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Factoring a Polynomial by Finding the GCF
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Factoring a Polynomial by Finding the GCF

Example 1

Factor: 9wxy3 - 21w2y4 + 12w3xy2

Solution

Step 1 Identify the terms of the polynomial. 9wxy3, - 21w2y4, 12w3xy2
Step 2 Factor each term. 9wxy3

- 21w2y4

12w3xy2

= 3 · 3 · w · x · y · y · y

= -1 · 3 · 7 · w · w · y · y · y · y

= 2 · 2 · 3 · w · w · w · x · y · y

Step 3 Find the GCF of the terms.

In the lists, the common factors are 3, w, y, and y.

The GCF is 3 · w · y · y = 3wy2.

 
Step 4 Rewrite each term using the GCF.

To help keep the signs straight, write the subtraction of 21w2y4 as addition of -21w2y4.

Rewrite each term using 3wy2 as a factor.

 

=

=

9wxy3 - 21w2y4 + 12w3xy2

9wxy3 + (-21w2y4) + 12w3xy2

3wy2 · 3xy + 3wy2 · (-7wy2) + 3wy2 · 4w2x

Step 5 Factor out the GCF.

Factor 3wy2.

 

=

 

3wy2(3xy -7wy2 + 4w2x)

 

Thus, 9wxy3 - 21w2y4 + 12w3xy2 = 3wy2(3xy -7wy2 + 4w2x)

You can multiply to check the factorization. We leave the check to you.

Note:

Another way to decide which terms belong inside the parentheses is to ask:

“3wy2 times what gives 9wxy3?” Answer: 3xy

“3wy2 times what gives -21w2y4?” Answer: -7wy2

“3wy2 times what gives 12w3xy2?” Answer: 4w2x

 

Example 2

Rewrite 5 - x as a product by factoring out -1.

Solution

Identify the terms of the polynomial. 5 and -x
Rewrite each term using -1 as a factor.

5

-x

= -1 · -5

= -1 · x

We can write:  
Factor out -1-   = -1(-5 + x)
We usually write terms with variables first. So, we use the Commutative Property of Addition to rearrange the terms inside the parentheses.

So we can write 5 - x as -1(x - 5).

We multiply to check the factorization.

  = -1(x - 5)
Is

Is

Is

-1(x - 5)

-1 · x + (-1) · (-5)

- x + 5

= 5 - x ?

= 5 - x ?

= 5 - x ? Yes

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