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Factoring Using the Distributive Property

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**Distributive Property**

Apply the distributive property to 3c(4c – 2). In this example, the result of the multiplication, 12c2 – 6c, is the product. The factors in this problem were the monomial, 3c, and the polynomial, (4c-2). 3c(4c-2) is the factored form of 12c2 – 6c. Polynomials can be written in factored form by reversing the process of the distributive property.

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Factoring Factor the following polynomial using the distributive property. Step 1: Find the GCF for both terms. 9m3n2 = 3 3 m m m n n 24mn4 = 2 2 2 3 m n n n n The GCF is 3mn2.

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Factoring Step 2: Divide each term of the polynomial by the GCF.

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Factoring Step 3: Write the polynomial as the product of the GCF and the remaining factor of each term using the distributive property.

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**Factoring Polynomials**

What happens when the distributive property is applied to a problem such as (2a + 3b)(5c + 8d)? This problem can be rewritten by distributing each term in the first set of parentheses with each term in the second set of parentheses as follows. We now have a polynomial with four terms and there doesn’t appear to be a factor that is common to all four terms.

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**Factoring Polynomials**

How can we factor 4ab + 2ac + 8xb + 4xc? The answer is by grouping terms which do have something in common. Often, this can be done in more than one way. For example: or Next, find the greatest common factor for the polynomial in each set of parentheses. The GCF for (4ab+2ac) is 2a. The GCF for (8xb + 4xc) is 4x. The GCF for (4ab +8xb) is 4b. The GCF for (2ac + 4xc) is 2c.

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**Factoring Polynomials**

Divide each polynomial in parentheses by the GCF.

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**Factoring Polynomials**

Write each of the polynomials in parentheses as the product of the GCF and the remaining polynomial. Apply the distributive property to any common factors. Factor further if necessary. Notice that it did not matter how the terms were originally grouped, the factored forms of the polynomials are identical.

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**You Try It 1. Find the GCF of the terms in each expression.**

a. 4x2 + 6xy b. 60a2 + 30ab – 90ac 2. Factor each polynomial. a. 6tt + 42ts b. 7a2 –4ab + 12b4 – 21ab3

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**Problem 1 4x2 = 2 2 x x The GCF is 2x . 6xy = 2 3 x y**

60a2 = 2 3 a a 30ab = a b 90ac = 2 a c The GCF is 30a .

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**Problem 2 6t2 = 2 3 t t 42ts = 2 3 7 t s The GCF is 6t.**

Divide each term by the GCF. Write the polynomial as the product of the GCF and the remaining factor of each term using the distributive property.

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Problem 2 Use grouping symbols to group terms which have a common factor. Divide the polynomial in each set of parentheses by its common factor. Factor out a (-1) from the second set of parentheses. Apply the distributive property.

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