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**Extracting Factors from Polynomials**

Learn to extract the greatest common factor from a polynomial.

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Extracting Factors To factor a polynomial, we first begin by determining if the polynomial has a monomial factor other than 1. We need to check to see if the terms of the polynomial have a GCF (greatest common factor). If so, we can extract that monomial factor by dividing the polynomial by that factor. The quotient from that division is the second factor of the polynomial. FHS Polynomials

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Finding the GCF To find the greatest common factor (GCF) of two (or more) terms in a polynomial: Find the prime factorization of the coefficient of each term and then expand each monomial term. Find all of the common factors. Multiply these common factors together to get the greatest common factor (GCF). FHS Polynomials

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Prime Factorization To review how to find the prime factorization of a number, let’s look at a couple of examples. 45 5 9 3 75 3 25 5 Prime Factorization of 45 is 3·3·5 Prime Factorization of 75 is 3·5·5 FHS Polynomials

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Expanding a Monomial To expand a monomial, we find the prime factorization of the coefficient, and write the variables without exponents. For example: 24x2y3 = 15a2b = 8xyz = 2 · 2 · 2 · 3 · x · x · y · y · y 3 · 5 · a · a · b 2 · 2 · 2 · x · y · z FHS Polynomials

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**Finding the GCF 15x + 45x2 15x = 3 · 5 · x 45x2 = 3 · 3 · 5 · x · x**

To find the GCF of the terms in the polynomial, expand each term and find the common factors: Let’s look at this example: 15x + 45x2 15x = 3 · 5 · x 45x2 = 3 · 3 · 5 · x · x GCF = 3 · 5 · x = 15x FHS Polynomials

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**Factoring a Polynomial**

Once you have found the GCF, that will be the first factor. It is written in front of a set of parentheses for the paired factor. The numbers and variables that are left after the GCF has been removed go on the inside of the parentheses. This becomes the paired factor. 15x = 3 · 5 · x 45x2 = 3 · 3 · 5 · x · x The GCF was 15x 1 15x + 45x2 = 15x ( ) 3x FHS Polynomials

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**Finding the GCF 4n 4 + 6n 3 – 8n 2 Let’s try another example**

4n 4 = 2 · 2 · n · n · n · n 6n 3 = 2 · 3 · n · n · n 8n 2 = 2 · 2 · 2 · n · n GCF = 2 · n · n = 2n 2 4n 4 + 6n 3 – 8n 2 = 2n 2( – ) 2n 2 3n 4 FHS Polynomials

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