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Mathematics LibreTexts

6.2E: Exercises

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    30863
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    Practice Makes Perfect

    Factor Trinomials of the Form \(x^2+bx+c\)

    In the following exercises, factor each trinomial of the form \(x^2+bx+c\).

    1. \(p^2+11p+30\)

    Answer

    \((p+5)(p+6)\)

    2. \(w^2+10x+21\)

    3. \(n^2+19n+48\)

    Answer

    \((n+3)(n+16)\)

    4. \(b^2+14b+48\)

    5. \(a^2+25a+100\)

    Answer

    \((a+5)(a+20)\)

    6. \(u^2+101u+100\)

    7. \(x^2−8x+12\)

    Answer

    \((x−2)(x−6)\)

    8. \(q^2−13q+36\)

    9. \(y^2−18y+45\)

    Answer

    \((y−3)(y−15)\)

    10. \(m^2−13m+30\)

    11. \(x^2−8x+7\)

    Answer

    \((x−1)(x−7)\)

    12. \(y^2−5y+6\)

    13. \(5p−6+p^2\)

    Answer

    \((p−1)(p+6)\)

    14. \(6n−7+n^2\)

    15. \(8−6x+x^2\)

    Answer

    \((x−4)(x−2)\)

    16. \(7x+x^2+6\)

    17. \(x^2−12−11x\)

    Answer

    \((x−12)(x+1)\)

    18. \(−11−10x+x^2\)

    In the following exercises, factor each trinomial of the form \(x^2+bxy+cy^2\).

    19. \(x^2−2xy−80y^2\)

    Answer

    \((x+8y)(x−10y)\)

    20. \(p^2−8pq−65q^2\)

    21. \(m^2−64mn−65n^2\)

    Answer

    \((m+n)(m−65n)\)

    22. \(p^2−2pq−35q^2\)

    23. \(a^2+5ab−24b^2\)

    Answer

    \((a+8b)(a−3b)\)

    24. \(r^2+3rs−28s^2\)

    25. \(x^2−3xy−14y^2\)

    Answer

    Prime

    26. \(u^2−8uv−24v^2\)

    27. \(m^2−5mn+30n^2\)

    Answer

    Prime

    28. \(c^2−7cd+18d^2\)

    Factor Trinomials of the Form \(ax^2+bx+c\) Using Trial and Error

    In the following exercises, factor completely using trial and error.

    29. \(p^3−8p^2−20p\)

    Answer

    \(p(p−10)(p+2)\)

    30. \(q^3−5q^2−24q\)

    31. \(3m^3−21m^2+30m\)

    Answer

    \(3m(m−5)(m−2)\)

    32. \(11n^3−55n^2+44n\)

    33. \(5x^4+10x^3−75x^2\)

    Answer

    \(5x^2(x−3)(x+5)\)

    34. \(6y^4+12y^3−48y^2\)

    35. \(2t^2+7t+5\)

    Answer

    \((2t+5)(t+1)\)

    36. \(5y^2+16y+11\)

    37. \(11x^2+34x+3\)

    Answer

    \((11x+1)(x+3)\)

    38. \(7b^2+50b+7\)

    39. \(4w^2−5w+1\)

    Answer

    \((4w−1)(w−1)\)

    40. \(5x^2−17x+6\)

    41. \(4q^2−7q−2\)

    Answer

    \((4q+1)(q−2)\)

    42. \(10y^2−53y−111\)

    43. \(6p^2−19pq+10q^2\)

    Answer

    \((2p−5q)(3p−2q)\)

    44. \(21m^2−29mn+10n^2\)

    45. \(4a^2+17ab−15b^2\)

    Answer

    \((4a−3b)(a+5b)\)

    46. \(6u^2+5uv−14v^2\)

    47. \(−16x^2−32x−16\)

    Answer

    \(−16(x+1)(x+1)\)

    48. \(−81a^2+153a+18\)

    49. \(−30q^3−140q^2−80q\)

    Answer

    \(10q(3q+2)(q+4)\)

    50. \(−5y^3−30y^2+35y\)

    Factor Trinomials of the Form \(ax^2+bx+c\) using the ‘ac’ Method

    In the following exercises, factor using the ‘ac’ method.

