FACTORING BY GROUPING WORKSHEET

Factor each polynomial :

1)  ax + bx + ay + by 

2)  x² + xt + ax + at

3)  2y³ + 4y² + y + 2

4)  xy - 5y - 2x + 10

5)  6ax + 3bd - 2ad - 9bx

6)  3x³ + 3x² - 6x - 6

7)  2ax² - cx² + 6a - 3c

8)  4a² + 5ab - 10b - 8a

9)  a²x + 3a²y - 9x - 27y

10)  x³y² + 6x²y² + 8xy² + 48y²

11) 12a³ - 9a² + 4a - 3

12)  2p³ + 5p² + 6p + 15

13)  3n³ - 4n² + 9n - 12

14)  224 az + 56 ac − 84 yz − 21 yc

15) 35 xy − 5 x − 56 y + 8

16) mz - 5 mh² - 5 nz + 25 nh²

17)  12xy - 28x - 15y + 35

18) 40xy + 30x - 100 y - 75

19)  75 a² c - 45 a² d - 30 bc + 18bd

Detailed Answer Key

1. Answer : 

ax + bx + ay + by  =  (ax + bx) + (ay + by)

=  x(a + b) + y(a + b) 

=  (x + y)(a + b)

2. Answer :

x² + xt + ax + at  =  (x² + xt) + (ax + at)

=  x(x + t) + a(x + t)

=  (x + t)(x + a)

3. Answer :

2y³ + 4y² + y + 2  =  (2y³ + 4y²) + (y + 2)

=  2y²(y + 2) + 1(y + 2)

=  (y + 2)(2y² + 1)

4. Answer :

xy - 5y - 2x + 10  =  (xy - 5y) + (-2x + 10)

=  y(x - 5) - 2(x - 5)

=  (x - 5)(y - 2)

5. Answer :

6ax + 3bd - 2ad - 9bx  =  6ax - 9bx - 2ad + 3bd

=  (6ax - 9bx) + (-2ad + 3bd)

=  3x(2a - 3b) - d(2a - 3b)

=  (2a - 3b)(3x - d)

6. Answer :

3x³ + 3x² - 6x - 6  =  (3x³ + 3x²) + (-6x - 6)

=  3x²(x + 1) - 6(x + 1)

=  (x + 1)(3x² - 6)

=  (3x² - 6)(x + 1)

=  3(x² - 2)(x + 1)

7. Answer :

2ax² - cx² + 6a - 3c  =  (2ax² - cx²) + (6a - 3c)

=  x²(2a - c) + 3(2a - c)

=  (2a - c)(x² + 3)

8. Answer :

4a² + 5ab - 10b - 8a  =  4a² + 5ab - 8a - 10b

=  (4a² + 5ab) + (-8a - 10b)

=  a(4a + 5b) - 2(4a + 5b)

=   (4a + 5b)(a - 2)

9. Answer :

a²x + 3a²y - 9x - 27y  =  (a²x + 3a²y) + (-9x - 27y)

=  a²(x + 3y) - 9(x + 3y)

=  (x + 3y)(a² - 9)

=  (x + 3y)(a² - 3²)

=  (x + 3y)(a + 3)(a - 3)

10. Answer :

x³y² + 6x²y² + 8xy² + 48y² :

=  (x³y² + 6x²y²) + (8xy² + 48y²)

=  x²y²(x + 6) + 8y²(x + 6)

=  (x + 6)(x²y² + 8y²)

=  (x²y² + 8y²)(x + 6)

=  y²(x² + 8)(x + 6)

11. Answer :

12a³ - 9a² + 4a - 3

Factoring 3a² from first two terms and factoring 1 from third and fourth term, we get

= 3a²(4a - 3) + (4a - 3)

= 3a² (4a - 3) + 1 (4a - 3)

= (3a² + 1)(4a - 3)

12. Answer :

= 2p³ + 5p² + 6p + 15

Factoring p² from first two terms and 3 from 3rd and 4th term, we get

= p² (2p + 5) + 3(2p + 5)

= (p² + 3) (2p + 5)

13. Answer :

= 3n³ - 4n² + 9n - 12

= n²(3n - 4) + 3(3n - 4)

= (n²+ 3) (3n - 4)

14. Answer :

= 224 az + 56 ac − 84 yz − 21 yc

= 4(56) az + 56 ac − 4(21) yz − 21 yc

= 56 a (4z + c) − 21y (4z + c)

= (56 a − 21y) (4z + c)

15. Answer :

= 35 xy − 5 x − 56 y + 8

= 7y(5x) − 5 x − 7(8)y + 8

= 5x(7y - 1) - 8(7y - 1)

= (5x - 8)(7y - 1)

16. Answer :

= mz - 5 mh² - 5 nz + 25 nh²

= m (z - 5 h²) - 5 n(z - 5 h²)

= (m - 5 n)(z - 5 h²)

17. Answer :

= 12xy - 28x - 15y + 35

= 3x(4y) - 7(4x) - 3(5y) + 5(7)

= 4x(3y - 7) - 5(3y - 7)

= (4x - 5)(3y - 7)

18. Answer :

= 40xy + 30x - 100 y - 75

= 4(10xy) + 3(10x) - 25(4 y) - 25(3)

= 10x(4y + 3) - 25(4y + 3)

= (10x - 25) (4y + 3)

= 5(2x - 5)(4y + 3)

19. Answer :

= 75 a² c - 45 a² d - 30 bc + 18bd

= 3(25 a² c - 15 a² d - 10 bc + 6bd)

= 3[5a²(5c - 3d) - 2b(5c - 3d)]

= 3(5a²- 2b) (5c - 3d)

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