FACTORING BY GCF

Recall that the Distributive Property states that

ab + ac  =  a(b + c)

The Distributive Property allows you to factor out the GCF of the terms in a polynomial to write a factored form of the polynomial.

A polynomial is in its factored form when it is written as a product of monomials and polynomials that cannot be factored further. The expression 2(3x - 4x) is not fully factored because the terms in the parentheses have a common factor of x.

Factoring by Using the GCF

Factor each polynomial. Check your answer.

Example 1 : 

4x² - 3x

Solution : 

Find the GCF :

4x²  =  2 ⋅ 2 ⋅ x ⋅ x

3x  =  3 ⋅ x

The GCF of 4x² and 3x is x.

Write terms as products using the GCF as a factor.

=  4x(x) - 3(x)

Use the Distributive Property to factor out the GCF.

=  x(4x - 3)

Check : 

Multiply to check your answer.

x(4x - 3)  =  x(4x) - x(3)

=  4x² - 3x

The product is the original polynomial.

Example 2 : 

10y³ + 20y² - 5y

Solution : 

Find the GCF :

10y³  =  2 ⋅ 5 ⋅ y ⋅ y ⋅ y

20y²  =  2 ⋅ 2 ⋅ 5 ⋅ y ⋅ y

5y  =  5 ⋅ y

The GCF of 10y³, 20y² and 5y is 5y.

Write terms as products using the GCF as a factor.

10y³ + 20y² - 5y  =  2y²(5y) + 4y(5y) - 1(5y)

Use the Distributive Property to factor out the GCF.

=  5y(2y² + 4y - 1)

Check : 

Multiply to check your answer.

5y(2y² + 4y - 1)  =  5y(2y²) + 5y(4y) + 5y(-1)

=  10y³ + 20y² - 5y

The product is the original polynomial.

Example 3 : 

-12a - 8a²

Solution : 

Both coefficients are negative. Factor out -1.

-12a - 8a²  =  -1(12a + 8a²)

Find the GCF :

12a  =  2 ⋅ 2 ⋅ 3 ⋅ a

8a²  =  2 ⋅ 2 ⋅ 2 ⋅ a ⋅ a

The GCF of 12a and 8a² is 4a. 

Write terms as products using the GCF as a factor.

=  -1[3(4a) + 2a(4a)]

Use the Distributive Property to factor out the GCF.

=  -1[4a(3 + 2a)]

=  -4a(3 + 2a)

Check :

Multiply to check your answer.

-4a(3 + 2a)  =  -4a(3) - 4a(2a)

=  -12a - 8a²

The product is the original polynomial.

Example 4 : 

5x² + 7

Solution : 

Find the GCF :

5x²  =  5 ⋅ x ⋅ x

7  =  7

There are no common factors other than 1.

The polynomial cannot be factored.

Factoring Out a Common Binomial Factor

Factor each expression.

Example 5 : 

7(x - 3) - 2x(x - 3)

Solution : 

(x - 3) is a common binomial factor.

=  7(x - 3) - 2x(x - 3)

Factor out (x - 3).

=  (x - 3)(7 - 2x)

Example 6 : 

-y(y² + 5) + (y² + 5)

Solution : 

(y² + 5) is a common binomial factor.

=  -y(y² + 5) + (y² + 5)

=  -y(y² + 5) + 1(y² + 5)

Factor out (y² + 5).

=  (y² + 5)(-y + 1)

=  (y² + 5)(1 - y)

Example 7 : 

9n(n + 4) - 5(4 + n)

Solution : 

Addition is always commutative. So,

4 + n  =  n + 4

Then, 

=  9n(n + 4) - 5(n + 4)

(n + 4) is a common binomial factor.

=  9n(n + 4) - 5(n + 4)

Factor out (n + 4).

=  (n + 4)(9n - 5)

Example 8 : 

-3y²(y + 2) + 4(y - 7)

Solution : 

There are no common factors.

The expression cannot be factored.

Factoring by Grouping

Factor each polynomial by grouping. Check your answer.

Example 9 : 

12x³ - 9x² + 20x - 15

Solution : 

Group terms that have a common number or variable as a factor.

=  (12x³ - 9x²) + (20x - 15)

Factor out the GCF of each group.

=  3x²(4x - 3) + 5(4x - 3)

(4x - 3) is a common factor.

=  3x²(4x - 3) + 5(4x - 3)

Factor out (4x - 3).

=  (4x - 3)(3x² + 5)

Check : 

Multiply to check your answer.

(4x - 3)(3x² + 5)  =  4x(3x²) + 4x(5) - 3(3x²) - 3(5)

=  12x³ + 20x - 9x² - 15

=  12x³ - 9x² + 20x - 15

The product is the original polynomial.

Example 10 : 

9a³ + 18a² + a + 2

Solution : 

Group terms that have a common number or variable as a factor.

=  (9a³ + 18a²) + (a + 2)

Factor out the GCF of each group.

=  9a²(a + 2) + 1(a + 2)

(a + 2) is a common factor.

=  9a²(a + 2) + 1(a + 2)

Factor out (a + 2).

=  (a + 2)(9a² + 1)

Check : 

Multiply to check your answer.

(a + 2)(9a² + 1)  =  a(9a²) + a(1) + 2(9a²) + 2(1)

=  9a³ + a + 18a² + 2

=  9a³ + 18a² + a + 2

The product is the original polynomial.

Factoring with Opposites

Example 11 : 

Factor 3x³ - 15x² + 10 - 2x by grouping.

Solution :

=  3x³ - 15x² + 10 - 2x

Group terms.

=  (3x³ - 15x²) + (10 - 2x)

Factor out the GCF of each group.

=  3x²(x - 5) + 2(5 - x)

Write (5 - x) as -1(x - 5).

=  3x²(x - 5) + 2(-1)(x - 5)

=  3x²(x - 5) - 2(x - 5)

(x - 5) is a common factor. 

=  3x²(x - 5) - 2(x - 5)

Factor out (x - 5).

=  (x - 5)(3x² - 2)

Science Application

Example 12 :

Lily’s calculator is powered by solar energy. The area of the solar panel is (7x² + x) cm². Factor this polynomial to find possible expressions for the dimensions of the solar panel.

Solution : 

A  =  7x² + x

The GCF of 7x² and x is x.

Write each term as a product using the GCF as a factor.

=  7x(x) + 1(x)

Use the Distributive Property to factor out the GCF.

=  x(7x + 1)

Possible expressions for the dimensions of the solar panel are x cm and (7x + 1) cm.

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