Factoring a Polynomial by Finding the GCF
To factor a polynomial such as 6wx2 + 18wxy + 3wx + 9wy, we may
need to use several factoring techniques.
The first step is to factor out the GCF of the monomial terms.
Procedure —
To Factor Out the GCF of a Polynomials
Step 1 Identify the terms of the polynomial.
Step 2 Factor each term.
Step 3 Find the GCF of the terms.
Step 4 Rewrite each term using the GCF.
Step 5 Factor out the GCF.
If there is no factor (other than 1) common to each term, the GCF is 1.
To check the factorization, multiply the factors.
Example 1
Factor: 6x2 + 8xy2 - 2x
Solution
| Step 1 Identify the terms of the polynomial.
|
6x2, 8xy2, |
| Step 2 Factor each term. |
6x2
8xy
-2x |
= 2
· 3 · x · x
= 2
· 2 · 2 · x · y · y
= -1 · 2 · x |
| Step 3 Find the GCF of the terms.
In the lists, the common factors are 2 and x.
The greatest common factor (GCF) is 2 · x
= 2x.
Step 4 Rewrite each term using the GCF. |
|
|
| To help keep the signs straight, write the
subtraction of 2x as the addition of -2x. |
= |
6x2 + 8xy2 - 2x 6x2 + 8xy2
+ (- 2x) |
| Rewrite each term using 2x as
a factor. |
= |
2x · 3x +
2x · 4y2 + 2x · (-1) |
| Step 5 Factor out the GCF.
Factor out 2x. |
= |
2x(3x + 4y2 -1) |
Thus, 6x2 + 8xy2 - 2x = 2x(3x + 4y2 - 1)
We can multiply to check the factorization. We use the distributive
property.
| Is Is
Is |
2x(3x + 4y2 -1) 2x · 3x +
2x · 4y2 + 2x · (-1)
6x2 + 8xy2 - 2x |
6x2 + 8xy2 - 2x ? 6x2 + 8xy2 - 2x
?
6x2 + 8xy2 - 2x ? Yes |
|
