8.2E: Exercises
- Page ID
- 49946
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Practice Makes Perfect
Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
1. \(10p^3q,12pq^2\)
- Answer
-
\(2pq\)
2. \(8a^2b^3,10ab^2\)
3. \(12m^2n^3,30m^5n^3\)
- Answer
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\(6m^2n^3\)
4. \(28x^2y^4,42x^4y^4\)
5. \(10a^3,12a^2,14a\)
- Answer
-
\(2a\)
6. \(20y^3,28y^2,40y\)
7. \(35x^3y^2,10x^4y,5x^5y^3\)
- Answer
-
\(5x^3y\)
8. \(27p^2q^3,45p^3q^4,9p^4q^3\)
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
9. \(6m+9\)
- Answer
-
\(3(2m+3)\)
10. \(14p+35\)
11. \(9n−63\)
- Answer
-
\(9(n−7)\)
12. \(45b−18\)
13. \(3x^2+6x−9\)
- Answer
-
\(3(x^2+2x−3)\)
14. \(4y^2+8y−4\)
15. \(8p^2+4p+2\)
- Answer
-
\(2(4p^2+2p+1)\)
16. \(10q^2+14q+20\)
17. \(8y^3+16y^2\)
- Answer
-
\(8y^2(y+2)\)
18. \(12x^3−10x\)
19. \(5x^3−15x^2+20x\)
- Answer
-
\(5x(x^2−3x+4)\)
20. \(8m^2−40m+16\)
21. \(24x^3−12x^2+15x\)
- Answer
-
\(3x(8x^2−4x+5)\)
22. \(24y^3−18y^2−30y\)
23. \(12xy^2+18x^2y^2−30y^3\)
- Answer
-
\(6y^2(2x+3x^2−5y)\)
24. \(21pq^2+35p^2q^2−28q^3\)
25. \(20x^3y−4x^2y^2+12xy^3\)
- Answer
-
\(4xy(5x^2−xy+3y^2)\)
26. \(24a^3b+6a^2b^2−18ab^3\)
27. \(−2x−4\)
- Answer
-
\(−2(x+4)\)
28. \(−3b+12\)
29. \(−2x^3+18x^2−8x\)
- Answer
-
\(−2x(x^2−9x+4)\)
30. \(−5y^3+35y^2−15y\)
31. \(−4p^3q−12p^2q^2+16pq^2\)
- Answer
-
\(−4pq(p^2+3pq−4q)\)
32. \(−6a^3b−12a^2b^2+18ab^2\)
33. \(5x(x+1)+3(x+1)\)
- Answer
-
\((x+1)(5x+3)\)
34. \(2x(x−1)+9(x−1)\)
35. \(3b(b−2)−13(b−2)\)
- Answer
-
\((b−2)(3b−13)\)
36. \(6m(m−5)−7(m−5)\)
Factor by Grouping
In the following exercises, factor by grouping.
37. \(ab+5a+3b+15\)
- Answer
-
\((b+5)(a+3)\)
38. \(cd+6c+4d+24\)
39. \(8y^2+y+40y+5\)
- Answer
-
\((y+5)(8y+1)\)
40. \(6y^2+7y+24y+28\)
41. \(uv−9u+2v−18\)
- Answer
-
\((u+2)(v−9)\)
42. \(pq−10p+8q−80\)
43. \(u^2−u+6u−6\)
- Answer
-
\((u−1)(u+6)\)
44. \(x^2−x+4x−4\)
45. \(9p^2−15p+12p−20\)
- Answer
-
\((3p−5)(3p+4)\)
46. \(16q^2+20q−28q−35\)
47. \(mn−6m−4n+24\)
- Answer
-
\((n−6)(m−4)\)
48. \(r^2−3r−r+3\)
49. \(2x^2−14x−5x+35\)
- Answer
-
\((x−7)(2x−5)\)
50. \(4x^2−36x−3x+27\)
Mixed Practice
In the following exercises, factor.
51. \(−18xy^2−27x^2y\)
- Answer
-
\(−9xy(3x+2y)\)
52. \(−4x^3y^5−x^2y^3+12xy^4\)
53. \(3x^3−7x^2+6x−14\)
- Answer
-
\((x^2+2)(3x−7)\)
54. \(x^3+x^2−x−1\)
55. \(x^2+xy+5x+5y\)
- Answer
-
\((x+y)(x+5)\)
56. \(5x^3−3x^2+5x−3\)
Writing Exercises
57. What does it mean to say a polynomial is in factored form?
- Answer
-
Answers will vary.
58. How do you check result after factoring a polynomial?
59. The greatest common factor of \(36\) and \(60\) is \(12\). Explain what this means.
- Answer
-
Answers will vary.
60. What is the GCF of \(y^4,\space y^5\), and \(y^{10}\)? Write a general rule that tells you how to find the GCF of \(y^a,\space y^b\), and \(y^c\).
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.