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Mathematics LibreTexts

6.4E: Exercises

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    30867
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    Practice Makes Perfect

    Recognize and Use the Appropriate Method to Factor a Polynomial Completely

    In the following exercises, factor completely.

    1. \(2n^2+13n−7\)

    Answer

    \((2n−1)(n+7)\)

    2. \(8x^2−9x−3\)

    3. \(a^5+9a^3\)

    Answer

    \(a^3(a^2+9)\)

    4. \(75m^3+12m\)

    5. \(121r^2−s^2\)

    Answer

    \((11r−s)(11r+s)\)

    6. \(49b^2−36a^2\)

    7. \(8m^2−32\)

    Answer

    \(8(m−2)(m+2)\)

    8. \(36q^2−100\)

    9. \(25w^2−60w+36\)

    Answer

    \((5w−6)^2\)

    10. \(49b^2−112b+64\)

    11. \(m^2+14mn+49n^2\)

    Answer

    \((m+7n)^2\)

    12. \(64x^2+16xy+y^2\)

    13. \(7b^2+7b−42\)

    Answer

    \(7(b+3)(b−2)\)

    14. \(30n^2+30n+72\)

    15. \(3x^4y−81xy\)

    Answer

    \(3(x−3)(x^2+3x+9)\)

    16. \(4x^5y−32x^2y\)

    17. \(k^4−16\)

    Answer

    \((k−2)(k+2)(k^2+4)\)

    18. \(m^4−81\)

    19. \(5x5y^2−80xy^2\)

    Answer

    \(5xy^2(x^2+4)(x+2)(x−2)\)

    20. \(48x^5y^2−243xy^2\)

    21. \(15pq−15p+12q−12\)

    Answer

    \(3(5p+4)(q−1)\)

    22. \(12ab−6a+10b−5\)

    23. \(4x^2+40x+84\)

    Answer

    \(4(x+3)(x+7)\)

    24. \(5q^2−15q−90\)

    25. \(4u^5v+4u^2v^3\)

    Answer

    \(u^2(u+1)(u^2−u+1)\)

    26. \(5m^4n+320mn^4\)

    27. \(4c^2+20cd+81d^2\)

    Answer

    prime

    28. \(25x^2+35xy+49y^2\)

    29. \(10m^4−6250\)

    Answer

    \(10(m−5)(m+5)(m^2+25)\)

    30. \(3v^4−768\)

    31. \(36x^2y+15xy−6y\)

    Answer

    \(3y(3x+2)(4x−1)\)

    32. \(60x^2y−75xy+30y\)

    33. \(8x^3−27y^3\)

    Answer

    \((2x−3y)(4x^2+6xy+9y^2)\)

    34. \(64x^3+125y^3\)

    35. \(y^6−1\)

    Answer

    \((y+1)(y−1)(y^2−y+1)\)

    36. \(y^6+1\)

    37. \(9x^2−6xy+y^2−49\)

    Answer

    \((3x−y+7)(3x−y−7)\)

    38. \(16x^2−24xy+9y^2−64\)

    39. \((3x+1)^2−6(3x−1)+9\)

    Answer

    \((3x−2)2\)

    40. \((4x−5)^2−7(4x−5)+12\)

    Writing Exercises

    41. Explain what it mean to factor a polynomial completely.

    Answer

    Answers will vary.

    42. The difference of squares \(y^4−625\) can be factored as \((y^2−25)(y^2+25)\). But it is not completely factored. What more must be done to completely factor.

    43. Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, ‘ac’ method, special products) which is the easiest for you? Which is the hardest? Explain your answers.

    Answer

    Answers will vary.

    44. Create three factoring problems that would be good test questions to measure your knowledge of factoring. Show the solutions.

    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, 1 row 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 statement: recognize and use the appropriate method to factor a polynomial completely. The remaining columns are blank.

    b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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