Factoring a Polynomial by Finding the GCF
Example
Factor: 6x^{4}y^{2}  30x^{2}y^{3}  2x^{2}y
Solution
Step 1 Identify the terms of the polynomial.
Step 2 Factor each term. 
6x^{4}y^{2},  30x^{2}y^{3},  2x^{2}y 
Each term has a negative
coefficient. So, we include 1 as a factor of each term. 
6x^{4}y^{2}
30x^{2}y^{3}
2x^{2}y 
= 1 Â· 2 Â· 3 Â· x
Â· x Â· x Â· x Â· y Â· y = 1 Â· 2 Â· 3 Â· 5
Â· x Â· x Â· y Â· y Â· y
= 1 Â· 2 Â· x Â· x Â· 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 Â· 2 Â· x
Â· x Â· y
= 2x^{2}y 

Step 4 Rewrite each term using the GCF.
To avoid an error with the signs, write each subtraction as an addition of
the opposite. 
6x^{4}y^{2}  30x^{2}y^{3}  2x^{2}y
= 6x^{4}y^{2} + (30x^{2}y^{3}) + (2x^{2}y) 
Rewrite each term
using 2x^{2}y as a
factor. 
= 2x^{2}y Â· 3x^{2}y
+ (2x^{2}y) Â· 15y^{2}
+ (2x^{2}y) Â· 1 
Step 5 Factor out the GCF.
Factor out 2x^{2}y. 
= 2x^{2}y(3x^{2}y + 15y^{2}
+ 1) 
Thus, 6x^{4}y^{2}  30x^{2}y^{3}  2x^{2}y
= 2x^{2}y(3x^{2}y + 15y^{2} + 1).
You can multiply to check the factorization. We leave the check to you.Note:
Note that the third term, 2x^{2}y, is the common factor.
So we write that term as 2x^{2}y Â· 1.
We can also factor the polynomial
using +2x^{2}y as the common factor.
Then we have:
6x^{4}y^{2} 
= 30x^{2}y^{3}  2x^{2}y
= 2x^{2}y(3x^{2}y  15y^{2}  1)

