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2.EB: Exercises for Factoring Polynomial Expressions and Solving Polynomial Equations

  • Page ID
    95200
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    Greatest Common Factor and Factor by Grouping

    In the following exercises, find the greatest common factor.

    1. \(12a^2b^3,\space 15ab^2\)
    Answer

    \(3ab^2\)

    2. \(12m^2n^3,42m^5n^3\)

    3. \(15y^3,\space 21y^2,\space 30y\)

    Answer

    \(3y\)

    4. \(45x^3y^2,\space 15x^4y,\space 10x^5y^3\)

    In the following exercises, factor the greatest common factor from each polynomial.

    5. \(35y+84\)

    Answer

    \(7(5y+12)\)

    6. \(6y^2+12y−6\)

    7. \(18x^3−15x\)

    Answer

    \(3x(6x^2−5)\)

    8. \(15m^4+6m^2n\)

    9. \(4x^3−12x^2+16x\)

    Answer

    \(4x(x^2−3x+4)\)

    10. \(−3x+24\)

    11. \(−3x^3+27x^2−12x\)

    Answer

    \(−3x(x^2−9x+4)\)

    12. \(3x(x−1)+5(x−1)\)

    In the following exercises, factor by grouping.

    13. \(ax−ay+bx−by\)

    Answer

    \((a+b)(x−y)\)

    14. \(x^2y−xy^2+2x−2y\)

    15. \(x^2+7x−3x−21\)

    Answer

    \((x−3)(x+7)\)

    16. \(4x^2−16x+3x−12\)

    17. \(m^3+m^2+m+1\)

    Answer

    \((m^2+1)(m+1)\)

    18. \(5x−5y−y+x\)

    Factor \(ax^2+bx+c \text{ when } a=1\)

    In the following exercises, factor each trinomial completely.

    1. \(a^2+14a+33\)

    Answer

    \((a+3)(a+11)\)

    2. \(k^2−16k+60\)

    3. \(m^2+3m−54\)

    Answer

    \((m+9)(m−6)\)

    4. \(x^2−3x−10\)

    5. \(x^2+12xy+35y^2\)

    Answer

    \((x+5y)(x+7y)\)

    6. \(r^2+3rs−28s^2\)

    7. \(a^2+4ab−21b^2\)

    Answer

    \((a+7b)(a−3b)\)

    8. \(p^2−5pq−36q^2\)

    9. \(m^2−5mn+30n^2\)

    Answer

    Prime

    10. \(x^3+5x^2−24x\)

    11. \(3y^3−21y^2+30y\)

    Answer

    \(3y(y−5)(y−2)\)

    12. \(5x^4+10x^3−75x^2\)

    Factor \(ax^2+bx+c \text{ when } a\neq 1\)

    In the following exercises, factor each trinomial completely.

    1. \(5y^2+14y+9\)

    Answer

    \((5y+9)(y+1)\)

    2. \(8x^2+25x+3\)

    3. \(10y^2−53y−11\)

    Answer

    \((5y+1)(2y−11)\)

    4. \(6p^2−19pq+10q^2\)

    5. \(−81a^2+153a+18\)

    Answer

    \(−9(9a−1)(a+2)\)

    6. \(2x^2+9x+4\)

    7. \(18a^2−9a+1\)

    Answer

    \((3a−1)(6a−1)\)

    8. \(15p^2+2p−8\)

    9. \(15x^2+6x−2\)

    Answer

    \((3x−1)(5x+2)\)

    10. \(8a^2+32a+24\)

    11. \(3x^2+3x−36\)

    Answer

    \(3(x+4)(x−3)\)

    12. \(48y^2+12y−36\)

    13. \(18a^2−57a−21\)

    Answer

    \(3(2a−7)(3a+1)\)

    14. \(3n^4−12n^3−96n^2\)

    15. \(x^4−13x^2−30\)

    Answer

    \((x^2−15)(x^2+2)\)

    16. \((x−3)^2−5(x−3)−36\)

    Factoring Special Products

    In the following exercises, factor completely.

