# 6.3E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

## 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?