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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 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 = 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 Â· 3xy + 3wy2 Â· (-7wy2) + 3wy2 Â· 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:

â€œ3wy2 times what gives 9wxy3?â€ Answer: 3xy

â€œ3wy2 times what gives -21w2y4?â€ Answer: -7wy2

â€œ3wy2 times what gives 12w3xy2?â€ 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 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)
 IsIs Is -1(x - 5) -1 Â· x + (-1) Â· (-5) - x + 5 = 5 - x ?= 5 - x ? = 5 - x ? Yes