1.2.EB: Exercises for Factoring Polynomial Expressions and Solving Polynomial Equations
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Greatest Common Factor and Factor by Grouping
In the following exercises, find the greatest common factor.
- \(12a^2b^3,\space 15ab^2\)
- Answer
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\(3ab^2\)
2. \(12m^2n^3,42m^5n^3\)
3. \(15y^3,\space 21y^2,\space 30y\)
- Answer
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\(3y\)
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
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\(7(5y+12)\)
6. \(6y^2+12y−6\)
7. \(18x^3−15x\)
- Answer
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\(3x(6x^2−5)\)
8. \(15m^4+6m^2n\)
9. \(4x^3−12x^2+16x\)
- Answer
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\(4x(x^2−3x+4)\)
10. \(−3x+24\)
11. \(−3x^3+27x^2−12x\)
- Answer
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\(−3x(x^2−9x+4)\)
12. \(3x(x−1)+5(x−1)\)
In the following exercises, factor by grouping.
13. \(ax−ay+bx−by\)
- Answer
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\((a+b)(x−y)\)
14. \(x^2y−xy^2+2x−2y\)
15. \(x^2+7x−3x−21\)
- Answer
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\((x−3)(x+7)\)
16. \(4x^2−16x+3x−12\)
17. \(m^3+m^2+m+1\)
- Answer
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\((m^2+1)(m+1)\)
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
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\((a+3)(a+11)\)
2. \(k^2−16k+60\)
3. \(m^2+3m−54\)
- Answer
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\((m+9)(m−6)\)
4. \(x^2−3x−10\)
5. \(x^2+12xy+35y^2\)
- Answer
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\((x+5y)(x+7y)\)
6. \(r^2+3rs−28s^2\)
7. \(a^2+4ab−21b^2\)
- Answer
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\((a+7b)(a−3b)\)
8. \(p^2−5pq−36q^2\)
9. \(m^2−5mn+30n^2\)
- Answer
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Prime
10. \(x^3+5x^2−24x\)
11. \(3y^3−21y^2+30y\)
- Answer
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\(3y(y−5)(y−2)\)
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
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\((5y+9)(y+1)\)
2. \(8x^2+25x+3\)
3. \(10y^2−53y−11\)
- Answer
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\((5y+1)(2y−11)\)
4. \(6p^2−19pq+10q^2\)
5. \(−81a^2+153a+18\)
- Answer
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\(−9(9a−1)(a+2)\)
6. \(2x^2+9x+4\)
7. \(18a^2−9a+1\)
- Answer
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\((3a−1)(6a−1)\)
8. \(15p^2+2p−8\)
9. \(15x^2+6x−2\)
- Answer
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\((3x−1)(5x+2)\)
10. \(8a^2+32a+24\)
11. \(3x^2+3x−36\)
- Answer
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\(3(x+4)(x−3)\)
12. \(48y^2+12y−36\)
13. \(18a^2−57a−21\)
- Answer
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\(3(2a−7)(3a+1)\)
14. \(3n^4−12n^3−96n^2\)
15. \(x^4−13x^2−30\)
- Answer
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\((x^2−15)(x^2+2)\)
16. \((x−3)^2−5(x−3)−36\)
Factoring Special Products
In the following exercises, factor completely.
1. \(25x^2+30x+9\)
- Answer
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\((5x+3)^2\)
2. \(36a^2−84ab+49b^2\)
3. \(40x^2+360x+810\)
- Answer
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\(10(2x+9)^2\)
4. \(5k^3−70k^2+245k\)
5. \(75u^4−30u^3v+3u^2v^2\)
- Answer
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\(3u^2(5u−v)^2\)
6. \(81r^2−25\)
7. \(169m^2−n^2\)
- Answer
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\((13m+n)(13m−n)\)
8. \(25p^2−1\)
9. \(9−121y^2\)
- Answer
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\((3+11y)(3−11y)\)
10. \(20x^2−125\)
11. \(169n^3−n\)
- Answer
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\(n(13n+1)(13n−1)\)
12. \(6p^2q^2−54p^2\)
13. \(24p^2+54\)
- Answer
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\(6(4p^2+9)\)
14. \(49x^2−81y^2\)
15. \(16z^4−1\)
- Answer
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\((2z−1)(2z+1)(4z^2+1)\)
16. \(48m^4n^2−243n^2\)
17. \(a^2+6a+9−9b^2\)
- Answer
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\((a+3−3b)(a+3+3b)\)
18. \(x^2−16x+64−y^2\)
19. \(a^3−125\)
- Answer
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\((a−5)(a^2+5a+25)\)
20. \(b^3−216\)
21. \(2m^3+54\)
- Answer
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\(2(m+3)(m^2−3m+9)\)
22.\(81m^3+3\)
General Strategy for Factoring Polynomials
In the following exercises, factor completely.
1. \(24x^3+44x^2\)
- Answer
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\(4x^2(6x+11)\)
2. \(24a^4−9a^3\)
3. \(16n^2−56mn+49m^2\)
- Answer
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\((4n−7m)^2\)
4. \(6a^2−25a−9\)
5. \(5u^4−45u^2\)
- Answer
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\(5u^2(u+3)(u−3)\)
6. \(n^4−81\)
7. \(64j^2+225\)
- Answer
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prime
8. \(5x^2+5x−60\)
9. \(b^3−64\)
- Answer
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\((b−4)(b^2+4b+16)\)
10. \(m^3+125\)
11. \(2b^2−2bc+5cb−5c^2\)
- Answer
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\((2b+5c)(b−c)\)
12. \(48x^5y^2−243xy^2\)
13. \(5q^2−15q−90\)
- Answer
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\(5(q+3)(q−6) \)
14. \(4u^5v+4u^2v^3\)
15. \(10m^4−6250\)
- Answer
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\(10(m−5)(m+5)(m^2+25)\)
16. \(60x^2y−75xy+30y\)
17. \(16x^2−24xy+9y^2−64\)
- Answer
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\((4x−3y+8)(4x−3y−8)\)
Polynomial Equations
In the following exercises, solve.
1. \((a−3)(a+7)=0\)
2. \((5b+1)(6b+1)=0\)
- Answer
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\(b=−\frac{1}{5},\space b=−\frac{1}{6}\)
3. \(6m(12m−5)=0\)
4. \((2x−1)^2=0\)
- Answer
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\(x=\frac{1}{2}\)
5. \(3m(2m−5)(m+6)=0\)
6. \(x^2+9x+20=0\)
- Answer
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\(x=−4,\space x=−5\)
7. \(y^2−y−72=0\)
8. \(2p^2−11p=40\)
- Answer
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\(p=−\frac{5}{2},p=8\)
9. \(q^3+3q^2+2q=0\)
10. \(144m^2−25=0\)
- Answer
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\(m=\frac{5}{12},\space m=−\frac{5}{12}\)
11. \(4n^2=36\)
12. \((x+6)(x−3)=−8\)
- Answer
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\(x=2,\space x=−5\)
13. \((3x−2)(x+4)=12\)
14. \(16p^3=24p^2+9p\)
- Answer
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\(p=0,\space p=\frac{3}{4}\)
15. \(2y^3+2y^2=12y\)