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Mathematics LibreTexts

6.1E: Exercises

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    30861
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    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

    \(6m^2n^3\)

    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. \(uv−9u+2v−18uv\)

    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.

    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: find the greatest common factor of 2 or more expressions, factor the greatest common factor from a polynomial, factor by grouping. The remaining columns are blank.

    ⓑ 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.

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