FACTORING MONOMIALS

Monomial is an algebraic expression consisting of one term. Factoring monomials is a process of breaking down a monomial into smaller terms. Even though factoring a polynomial usually shortens the polynomial, factoring a monomial expands it.

Example 1 :

25n²

Solution : 

25n²  =  5 ⋅ 5 ⋅ n ⋅ n

Example 2 :

18xy

Solution : 

18xy  =  2 ⋅ 3 ⋅ 3 ⋅ x ⋅ y

Example 3 :

12y

Solution : 

12y  =  2 ⋅ 2 ⋅ 3 ⋅ y

Example 4 :

21y²

Solution : 

21y²  =  3 ⋅ 7 ⋅ y ⋅ y

Example 5 :

81a

Solution : 

81a  =  3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ a

Example 6 :

92q

Solution : 

92q  =  2 ⋅ 2 ⋅ 23 ⋅ q

Example 7 :

36x³

Solution : 

36x³  =  2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ x ⋅ x ⋅ x

Example 8 :

24h

Solution : 

24h  =  2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ h

Example 9 :

48x²

Solution : 

48x²  =  2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ x ⋅ x

Example 10 :

x²y³

Solution : 

x²y³  =  x ⋅ x ⋅ y ⋅ y ⋅ y

Factor monomial from the following polynomials.

Example 11 :

9a² - 18a

Solution : 

= 9a² - 18a

Factoring 9a, we get

= 9a(a) - 2(9a)

= 9 a (a - 2)

Example 12 :

x³ - 5x²

Solution : 

= x³ - 5x²

Factoring the monomial x², we get

= x² (x - 5)

Example 13 :

9x² - 36

Solution : 

= 9x² - 36

= 9x² - 4(9)

Factoring 9, we get

= 9(x² - 4)

Here 4 can be written as 2²

= 9(x² - 2²)

Here x² - 2² looks like a² - b² = (a + b)(a - b)

= 9 (x + 2)(x - 2)

Example 14 :

8x² + 72x

Solution : 

= 8x² + 72x

Writing 72 as a product of 8.

= 8x² + 8(9)x

= 8 x(x + 9)

Example 15 :

8x² - 80x

Solution : 

= 8x² - 80x

Writing 80 as a product of 8.

= 8x² - 8(10)x

= 8 x (x - 10)

Example 16 :

16a⁵b³ + 32a⁴

Solution : 

= 16a⁵b³ + 32a⁴

= 16a⁵b³ + 16a⁴ (2)

= 16a⁴(ab³ + 2)

Example 17 :

11m² - 99m

Solution : 

= 11m² - 99m

= 11m² - 11m (9)

= 11m (m - 9)

Example 18 :

x² + x³ + x⁴ 

Solution : 

= x² + x³ + x⁴ 

Factoring x², we get

= x² (1 + x + x² )

Example 19 :

36x³ / 42 x² 

Solution : 

= 36x³ / 42 x² 

Both 36 and 42 are multiples of 6 and x² is in common. By factoring 6 x² from the numerator and denominator, we get

= 6x² (6x) / 6 x² (7)

Cancelling the common factor, we get 

= 6x/7

Example 20 :

16p² / 28 p

Solution : 

= 16p² / 28 p

Both 16 and 28 are multiples of 4 and p is in common. By factoring 4p from the numerator and denominator, we get

= 4p(4p) / 4p (7)

Cancelling the common factor, we get 

= 4p/7

Example 21 :

32n² / 24n

Solution : 

= 32n² / 24n

Both 32 and 24 are multiples of 8 and n is in common. By factoring 8n from the numerator and denominator, we get

= 8n(4n) / 8n (3)

Cancelling the common factor, we get 

= 4n/3

Example 22 :

70n² / 28n

Solution : 

= 70n² / 28n

Both 70 and 28 are multiples of 7 and n is in common. By factoring 7n from the numerator and denominator, we get

= 7n(10n) / 7n (4)

Cancelling the common factor, we get 

= 10n/4

= 2(5n)/2(2)

= 5n/2

Example 23 :

(2r - 4) / (r - 2)

Solution : 

= (2r - 4) / (r - 2)

= (2r - 2(2)) / (r - 2)

= 2(r - 2) / (r - 2)

Cancelling common factor, we get

= 2

Example 24 :

45/(10a - 25)

Solution : 

= 45/(10a - 25)

In the denominator 10 and 25 are multiples of 5. Factoring 5, we get

= 5(9)/(2⋅5a - 5⋅5)

= 5(9)/5(2a - 5)

= 9/(2a - 5)

Example 25 :

(15 a - 3)/24

Solution : 

= (15 a - 3)/24

In the numerator 15 and 3 are multiples of 3. Factoring 3, we get

= (5⋅3 a - 3)/24

= 3(5 a - 1)/24

= (5a - 1)/8

Example 26 :

(x - 4)/(3x² - 12x)

Solution : 

= (x - 4)/(3x² - 12x)

In the denominator, both 3x² and 12x can be expressed as a product of 3x.

= (x - 4)/3x(x - 4)

Cancelling common factor, we get

= 1/3x

Example 27 :

27/(27x + 18)

Solution : 

= 27/(27x + 18)

In the denominator 27 and 18 are the multiples of 9.

= 27/(3⋅9x + 2⋅9)

= 27/9(3x + 2)

Here 27 and 9 can be simplified using 9 times table.

= 3/(3x + 2)

Example 28 :

(4x - 4) / (6x - 20)

Solution : 

= (4x - 4) / (6x - 20)

Factoring 4 from the numerator and factoring 2 from the denominator, we get

= 4(x - 1) / 2(3x - 10)

= 2(x - 1)/(3x - 10)

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