FACTORING MONOMIALS WORKSHEET

Write the prime factorization of each monomial given below. Do not use exponents.

1)  25n²

2)  18xy

3)  12y

4)  21y²

5)  81a

6)  92q

7)  36x³

8)  24h

9)  48x²

10)  x²y³

11)  20b³c²

12)  82ab

13)  91yz²

14)  66y

15)  -18g⁴h

16)  15n/30n³

17)  8x²/12x⁴

18)  24x² y/32xy³

19)  (x + 1)³/(x + 1)⁵

20)  (2n - 3)/(3 - 2n)

21)  24x³ (4 - x)/18 x (4 - x)

22) (3x - 2)⁶/(3x - 2)⁴

23)   (a² + 6a) / (ac + 6c)

24)  12x³ (5 - 2x) / 18x (2x - 5)

25)  15y(y + 1) / 45y²(y + 1)

26)  (6 - 8x)/(4x² - 3x)

27)  (ax - ay + bx - by) / (ax - ay - bx + by)

28)  (xy + 3x - 2y - 6)/(y² + y - 6)

Answers

1. Answer : 

25n²  =  5 ⋅ 5 ⋅ n ⋅ n

We may write the repeating factors using exponential form.

=  5² ⋅ n²

2. Answer : 

18xy  =  2 ⋅ 3 ⋅ 3 ⋅ x ⋅ y

= 2 ⋅ 3² ⋅ xy

3. Answer : 

12y  =  2 ⋅ 2 ⋅ 3 ⋅ y

 = 2² ⋅ 3 ⋅ y

4. Answer : 

21y²  =  3 ⋅ 7 ⋅ y ⋅ y

= 3 ⋅ 7 ⋅ y²

5. Answer : 

81a  =  3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ a

6. Answer : 

92q  =  2 ⋅ 2 ⋅ 23 ⋅ q

7. Answer : 

36x³  =  2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ x ⋅ x ⋅ x

8. Answer : 

24h  =  2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ h

9. Answer : 

48x²  =  2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ x ⋅ x

10. Answer : 

x²y³  =  x ⋅ x ⋅ y ⋅ y ⋅ y

11. Answer : 

20b³c²  =  2 ⋅ 2 ⋅ 5 ⋅ b ⋅ b ⋅ b ⋅ c ⋅ c

12. Answer : 

82ab  =  2 ⋅ 41 ⋅ a ⋅ b

13. Answer : 

91yz²  =  7 ⋅ 13 ⋅ y ⋅ z ⋅ z

14. Answer : 

66y  =  2 ⋅ 3 ⋅ 11 ⋅ y

15. Answer : 

-18g⁴h  =  -1 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ g ⋅ g ⋅ g ⋅ g ⋅ h

16. Answer : 

= 15n/30n³

Both numerator and denominator are multiples of 15 and n.

= 15n / 15n(2n²)

= 1/2n²

17. Answer : 

= 8x²/12x⁴

Both numerator and denominator are multiples of 4 and x².

= 4x²(2)/4x² (3x²)

= 2/3x²

18. Answer : 

= 24x² y/32xy³

Both numerator and denominator are multiples of 8, x and  y.

= 8xy(3x) / 8xy(4y²)

= 3x/4y²

19. Answer : 

= (x + 1)³/(x + 1)⁵

In both numerator and denominator, we have (x + 1)³ in common.

= = (x + 1)³/(x + 1)³ . (x + 1)²

By cancelling the common factors, we get

= 1/(x + 1)²

20. Answer : 

= (2n - 3)/(3 - 2n)

Factoring - sign from the numerator or denominator, we get

= -(3 - 2n) / (3 - 2n)

By cancelling the common factor, we get

= -1

21. Answer : 

= 24x³ (4 - x)/18 x (4 - x)

By cancelling the common factor (4 - x), we get

= 24x³ /18 x

Now 24 and 18 both are multiples of 6x.

= 6x(4x²) /6x(3)

= 4x² / 3

22. Answer : 

= (3x - 2)⁶/(3x - 2)⁴

= (3x - 2)⁴  (3x - 2)²/(3x - 2)⁴

Cancelling common factors, we get

=  (3x - 2)²

23. Answer : 

= (a² + 6a) / (ac + 6c)

Factoring a from the numerator

= a(a + 6) / (ac + 6c)

Factoring c from the denominator 

= a(a + 6) / c(a + 6)

Cancelling the common factors, we get

= a/c

24. Answer : 

= 12x³ (5 - 2x) / 18x (2x - 5)

Factoring - sign from the numerator or denominator, we get

= -12x³ (2x - 5) / 18x (2x - 5)

= -12x³ / 18x

Both 12 and 18 are multiples of 6. Expressing this as factor of 6x, we get

= -6x(2x²) / 6x(3)

= -2x² / 3

25. Answer : 

= 15y(y + 1) / 45y²(y + 1)

Cancelling common factors, we get

= 15y / 45y²

Here 15 and 45 are multiples of 15.

= 15y/15y(3y)

= 1/3y

26. Answer : 

= (6 - 8x)/(4x² - 3x)

Factoring 2 form the numerator

= 2(3 - 4x)/(4x² - 3x)

Factoring x from the denominator, we get

= 2 (3 - 4x)/x(4x - 3)

Factoring negative sign from the numerator, we get

= -2(4x - 3)/x(4x - 3)

= -2/x

27. Answer : 

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

= [a(x - y) + b(x - y)] / [a(x - y) - b(x - y)]

= (a + b)(x - y) / (a - b)(x - y)

Cancelling the common factors, we get

= (a + b)/(a - b)

28. Answer : 

= (xy + 3x - 2y - 6)/(y² + y - 6)

= [x(y + 3) - 2(y + 3)]/(y² + 3y - 2y - 6)

= [x(y + 3) - 2(y + 3)]/[y(y + 3) - 2(y + 3)]

= [(x - 2)(y + 3)]/[(y - 2)(y + 3)]

= (x - 2) / (y - 2)

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