$$

$$10j(k+4)^2$$

14. $$64x^2y−96xy+36y$$

15. $$75u^4−30u^3v+3u^2v^2$$

$$3u^2(5u−v)^2$$

16. $$90p^4+300p^4q+250p^2q^2$$

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

17. $$25v^2−1$$

$$(5v−1)(5v+1)$$

18. $$169q^2−1$$

19. $$4−49x^2$$

$$(7x−2)(7x+2)$$

20. $$121−25s^2$$

21. $$6p^2q^2−54p^2$$

$$6p^2(q−3)(q+3)$$

22. $$98r^3−72r$$

23. $$24p^2+54$$

$$6(4p^2+9)$$

24. $$20b^2+140$$

25. $$121x^2−144y^2$$

$$(11x−12y)(11x+12y)$$

26. $$49x^2−81y^2$$

27. $$169c^2−36d^2$$

$$(13c−6d)(13c+6d)$$

28. $$36p^2−49q^2$$

29. $$16z^4−1$$

$$(2z−1)(2z+1)(4z^2+1)$$

30. $$m^4−n^4$$

31. $$162a^4b^2−32b^2$$

$$2b^2(3a−2)(3a+2)(9a^2+4)$$

32. $$48m^4n^2−243n^2$$

33. $$x^2−16x+64−y^2$$

$$(x−8−y)(x−8+y)$$

34. $$p^2+14p+49−q^2$$

35. $$a^2+6a+9−9b^2$$

$$(a+3−3b)(a+3+3b)$$

36. $$m^2−6m+9−16n^2$$

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

37. $$x^3+125$$

$$(x+5)(x^2−5x+25)$$

38. $$n^6+512$$

39. $$z^6−27$$

$$(z^2−3)(z^4+3z^2+9)$$

40. $$v^3−216$$

41. $$8−343t^3$$

$$(2−7t)(4+14t+49t^2)$$

42. $$125−27w^3$$

43. $$8y^3−125z^3$$

$$(2y−5z)(4y^2+10yz+25z^2)$$

44. $$27x^3−64y^3$$

45. $$216a^3+125b^3$$

$$(6a+5b)(36a^2−30ab+25b^2)$$

46. $$27y^3+8z^3$$

47. $$7k^3+56$$

$$7(k+2)(k^2−2k+4)$$

48. $$6x^3−48y^3$$

49. $$2x^2−16x^2y^3$$

$$2x^2(1−2y)(1+2y+4y^2)$$

50. $$−2x^3y^2−16y^5$$

51. $$(x+3)^3+8x^3$$

$$9(x+1)(x^2+3)$$

52. $$(x+4)^3−27x^3$$

53. $$(y−5)^3−64y^3$$

$$−(3y+5)(21y^2−30y+25)$$

54. $$(y−5)^3+125y^3$$

Mixed Practice

In the following exercises, factor completely.

55. $$64a^2−25$$

$$(8a−5)(8a+5)$$

56. $$121x^2−144$$

57. $$27q^2−3$$

$$3(3q−1)(3q+1)$$

58. $$4p^2−100$$

59. $$16x^2−72x+81$$

$$(4x−9)^2$$

60. $$36y^2+12y+1$$

61. $$8p^2+2$$

$$2(4p^2+1)$$

62. $$81x^2+169$$

63. $$125−8y^3$$

$$(5−2y)(25+10y+4y^2)$$

64. $$27u^3+1000$$

65. $$45n^2+60n+20$$

$$5(3n+2)^2$$

66. $$48q^3−24q^2+3q$$

67. $$x^2−10x+25−y^2$$

$$(x+y−5)(x−y−5)$$

68. $$x^2+12x+36−y^2$$

69. $$(x+1)^3+8x^3$$

$$(3x+1)(3x^2+1)$$

70. $$(y−3)^3−64y^3$$

## Writing Exercises

71. Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?

72. How do you recognize the binomial squares pattern?

73. Explain why $$n^2+25\neq (n+5)^2$$. Use algebra, words, or pictures.

74. Maribel factored $$y^2−30y+81$$ as $$(y−9)^2$$. Was she right or wrong? How do you know? 