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Mathematics LibreTexts

6.3E: Exercises

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    30865
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    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?

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