Factoring a Polynomial by Finding the GCF
Example 1Factor: 9wxy^{3}  21w^{2}y^{4} + 12w^{3}xy^{2}
Solution
Step 1 Identify the terms of the polynomial. 
9wxy^{3},  21w^{2}y^{4}, 12w^{3}xy^{2}

Step 2 Factor each term. 
9wxy^{3}  21w^{2}y^{4}
12w^{3}xy^{2} 
= 3 Â· 3 Â· w Â· x
Â· y Â· y Â· y = 1 Â·
3 Â· 7 Â· w Â· w Â· y Â· y Â· y Â· y
= 2 Â· 2 Â· 3 Â· w Â· w Â· w Â· x
Â· y Â· 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 Â· w
Â· y Â· y = 3wy^{2}. 

Step 4 Rewrite each term using the GCF. To help keep the signs
straight, write the
subtraction of 21w^{2}y^{4}
as addition of 21w^{2}y^{4}. Rewrite each term
using 3wy^{2} as a factor. 
=
= 
9wxy^{3}  21w^{2}y^{4} + 12w^{3}xy^{2}
9wxy^{3} + (21w^{2}y^{4}) + 12w^{3}xy^{2}
3wy^{2} Â· 3xy + 3wy^{2}
Â· (7wy^{2}) + 3wy^{2}
Â· 4w^{2}x 
Step 5 Factor out the GCF. Factor 3wy^{2}. 
= 
3wy^{2}(3xy 7wy^{2}
+ 4w^{2}x) 
Thus, 9wxy^{3}  21w^{2}y^{4} + 12w^{3}xy^{2}
= 3wy^{2}(3xy 7wy^{2}
+ 4w^{2}x)
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:
â€œ3wy^{2} times what gives 9wxy^{3}?â€
Answer: 3xy
â€œ3wy^{2} times what gives 21w^{2}y^{4}?â€
Answer: 7wy^{2}
â€œ3wy^{2} times what gives 12w^{3}xy^{2}?â€
Answer: 4w^{2}x
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 Â· 5
= 1 Â· 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 Â· x + (1) Â· (5)
 x + 5 
= 5  x ? = 5  x ?
= 5  x ? Yes 