    1. \(25x^2+30x+9\)

    Answer

    \((5x+3)^2\)

    2. \(36a^2−84ab+49b^2\)

    3. \(40x^2+360x+810\)

    Answer

    \(10(2x+9)^2\)

    4. \(5k^3−70k^2+245k\)

    5. \(75u^4−30u^3v+3u^2v^2\)

    Answer

    \(3u^2(5u−v)^2\)

    6. \(81r^2−25\)

    7. \(169m^2−n^2\)

    Answer

    \((13m+n)(13m−n)\)

    8. \(25p^2−1\)

    9. \(9−121y^2\)

    Answer

    \((3+11y)(3−11y)\)

    10. \(20x^2−125\)

    11. \(169n^3−n\)

    Answer

    \(n(13n+1)(13n−1)\)

    12. \(6p^2q^2−54p^2\)

    13. \(24p^2+54\)

    Answer

    \(6(4p^2+9)\)

    14. \(49x^2−81y^2\)

    15. \(16z^4−1\)

    Answer

    \((2z−1)(2z+1)(4z^2+1)\)

    16. \(48m^4n^2−243n^2\)

    17. \(a^2+6a+9−9b^2\)

    Answer

    \((a+3−3b)(a+3+3b)\)

    18. \(x^2−16x+64−y^2\)

    19. \(a^3−125\)

    Answer

    \((a−5)(a^2+5a+25)\)

    20. \(b^3−216\)

    21. \(2m^3+54\)

    Answer

    \(2(m+3)(m^2−3m+9)\)

    22.\(81m^3+3\)

    General Strategy for Factoring Polynomials

    In the following exercises, factor completely.

    1. \(24x^3+44x^2\)

    Answer

    \(4x^2(6x+11)\)

    2. \(24a^4−9a^3\)

    3. \(16n^2−56mn+49m^2\)

    Answer

    \((4n−7m)^2\)

    4. \(6a^2−25a−9\)

    5. \(5u^4−45u^2\)

    Answer

    \(5u^2(u+3)(u−3)\)

    6. \(n^4−81\)

    7. \(64j^2+225\)

    Answer

    prime

    8. \(5x^2+5x−60\)

    9. \(b^3−64\)

    Answer

    \((b−4)(b^2+4b+16)\)

    10. \(m^3+125\)

    11. \(2b^2−2bc+5cb−5c^2\)

    Answer

    \((2b+5c)(b−c)\)

    12. \(48x^5y^2−243xy^2\)

    13. \(5q^2−15q−90\)

    Answer

    \(5(q+3)(q−6) \)

    14. \(4u^5v+4u^2v^3\)

    15. \(10m^4−6250\)

    Answer

    \(10(m−5)(m+5)(m^2+25)\)

    16. \(60x^2y−75xy+30y\)

    17. \(16x^2−24xy+9y^2−64\)

    Answer

    \((4x−3y+8)(4x−3y−8)\)

    Polynomial Equations

    In the following exercises, solve.

    1. \((a−3)(a+7)=0\)

    2. \((5b+1)(6b+1)=0\)

    Answer

    \(b=−\frac{1}{5},\space b=−\frac{1}{6}\)

    3. \(6m(12m−5)=0\)

    4. \((2x−1)^2=0\)

    Answer

    \(x=\frac{1}{2}\)

    5. \(3m(2m−5)(m+6)=0\)

    6. \(x^2+9x+20=0\)

    Answer

    \(x=−4,\space x=−5\)

    7. \(y^2−y−72=0\)

    8. \(2p^2−11p=40\)

    Answer

    \(p=−\frac{5}{2},p=8\)

    9. \(q^3+3q^2+2q=0\)

    10. \(144m^2−25=0\)

    Answer

    \(m=\frac{5}{12},\space m=−\frac{5}{12}\)

    11. \(4n^2=36\)

    12. \((x+6)(x−3)=−8\)

    Answer

    \(x=2,\space x=−5\)

    13. \((3x−2)(x+4)=12\)

    14. \(16p^3=24p^2+9p\)

    Answer

    \(p=0,\space p=\frac{3}{4}\)

    15. \(2y^3+2y^2=12y\)


    This page titled 2.EB: Exercises for Factoring Polynomial Expressions and Solving Polynomial Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Katherine Skelton.