FACTORING MONOMIALS FROM POLYNOMIALS

Write each polynomial as the product of its greatest common monomial factor and a polynomial.

Problem 1 :

8x² + 12x

Solution :

8x² = 2 ∙ 2 ∙ 2 ∙ x ∙ x

12x = 2 ∙ 2 ∙ 3 ∙ x

Product of common factors:

= 2 ∙ 2 ∙ x

= 4x

Greatest common factor of 8x² + 12x is 4x.

Divide 8x² + 12x by 4x

= 4x (2x + 3)

Problem 2 :

6a⁴ – 3a³ + 9a²

Solution :

6a⁴ = 3 ∙ 2 ∙ a ∙ a ∙ a ∙ a

3a³ = 3 ∙ a ∙ a ∙ a

9a² = 3 ∙ 3 ∙ a ∙ a

Product of common factors:

= 3 ∙ a ∙ a

= 3a²

Greatest common factor of 6a⁴ – 3a³ + 9a² is 3a².

Divide 6a⁴ – 3a³ + 9a² by 3a²

So,

6a⁴ – 3a³ + 9a² = 3a² (2a² - a + 3)

Problem 3 :

5ab² - 15ab + 20a²b

Solution :

5ab² = 5 ∙ a ∙ b ∙ b

15ab = 5 ∙ 3 ∙ a ∙ b

20a²b = 5 ∙ 4 ∙ a ∙ a ∙ b

Product of common factors:

= 5 ∙ a ∙ b

= 5ab

Greatest common factor of 5ab² - 15ab + 20a²b is 5ab.

Divide 5ab² - 15ab + 20a²b by 5ab

So,

5ab² - 15ab + 20a²b = 5ab (b – 3 + 4a)

Problem 4 :

x³y³ - 2x³y² + x²y²

Solution :

x³y³ = x ∙ x ∙ x ∙ y ∙ y ∙ y

2x³y² = 2 ∙ x ∙ x ∙ x ∙ y ∙ y

x²y² = x ∙ x ∙ y ∙ y

Product of common factors:

= x ∙ x ∙ y ∙ y

= x²y²

Greatest common factor of x³y³ - 2x³y² + x²y² is x²y²

Divide x³y³ - 2x³y² + x²y² by x²y²

So,

 x³y³ - 2x³y² + x²y² = x²y² (xy – 2x + 1)

Problem 5 :

4a – 12ab + 16a²

Solution :

4a = 2 ∙ 2 ∙ a

12ab = 2 ∙ 2 ∙ 3 ∙ a ∙ b

16a² = 2 ∙ 2 ∙ 2 ∙ 2 ∙ a ∙ a

Product of common factors:

= 2 ∙ 2 ∙ a

= 4a

Greatest common factor of 4a – 12ab + 16a² is 4a.

Divide 4a – 12ab + 16a² by 4a

So,

4a – 12ab + 16a² = 4a (1 – 3b + 4a)

Problem 6 :

21a² - 14a + 7

Solution :

21a² = 7 ∙ 3 ∙ a ∙ a

14a = 7 ∙ 2 ∙ a

7 = 7

Product of common factors:

= 7

Greatest common factor of 21a² - 14a + 7 is 7.

Divide 21a² - 14a + 7 by 7

So,

21a² - 14a + 7 = 7 (3a² - 2a + 1)