FACTOR OUT A MONOMIAL

Example 1 :

Factor :

6x⁴ + 4x³y

Solution : 

Greatest common factor of 6x⁴ and 4x³y is 2x³. 

Divide 6x⁴ and 4x³y by 2x³.

6x⁴/2x³  =  3x

4x³y/2x³  =  2y

Write the quotients 3x and 2y inside the parenthesis and multiply by the greatest common factor 2x³.

2x³(3x + 2y)

So, 

6x⁴ + 4x³y  =  2x³(3x + 2y)

Example 2 :

Factor : 

12a² - 21ab

Solution : 

Greatest common factor of 12a² and -21ab is 3a. 

Divide 12a² and -21ab by 3a. 

12a²/3a  =  4a

-21ab/3a  =  -7b

Write the quotients 4a and -7b inside the parenthesis and multiply by the greatest common factor 3a.

3a(4a - 7b)

So, 

12a² - 21ab  =  3a(4a - 7b)

Example 3 :

Factor : 

6n³ - 3n⁵

Solution : 

Greatest common factor of 6n³ and -3n⁵ is 3n³. 

Divide 6n³ and -3n⁵ by 3n³. 

6n³/3n³  =  2n

-3n⁵/3n³  =  -n²

Write the quotients 2n and -n² inside the parenthesis and multiply by the greatest common factor 3n³.

3n³(2n - n²)

So, 

6n³ - 3n⁵  =  3n³(2n - n²)

Example 4 :

Factor : 

10y³ - 9y² + y

Solution : 

Greatest common factor of 10y³, -9y² and y is y. 

Divide 10y³ and -9y² and y by y. 

10y³/y  =  10y²

-9y²/y  =  -9y

y/y  =  1

Write the quotients 10y², -9y and y inside the parenthesis and multiply by the greatest common factor y.

y(10y² - 9y + 1)

So, 

10y³ - 9y² + y  =  y(10y² - 9y + 1)

Example 5 :

Factor :

15x³ + 10x²y - 25x²z

Solution : 

Greatest common factor of 15x³, 10x²y, -25x²z is 5x².

Divide 15x³, 10x²y and -25x²z by 5x².

15x³/5x²  =  3x

10x²y/5x²  =  2y

-25x²z/5x²  =  -5z

Write the quotients 3x, 2y and -5z inside the parenthesis and multiply by the greatest common factor 5x².

5x²(3x + 2y - 5z)

So, 

15x³ + 10x²y - 25x²z  =  5x²(3x + 2y - 5z)

Example 6 :

Factor : 

-8m⁷ + 24m⁶ + 12m⁵

Solution : 

Greatest common factor of -8m⁷, 24m⁶ and 12m⁵ is 4m⁵. 

Divide -8m⁷, 24m⁶ and 12m⁵ by 4m⁵. 

-8m⁷/4m⁵  =  -2m²

24m⁶/4m⁵  =  6m

12m⁵/4m⁵  =  3

Write the quotients -2m², 6m and 3 inside the parenthesis and multiply by the greatest common factor 4m⁵.

4m⁵(-2m² + 6m + 3)

So, 

-8m⁷ + 24m⁶ + 12m⁵  =  4m⁵(-2m² + 6m + 3)

Example 7 :

Factor :

-7u² - 21u³

Solution : 

Greatest common factor of -7u² and -21u³ is -7u².

Divide -7u² and -21u³ by -7u².

-7u²/(-7u²)  =  1

-21u³/(-7u²)  =  3u

Write the quotients 1 and 3u inside the parenthesis and multiply by the greatest common factor -7u².

-7u²(1 + 3u)

So, 

-7u² - 21u³  =  -7u²(1 + 3u)

Example 8 :

Factor :

28m²n² - 12m³n - 20m³n²

Solution : 

Greatest common factor of 28m²n², -12m³n, -20m³n² is 4m²n.

Divide 28m²n², -12m³n, -20m³n² by 4m²n.

28m²n²/4m²n  =  7n

-12m³n/4m²n  =  -3m

-20m³n²/4m²n  =  -5mn

Write the quotients 7n, -3m and -5mn inside the parenthesis and multiply by the greatest common factor 4m²n.

4m²n(7n - 3m - 5mn)

So, 

28m²n² - 12m³n - 20m³n²  =  4m²n(7n - 3m - 5mn)

Example 9 :

Factor :

16a⁵b³ + 32a⁴b

Solution : 

= 16a⁵b³ + 32a⁴b

Writing 16 and 32 as product of prime factors.

= 2⁴a⁵b³ + 2⁵a⁴b

Factoring 2⁴, we get

= 2⁴a⁴b (a b² + 1)

= 16a⁴b (a b² + 1)

Example 10 :

Factor :

16x² - 12x³- 18x⁴

Solution : 

= 16x² - 12x³- 18x⁴

16 = 2(8)

12 = 2(6)

18 = 2(9)

Every coefficients can be written as a product of 2.

= 2x² (8 - 6x- 9x²)

Example 11 :

Factor :

35a²b - 5ab

Solution : 

= 35a²b - 5ab

= 5ab(7a - 1)

Example 12 :

Factor :

4k² - 56k

Solution : 

= 4k² - 56k

Factoring 4k, we get

= 4k(k - 14)

Example 13 :

Factor :

6n² - 114n

Solution : 

= 6n² - 114n

Factoring 6n, we get

= 6n(n - 19)

Example 14 :

Factor :

9x² - 36x

Solution : 

= 9x² - 36x

Factoring 9x, we get

= 9x(x - 4)

Example 15 :

Factor :

8r⁵ - 20r⁴ - 12r³

Solution : 

= 8r⁵ - 20r⁴ - 12r³

Factoring 4, we get

= 4r³(2r² - 5r - 3)

Example 16 :

Factor :

x³ - 5x²

Solution : 

= x³ - 5x²

Factoring x², we get

= x²(x - 5)

Example 16 :

Factor :

6w⁴ - 10w³ + 2w

Solution : 

= 6w⁴ - 10w³ + 2w

Factoring 2w, we get

= 2w(3w³ - 5w² + 1)

Example 17 :

Factor :

-3p⁴ + 15p² + 6p

Solution : 

= -3p⁴ + 15p² + 6p

Factoring 3p, we get

= 3p(-p³ + 5p + 1)

Example 18 :

Factor :

4x² + 32xy

Solution : 

= 4x² + 32xy

Factoring 4x, we get

= 4x(x + 8y)

Example 19 :

Factor :

9a² - 153a

Solution : 

= 9a² - 153a

Factoring 9a, we get

= 9a(a - 17)

Example 20 :

Factor :

11x² - 165x

Solution : 

= 11x² - 165x

Factoring 11x, we get

= 11x(x - 15)

Example 21 :

Factor :

3v² - 60v

Solution : 

= 3v² - 60v

Factoring 3v, we get

= 3v(v - 20)

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