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Mathematics LibreTexts

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.

  1. 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+12y6

7. 18x315x

Answer

3x(6x25)

8. 15m4+6m2n

9. 4x312x2+16x

Answer

4x(x23x+4)

10. 3x+24

11. 3x3+27x212x

Answer

3x(x29x+4)

12. 3x(x1)+5(x1)

In the following exercises, factor by grouping.

13. axay+bxby

Answer

(a+b)(xy)

14. x2yxy2+2x2y

15. x2+7x3x21

Answer

(x3)(x+7)

16. 4x216x+3x12

17. m3+m2+m+1

Answer

(m2+1)(m+1)

18. 5x5yy+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. k216k+60

3. m2+3m54

Answer

(m+9)(m6)

4. x23x10

5. x2+12xy+35y2

Answer

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

6. r2+3rs28s2

7. a2+4ab21b2

Answer

(a+7b)(a3b)

8. p25pq36q2

9. m25mn+30n2

Answer

Prime

10. x3+5x224x

11. 3y321y2+30y

Answer

3y(y5)(y2)

12. 5x4+10x375x2

Factor ax2+bx+c when a1

In the following exercises, factor each trinomial completely.

1. 5y2+14y+9

Answer

(5y+9)(y+1)

2. 8x2+25x+3

3. 10y253y11

Answer

(5y+1)(2y11)

4. 6p219pq+10q2

5. 81a2+153a+18

Answer

9(9a1)(a+2)

6. 2x2+9x+4

7. 18a29a+1

Answer

(3a1)(6a1)

8. 15p2+2p8

9. 15x2+6x2

Answer

(3x1)(5x+2)

10. 8a2+32a+24

11. 3x2+3x36

Answer

3(x+4)(x3)

12. 48y2+12y36

13. 18a257a21

Answer

3(2a7)(3a+1)

14. 3n412n396n2

15. x413x230

Answer

(x215)(x2+2)

16. (x3)25(x3)36

Factoring Special Products

In the following exercises, factor completely.

1. 25x2+30x+9

Answer

(5x+3)2

2. 36a284ab+49b2

3. 40x2+360x+810

Answer

10(2x+9)2

4. 5k370k2+245k

5. 75u430u3v+3u2v2

Answer

3u2(5uv)2

6. 81r225

7. 169m2n2

Answer

(13m+n)(13mn)

8. 25p21

9. 9121y2

Answer

(3+11y)(311y)

10. 20x2125

11. 169n3n

Answer

n(13n+1)(13n1)

12. 6p2q254p2

13. 24p2+54

Answer

6(4p2+9)

14. 49x281y2

15. 16z41

Answer

(2z1)(2z+1)(4z2+1)

16. 48m4n2243n2

17. a2+6a+99b2

Answer

(a+33b)(a+3+3b)

18. x216x+64y2

19. a3125

Answer

(a5)(a2+5a+25)

20. b3216

21. 2m3+54

Answer

2(m+3)(m23m+9)

22.81m3+3

General Strategy for Factoring Polynomials

In the following exercises, factor completely.

1. 24x3+44x2

Answer

4x2(6x+11)

2. 24a49a3

3. 16n256mn+49m2

Answer

(4n7m)2

4. 6a225a9

5. 5u445u2

Answer

5u2(u+3)(u3)

6. n481

7. 64j2+225

Answer

prime

8. 5x2+5x60

9. b364

Answer

(b4)(b2+4b+16)

10. m3+125

11. 2b22bc+5cb5c2

Answer

(2b+5c)(bc)

12. 48x5y2243xy2

13. 5q215q90

Answer

5(q+3)(q6)

14. 4u5v+4u2v3

15. 10m46250

Answer

10(m5)(m+5)(m2+25)

16. 60x2y75xy+30y

17. 16x224xy+9y264

Answer

(4x3y+8)(4x3y8)

Polynomial Equations

In the following exercises, solve.

1. (a3)(a+7)=0

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

Answer

b=15, b=16

3. 6m(12m5)=0

4. (2x1)2=0

Answer

x=12

5. 3m(2m5)(m+6)=0

6. x2+9x+20=0

Answer

x=4, x=5

7. y2y72=0

8. 2p211p=40

Answer

p=52,p=8

9. q3+3q2+2q=0

10. 144m225=0

Answer

m=512, m=512

11. 4n2=36

12. (x+6)(x3)=8

Answer

x=2, x=5

13. (3x2)(x+4)=12

14. 16p3=24p2+9p

Answer

p=0, p=34

15. 2y3+2y2=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.

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