FACTORING EXPRESSIONS WORKSHEET

Factor each of the following expressions.


1) 4x + 24y

2) 2x2 – 8x

3) x2 – 3x

4) 3a + 6ab + 9a2

5) ab + bc - 2b

6) 3(x – 6) + d(x – 6)

7) 2x(x + 4) + 3x + 12

8) 4x2y3 + 6x3y2

9) x2 + 5x + 6

10) x2 - 15x + 56

11) x2 + 2x - 15

12) x2 - x - 20

13) 2x2 + 9x + 9

14) 2x2 + 9x + 9

15) x2 + 8x + 16

16) x2 + 6xy + 9y2

17)  4a2 - 20ab + 25b2

18)  9m2 - 16y2

19)  a2b2 - c2d2

20)  x4 - y4

21)  x3 + 3x2 + 6x + 18

22)  x3 + 2x2 - 9x - 18

1. Answer :

4x + 24y

Greatest common divisor of 4x and 24y is 4.  

Divide 4x and 24y by 4. The results are x and 6y. 

4x + 24y = 4(x + 6y)

2. Answer :

2x2 – 8x

GCD of 2x2 and 8x is 2x.  

Divide 2x2 and 8x by 2x. The results x and 4. 

2x2 – 8x = 2x(x - 4)

3. Answer :

x2 – 3x

GCD of x2 and 3x is x.  

Divide x2 and 3x by x. The results x and 3. 

x2 – 3x = x(x - 3)

4. Answer :

3a + 6ab + 9a2

GCD of 3a, 6ab and 9a2 is 1, 2b and 3a.

3a + 6ab + 9a= 3a(1 + 2b + 3a)

5. Answer : 

ab + bc - 2b

GCD of ab, bc and 2b is b. 

Divide ab, bc and 2b by b. The results are a, c and -2.

ab + bc - 2b = b(a + c – 2)

6. Answer :

3(x – 6) + d(x – 6)

GCD of 3(x – 6) and d(x – 6) is (x - 6). 

Divide 3(x – 6) and d(x – 6) by (x - 6). The results are 3 and d. 

3(x – 6) + d(x – 6) = (x – 6)(3 + d)

7. Answer :

2x(x + 4) + 3x + 12

In the expression above, GCD of 3x and 12 is 3. 

Divide 3x and 12 by 3. The results are x and 4.

2x(x + 4) + 3x + 12 = 2x(x + 4) + 3(x + 4)

GCD of 2x(x + 4) and 3(x + 4) is (x + 4). 

Divide 2x(x + 4) and 3(x + 4) by (x + 4). The results are 2x and 3.

2x(x + 4) + 3x + 12 = (x + 4)(2x + 3)

8. Answer :

4x2y3 + 6x3y2

GCD of 4x2y3 + 6x3yis 2x2y2

Divide 4x2y3 and 6x3y2 is 2x2y2. The results are 2y and 3x.

4x2y3 + 6x3y2 = 2xy(2x + 3y)

9. Answer : 

x2 + 5x + 6

Find two factors of +6 such that the product is +6 and sum is +5. 

Then the two factors of +6 are +2 and +3.

Split 5x into two terms using the factors +2 and +3.

x2 + 5x + 6 = x2 + 2x + 3x + 6

= x(x + 2) + 3(x + 2)

= (x + 2)(x + 3)

10. Answer : 

x2 - 15x + 56

Find two factors of +56 such that the product is 56 and sum is -15. 

Then the two factors of -15 are -7 and -8.

Split -15x into two terms using the factors -7 and -8.

x2 - 15x + 56 = x2 - 7x - 8x + 56

= x(x - 7) - 8(x - 7)

= (x - 7)(x - 8)

11. Answer : 

x2 + 2x - 15

Find two factors of -15 such that the product is -15 and sum is +2. 

Then the two factors of -15 are -3 and +5.

Split -2x into two terms using the factors -3 and +5.

x2 + 2x - 15 = x2 - 3x + 5x - 15

= x(x - 3) + 5(x - 3)

= (x - 3)(x + 5)

12. Answer : 

x2 - x - 20

Find two factors of -20 such that the product is -20 and sum is -1. 

Then the two factors of -20 are -5 and +4.

Split -x into two terms using the factors -5 and +4.

x2 - x - 20 = x2 - 5x + 4x - 20

= x(x - 5) + 4(x - 5)

= (x - 5)(x + 4)

13. Answer : 

2x2 + 9x + 9

The product of 2 and 9 is +18.

Find two factors of +18 such that the product is +18 and sum is +9. 

Then the two factors of +18 are +3 and +6.

Split 9x into two terms using the factors +3 and +6.

2x2 + 9x + 9 = 2x2 + 3x + 6x + 9

= x(2x + 3) + 3(2x + 3)

= (2x + 3)(x + 3)

14. Answer : 

3x2 + x - 4

The product of 3 and -4 is -12.

Find two factors of -12 such that the product is -12 and sum is +1. 

Then the two factors of -12 are -3 and +4.

Split +x into two terms using the factors -3 and +4.

3x2 + x - 4 = 3x2 - 3x + 4x - 4

= 3x(x - 1) + 4(x - 1)

= (x - 1)(3x + 4)

x2 + 8x + 16

15. Answer : 

x2 + 8x + 16 = x2 + 2(x)(4) + 42

= (x + 4)2

= (x + 4)(x + 4)

16. Answer : 

x2 + 6xy + 9y= x2 + 2(x)(3y) + (3y)2

= (x + 3y)2

= (x + 3y)(x + 3y)

17. Answer : 

4a2 - 20ab + 25b2 = 22a2 - 20ab + 52b2

= (2a)2 - 20ab + (5b)2

= (2a)2 - 2(2a)(5b) + (5b)2

= (2a - 5b)2

= (2a - 5b)(2a - 5b)

18. Answer : 

9m2 - 16y2 = 32m2 - 42n2

= (3m)2 - (4n2)

= (3m + 4n)(3m - 4n)

19. Answer : 

a2b2 - c2d2 = (ab)- (cd)2

= (ab + cd)(ab - cd)

20. Answer : 

x4 - y= (x2)2 - (y2)2

Let a = x2 and b = y2.

= a2 - b2

= (a + b)(a - b)

Substitute a = x2 and b = y2.

= (x2 + y2)(x2 - y2)

=  (x2 + y2)(x + y)(x - y)

21. Answer :

x3 + 3x2 + 6x + 18

GCD of xand 3x2 is x2

Divide x3 and 3x2 by x2. The results are x and 3.

Similarly GCD of 6x and 18 is 6 

Divide 6x and 18 by 6. The results are x and 3.

x3 + 3x2 + 6x + 18 = x2(x + 3) + 6(x + 3)

= (x + 3)(x2 + 6)

22. Answer : 

x3 + 2x2 - 9x - 18

GCD of xand 2x2 is x2

Divide x3 and 2x2 by x2. The results are x and 2.

Similarly GCD of -9x and -18 is -9 

Divide -9x and -18 by -9. The results are x and 2.

x3 + 2x2 - 9x - 18 = x2(x + 2) - 9(x + 2)

= (x + 2)(x2 - 9)

= (x + 2)(x2 - 32)

= (x + 2)(x + 3)(x - 3)

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