Factoring a Polynomial by Finding the GCF
Example
Factor: -6x4y2 - 30x2y3 - 2x2y
Solution
| Step 1 Identify the terms of the polynomial.
Step 2 Factor each term. |
-6x4y2, - 30x2y3, - 2x2y |
| Each term has a negative
coefficient. So, we include -1 as a factor of each term. |
-6x4y2
-30x2y3
-2x2y |
= -1 · 2 · 3 · x
· x · x · x · y · y = -1 · 2 · 3 · 5
· x · x · y · y · y
= -1 · 2 · x · x · y |
| Step 3 Find the GCF of the terms.
In the lists, the common factors are -1, 2, x, x, and y.
So, a common factor of each term is:
-1 · 2 · x
· x · y
= -2x2y |
|
| Step 4 Rewrite each term using the GCF.
To avoid an error with the signs, write each subtraction as an addition of
the opposite. |
-6x4y2 - 30x2y3 - 2x2y
= -6x4y2 + (-30x2y3) + (-2x2y) |
| Rewrite each term
using -2x2y as a
factor. |
= -2x2y · 3x2y
+ (-2x2y) · 15y2
+ (-2x2y) · 1 |
| Step 5 Factor out the GCF.
Factor out -2x2y. |
= -2x2y(3x2y + 15y2
+ 1) |
Thus, -6x4y2 - 30x2y3 - 2x2y
= -2x2y(3x2y + 15y2 + 1).
You can multiply to check the factorization. We leave the check to you.Note:
Note that the third term, -2x2y, is the common factor.
So we write that term as -2x2y · 1.
We can also factor the polynomial
using +2x2y as the common factor.
Then we have:
| -6x4y2 |
= 30x2y3 - 2x2y
= 2x2y(-3x2y - 15y2 - 1)
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