    51. \(5n^2+21n+4\)

    Answer

    \((5n+1)(n+4)\)

    52. \(8w^2+25w+3\)

    53. \(4k^2−16k+15\)

    Answer

    \((2k−3)(2k−5)\)

    54. \(5s^2−9s+4\)

    55. \(6y^2+y−15\)

    Answer

    \((3y+5)(2y−3)\)

    56. \(6p^2+p−22\)

    57. \(2n^2−27n−45\)

    Answer

    \((2n+3)(n−15)\)

    58. \(12z^2−41z−11\)

    59. \(60y^2+290y−50\)

    Answer

    \(10(6y−1)(y+5)\)

    60. \(6u^2−46u−16\)

    61. \(48z^3−102z^2−45z\)

    Answer

    \(3z(8z+3)(2z−5)\)

    62. \(90n^3+42n^2−216n\)

    63. \(16s^2+40s+24\)

    Answer

    \(8(2s+3)(s+1)\)

    64. \(24p^2+160p+96\)

    65. \(48y^2+12y−36\)

    Answer

    \(12(4y−3)(y+1)\)

    66. \(30x^2+105x−60\)

    Factor Using Substitution

    In the following exercises, factor using substitution.

    67. \(x^4−x^2−12\)

    Answer

    \((x^2+3)(x^2−4)\)

    68. \(x^4+2x^2−8\)

    69. \(x^4−3x^2−28\)

    Answer

    \((x^2−7)(x^2+4)\)

    70. \(x^4−13x^2−30\)

    71. \((x−3)^2−5(x−3)−36\)

    Answer

    \((x−12)(x+1)\)

    72. \((x−2)^2−3(x−2)−54\)

    73. \((3y−2)^2−(3y−2)−2\)

    Answer

    \((3y−4)(3y−1)\)

    74. \((5y−1)^2−3(5y−1)−18\)

    Mixed Practice

    In the following exercises, factor each expression using any method.

    75. \(u^2−12u+36\)

    Answer

    \((u−6)(u−6)\)

    76. \(x^2−14x−32\)

    77. \(r^2−20rs+64s^2\)

    Answer

    \((r−4s)(r−16s)\)

    78. \(q^2−29qr−96r^2\)

    79. \(12y^2−29y+14\)

    Answer

    \((4y−7)(3y−2)\)

    80. \(12x^2+36y−24z\)

    81. \(6n^2+5n−4\)

    Answer

    \((2n−1)(3n+4)\)

    82. \(3q^2+6q+2\)

    83. \(13z^2+39z−26\)

    Answer

    \(13(z^2+3z−2)\)

    84. \(5r^2+25r+30\)

    85. \(3p^2+21p\)

    Answer

    \(3p(p+7)\)

    86. \(7x^2−21x\)

    87. \(6r^2+30r+36\)

    Answer

    \(6(r+2)(r+3)\)

    88. \(18m^2+15m+3\)

    89. \(24n^2+20n+4\)

    Answer

    \(4(2n+1)(3n+1)\)

    90. \(4a^2+5a+2\)

    91. \(x^4−4x^2−12\)

    Answer

    \((x^2+2)(x^2−6)\)

    92. \(x^4−7x^2−8\)

    93. \((x+3)^2−9(x+3)−36\)

    Answer

    \((x−9)(x+6)\)

    94. \((x+2)^2−25(x+2)−54\)

    Writing Exercises

    95. Many trinomials of the form \(x^2+bx+c\) factor into the product of two binomials \((x+m)(x+n)\). Explain how you find the values of \(m\) and \(n\).

    Answer

    Answers will vary.

    96. Tommy factored \(x^2−x−20\) as \((x+5)(x−4)\). Sara factored it as \((x+4)(x−5)\). Ernesto factored it as \((x−5)(x−4)\). Who is correct? Explain why the other two are wrong.

    97. List, in order, all the steps you take when using the “\(ac\)” method to factor a trinomial of the form \(ax^2+bx+c\).

    Answer

    Answers will vary.

    98. How is the “\(ac\)” method similar to the “undo FOIL” method? How is it different?

    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, 4 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 trinomials of the form x squared plus bx plus c, factor trinomials of the form a x squared plus b x plus c using trial and error, factor trinomials of the form a x squared plus bx plus c with using the “ac” method, factor using substitution.

    b. After reviewing this checklist, what will you do to become confident for all objectives?