Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

6.4.3 Factoring the GCF from Polynomials

6.4.3 • Factoring the GCF from Polynomials

Activity

To factor a GCF from a polynomial and represent it in factored form, we must first find the GCF.

Let’s look at an example: $7 z^{5} - 21 s z^{3}$ .

Compare your answer:

Step 1 - Find the GCF of the terms in the polynomial:

The GCF of the terms is $7 z^{3}$ .

Step 2 - Apply the Distributive Property “in reverse” to factor out the GCF from each term.

$7 z^{5} - 21 s z^{3} = 7 z^{3} \cdot z^{2} - 7 z^{3} \cdot 3 s$

$= 7 z^{3} (z^{2} - 3 s)$

The factored form is $7 z^{3} (z^{2} - 3 s)$ .

Work individually on the following problems.

Factor the GCF from each polynomial. Remember that if the leading coefficient is negative, then the GCF will be negative.

1. $5 x^{3} - 25 x^{2}$

Compare your answer:

$5 x^{2} (x - 5)$

2. $8 x^{3} y - 10 x^{2} y^{2} + 12 x y^{3}$

Compare your answer:

$2 x y (4 x^{2} - 5 x y + 6 y^{2})$

3. $- 5 y^{3} + 35 y^{2} - 15 y$

Compare your answer:

$- 5 y (y^{2} - 7 y + 3)$

4. $15 x^{3} y - 3 x^{2} y^{2} + 6 x y^{3}$

Compare your answer:

$3 x y (5 x^{2} - x y + 2 y^{2})$

5. $- 4 p^{3} q - 12 p^{2} q^{2} + 16 p q^{2}$

Compare your answer:

$- 4 p q (p^{2} + 3 p q - 4 q)$

6. $5 x (x + 1) + 3 (x + 1)$

Compare your answer:

$(x + 1) (5 x + 3)$

7. $3 b (b - 2) - 13 (b - 2)$

Compare your answer:

$(b - 2) (3 b - 13)$

8. $6 m (m - 5) - 7 (m - 5)$

Compare your answer:

$(m - 5) (6 m - 7)$

Self Check

Factor:

8a^3b + 2a^2b^2 − 6ab^3

.

$2ab(4a^2 + ab - 3b^2)$
$2b(4a^3 + a^2b - 3ab^2)$
$ab(8a^2 + 2ab - 6b^2)$
$-2ab(4a^2 + ab - 3b^2)$

Additional Resources

Factoring the GCF from Polynomials

It is sometimes useful to represent a number as a product of factors, for example, 12 as $2 \cdot 6$ or $3 \cdot 4$ . In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as $3 x^{2} + 15 x$ , and end with its factors, $3 x (x + 5)$ . To do this, we apply the Distributive Property “in reverse.” We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”

DISTRIBUTIVE PROPERTY

If $a$ , $b$ , and $c$ are real numbers, then

$a (b + c) = a b + a c$ and $a b + a c = a (b + c)$

The form on the left is used to multiply. The form on the right is used to factor.

So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!

Example 1

Factor: $8 m^{3} - 12 m^{2} n + 20 m n^{2}$ .

How to factor the greatest common factor from a polynomial:

Step 1 - Find the GCF of all the terms of the polynomial. For this problem, find the GCF of $8 m^{3}$ , $12 m^{2} n$ , and $20 m n^{2}$ .

Step 2 - Rewrite each term as a product using the GCF. In this case, rewrite $8 m^{3}$ , $12 m^{2} n$ , and $20 m n^{2}$ as products of their GCF, $4 m$ .

$8 m^{3} = 4 m \cdot 2 m^{2}$
$12 m^{2} n = 4 m \cdot 3 m n$
$20 m n^{2} = 4 m \cdot 5 n^{2}$

$8 m^{3} - 12 m^{2} n + 20 m n^{2}$

$4 m \cdot 2 m^{2} - 4 m \cdot 3 m n + 4 m \cdot 5 n^{2}$

Step 3 - Use the “reverse” Distributive Property to factor the expression.

$4 m (2 m^{2} - 3 m n + 5 n^{2})$

Step 4 - Check by multiplying the factors.

$4 m (2 m^{2} - 3 m n + 5 n^{2})$

$4 m \cdot 2 m^{2} - 4 m \cdot 3 m n + 4 m \cdot 5 n^{2}$

$8 m^{3} - 12 m^{2} n + 20 m n^{2}$

When the leading coefficient is negative, we factor the negative out as part of the GCF.

Example 2

Factor: $- 4 a^{3} + 36 a^{2} - 8 a$ .

The leading coefficient is negative, so the GCF will be negative.

Step 1 - Rewrite each term using the GCF, $- 4 a$ .

$- 4 a \cdot a^{2} - (- 4 a) \cdot 9 a + (- 4 a) \cdot 2$

Step 2 - Factor the GCF.

$- 4 a (a^{2} - 9 a + 2)$

Step 3 - Check.

$- 4 a (a^{2} - 9 a + 2)$

$- 4 a \cdot a^{2} - (- 4 a) \cdot 9 a + (- 4 a) \cdot 2$

$- 4 a^{3} + 36 a^{2} - 8 a$

So far, our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.

Example 3

Factor: $3 y (y + 7) - 4 (y + 7)$ .

The GCF is the binomial $y + 7$ .

$3 y (y + 7) - 4 (y + 7)$

Step 1 - Factor the GCF, $(y + 7)$ .

$(y + 7) (3 y - 4)$

Step 2 - Check on your own by multiplying.

Try it

Factoring the GCF from Polynomials

Factor.

1. $9 x y^{2} + 6 x^{2} y^{2} + 21 y^{3}$

2. $- 4 b^{3} + 16 b^{2} - 8 b$

3. $4 m (m + 3) - 7 (m + 3)$

Here is how to factor the GCF from polynomials.

$3 y^{2} (3 x + 2 x^{2} + 7 y)$

$- 4 b (b^{2} - 4 b + 2)$

$(m + 3) (4 m - 7)$