6.4E: Exercises

Last updated
Save as PDF

Page ID: 30320

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

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

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

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

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

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

5. $100x^2−20x+1$

Answer: $(10x−1)^2$

6. $64z^2−16z+1$

7. $25n^2−120n+144$

Answer: $(5n−12)^2$

8. $4p^2−52p+169$

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

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

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

11. $100y^2−52y+1$

Answer: $(50y−1)(2y−1)$

12. $64m^2−34m+1$

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

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

14. $64x^2y−96xy+36y$

15. $75u^4−30u^3v+3u^2v^2$

Answer: $3u^2(5u−v)^2$

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. $25v^2−1$

Answer: $(5v−1)(5v+1)$

18. $169q^2−1$

19. $4−49x^2$

Answer: $(7x−2)(7x+2)$

20. $121−25s^2$

21. $6p^2q^2−54p^2$

Answer: $6p^2(q−3)(q+3)$

22. $98r^3−72r$

23. $24p^2+54$

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

24. $20b^2+140$

25. $121x^2−144y^2$

Answer: $(11x−12y)(11x+12y)$

26. $49x^2−81y^2$

27. $169c^2−36d^2$

Answer: $(13c−6d)(13c+6d)$

28. $36p^2−49q^2$

29. $16z^4−1$

Answer: $(2z−1)(2z+1)(4z^2+1)$

30. $m^4−n^4$

31. $162a^4b^2−32b^2$

Answer: $2b^2(3a−2)(3a+2)(9a^2+4)$

32. $48m^4n^2−243n^2$

33. $x^2−16x+64−y^2$

Answer: $(x−8−y)(x−8+y)$

34. $p^2+14p+49−q^2$

35. $a^2+6a+9−9b^2$

Answer: $(a+3−3b)(a+3+3b)$

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. $x^3+125$

Answer: $(x+5)(x^2−5x+25)$

38. $n^6+512$

39. $z^6−27$

Answer: $(z^2−3)(z^4+3z^2+9)$

40. $v^3−216$

41. $8−343t^3$

Answer: $(2−7t)(4+14t+49t^2)$

42. $125−27w^3$

43. $8y^3−125z^3$

Answer: $(2y−5z)(4y^2+10yz+25z^2)$

44. $27x^3−64y^3$

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

Answer: $(6a+5b)(36a^2−30ab+25b^2)$

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

47. $7k^3+56$

Answer: $7(k+2)(k^2−2k+4)$

48. $6x^3−48y^3$

49. $2x^2−16x^2y^3$

Answer: $2x^2(1−2y)(1+2y+4y^2)$

50. $−2x^3y^2−16y^5$

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

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

52. $(x+4)^3−27x^3$

53. $(y−5)^3−64y^3$

Answer: $−(3y+5)(21y^2−30y+25)$

54. $(y−5)^3+125y^3$

Mixed Practice

In the following exercises, factor completely.

55. $64a^2−25$

Answer: $(8a−5)(8a+5)$

56. $121x^2−144$

57. $27q^2−3$

Answer: $3(3q−1)(3q+1)$

58. $4p^2−100$

59. $16x^2−72x+81$

Answer: $(4x−9)^2$

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

61. $8p^2+2$

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

62. $81x^2+169$

63. $125−8y^3$

Answer: $(5−2y)(25+10y+4y^2)$

64. $27u^3+1000$

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

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

66. $48q^3−24q^2+3q$

67. $x^2−10x+25−y^2$

Answer: $(x+y−5)(x−y−5)$

68. $x^2+12x+36−y^2$

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

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

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?

Answer: Answers will vary.

72. How do you recognize the binomial squares pattern?

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

Answer: Answers will vary.

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

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

$This table has 4 columns 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: factor perfect square trinomials, factor differences of squares, factor sums and differences of cubes. The remaining columns are blank.$

b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Search

Text Color

Text Size

Margin Size

Font Type