© William James Calhoun, : Factoring Using the Distributive Property OBJECTIVES: You must use GCF and distributive tools to factor polynomials and use grouping techniques to factor polynomials with four or more terms. When we used the distributive property, we turned 4(x + 5) into 4x Now, we will be using the second stage of factoring to “undistribute” polynomials, turning 4x + 20 into 4(x + 5). To do this, find the GCF of all the terms in the polynomial. Undistribute the GCF from each term and put it in front of a new set of parenthesis. Leave the remainders inside the parenthesis.

© William James Calhoun, : Factoring Using the Distributive Property EXAMPLE 1: Use the distributive property to factor each polynomial. A. 12mn m 2 n 2 B. 20abc + 15a 2 c - 5ac List the factors of 12mn 2.List the factors of 20abc. List the factors of 18m 2 n 2. 12mn 2 = 2 · 2 · 3 · m · n · n 18m 2 n 2 = 2 · 3 · 3 · m · m · n · n Find the GCF. 23mnxxxxn 6mn 2 Write the 6mn 2, open a set of parenthesis, pull the GCF from the two monomials, write what is left behind, then close the parenthesis. 6mn 2 (2 - 3m) List the factors of 18m 2 n 2. 20abc = 2 · 2 · 5 · a · b · c 15a 2 c = 3 · 5 · a · a · c Find the GCF. 5axxc 5ac Write the 5ac, open a set of parenthesis, pull the GCF from the three monomials, write what is left behind, then close the parenthesis. 5ac(4b + 3a - 1) List the factors of 5ac. 5ac = 5 · a · c

© William James Calhoun, : Factoring Using the Distributive Property EXAMPLE 3: Factor 12ac + 21ad + 8bc + 14bd. Any time you encounter a polynomial with four terms and are asked to factor it, there will be only one option open to you. Group the first two terms. Group the second two terms. Pull out the GCF of the first two. Make sure the signs of the second pair of terms is the same as the signs of the first pair. If so, write a plus sign. If not, write a minus sign and change the signs of both the second pair. 3a(4c + 7d) Pull out the GCF of the second two. + 2b(4c + 7d) Now, you have two terms with something the same in them… Both terms have a (4c + 7d). Pull the (4c + 7d) out, {{ (4c + 7d) open parenthesis, ( write what is left behind, and close the parenthesis. 3a+2b) (4c + 7d)(3a + 2b) This rule will change in higher mathematics, but for now, you get the easy life. This next process will work for any 4-nomials you encounter.

© William James Calhoun, : Factoring Using the Distributive Property EXAMPLE 4: Factor 15x - 3xy + 4y Here is one where the signs will not line up right. You will need to pull out a negative sign from the second pairing. Group the first two terms. Group the second two terms. Pull out the GCF of the first two. Make sure the signs of the second pair of terms is the same as the signs of the first pair. If so, write a plus sign. If not, write a minus sign and change the signs of both the second pair. 3x(5 - y) Pull out the GCF of the second two. + 4(y - 5) Now, you have two terms with something the same in them… Both terms have a (5 - y). Pull the (5 - y) out, open parenthesis, ( write what is left behind, and close the parenthesis. 3x-4) (5 - y)(3x - 4) (5 - y) {{ 3x(5 - y) - 4(-y + 5) 3x(5 - y) - 4(5 - y)

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