2.EB: Exercises for Factoring Polynomial Expressions and Solving Polynomial Equations
( \newcommand{\kernel}{\mathrm{null}\,}\)
Greatest Common Factor and Factor by Grouping
In the following exercises, find the greatest common factor.
- 12a2b3, 15ab2
- Answer
-
3ab2
2. 12m2n3,42m5n3
3. 15y3, 21y2, 30y
- Answer
-
3y
4. 45x3y2, 15x4y, 10x5y3
In the following exercises, factor the greatest common factor from each polynomial.
5. 35y+84
- Answer
-
7(5y+12)
6. 6y2+12y−6
7. 18x3−15x
- Answer
-
3x(6x2−5)
8. 15m4+6m2n
9. 4x3−12x2+16x
- Answer
-
4x(x2−3x+4)
10. −3x+24
11. −3x3+27x2−12x
- Answer
-
−3x(x2−9x+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. x2y−xy2+2x−2y
15. x2+7x−3x−21
- Answer
-
(x−3)(x+7)
16. 4x2−16x+3x−12
17. m3+m2+m+1
- Answer
-
(m2+1)(m+1)
18. 5x−5y−y+x
Factor ax2+bx+c when a=1
In the following exercises, factor each trinomial completely.
1. a2+14a+33
- Answer
-
(a+3)(a+11)
2. k2−16k+60
3. m2+3m−54
- Answer
-
(m+9)(m−6)
4. x2−3x−10
5. x2+12xy+35y2
- Answer
-
(x+5y)(x+7y)
6. r2+3rs−28s2
7. a2+4ab−21b2
- Answer
-
(a+7b)(a−3b)
8. p2−5pq−36q2
9. m2−5mn+30n2
- Answer
-
Prime
10. x3+5x2−24x
11. 3y3−21y2+30y
- Answer
-
3y(y−5)(y−2)
12. 5x4+10x3−75x2
Factor ax2+bx+c when a≠1
In the following exercises, factor each trinomial completely.
1. 5y2+14y+9
- Answer
-
(5y+9)(y+1)
2. 8x2+25x+3
3. 10y2−53y−11
- Answer
-
(5y+1)(2y−11)
4. 6p2−19pq+10q2
5. −81a2+153a+18
- Answer
-
−9(9a−1)(a+2)
6. 2x2+9x+4
7. 18a2−9a+1
- Answer
-
(3a−1)(6a−1)
8. 15p2+2p−8
9. 15x2+6x−2
- Answer
-
(3x−1)(5x+2)
10. 8a2+32a+24
11. 3x2+3x−36
- Answer
-
3(x+4)(x−3)
12. 48y2+12y−36
13. 18a2−57a−21
- Answer
-
3(2a−7)(3a+1)
14. 3n4−12n3−96n2
15. x4−13x2−30
- Answer
-
(x2−15)(x2+2)
16. (x−3)2−5(x−3)−36
Factoring Special Products
In the following exercises, factor completely.
1. 25x2+30x+9
- Answer
-
(5x+3)2
2. 36a2−84ab+49b2
3. 40x2+360x+810
- Answer
-
10(2x+9)2
4. 5k3−70k2+245k
5. 75u4−30u3v+3u2v2
- Answer
-
3u2(5u−v)2
6. 81r2−25
7. 169m2−n2
- Answer
-
(13m+n)(13m−n)
8. 25p2−1
9. 9−121y2
- Answer
-
(3+11y)(3−11y)
10. 20x2−125
11. 169n3−n
- Answer
-
n(13n+1)(13n−1)
12. 6p2q2−54p2
13. 24p2+54
- Answer
-
6(4p2+9)
14. 49x2−81y2
15. 16z4−1
- Answer
-
(2z−1)(2z+1)(4z2+1)
16. 48m4n2−243n2
17. a2+6a+9−9b2
- Answer
-
(a+3−3b)(a+3+3b)
18. x2−16x+64−y2
19. a3−125
- Answer
-
(a−5)(a2+5a+25)
20. b3−216
21. 2m3+54
- Answer
-
2(m+3)(m2−3m+9)
22.81m3+3
General Strategy for Factoring Polynomials
In the following exercises, factor completely.
1. 24x3+44x2
- Answer
-
4x2(6x+11)
2. 24a4−9a3
3. 16n2−56mn+49m2
- Answer
-
(4n−7m)2
4. 6a2−25a−9
5. 5u4−45u2
- Answer
-
5u2(u+3)(u−3)
6. n4−81
7. 64j2+225
- Answer
-
prime
8. 5x2+5x−60
9. b3−64
- Answer
-
(b−4)(b2+4b+16)
10. m3+125
11. 2b2−2bc+5cb−5c2
- Answer
-
(2b+5c)(b−c)
12. 48x5y2−243xy2
13. 5q2−15q−90
- Answer
-
5(q+3)(q−6)
14. 4u5v+4u2v3
15. 10m4−6250
- Answer
-
10(m−5)(m+5)(m2+25)
16. 60x2y−75xy+30y
17. 16x2−24xy+9y2−64
- 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=−15, b=−16
3. 6m(12m−5)=0
4. (2x−1)2=0
- Answer
-
x=12
5. 3m(2m−5)(m+6)=0
6. x2+9x+20=0
- Answer
-
x=−4, x=−5
7. y2−y−72=0
8. 2p2−11p=40
- Answer
-
p=−52,p=8
9. q3+3q2+2q=0
10. 144m2−25=0
- Answer
-
m=512, m=−512
11. 4n2=36
12. (x+6)(x−3)=−8
- Answer
-
x=2, x=−5
13. (3x−2)(x+4)=12
14. 16p3=24p2+9p
- Answer
-
p=0, p=34
15. 2y3+2y2=12y