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Extracting Factors from Polynomials Learn to extract the greatest common factor from a polynomial.

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FHSPolynomials2 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.

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FHSPolynomials3 Finding the GCF To find the greatest common factor (GCF) of two (or more) terms in a polynomial: 1.Find the prime factorization of the coefficient of each term and then expand each monomial term. 2.Find all of the common factors. 3.Multiply these common factors together to get the greatest common factor (GCF).

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FHSPolynomials4 Prime Factorization To review how to find the prime factorization of a number, let’s look at a couple of examples Prime Factorization of 45 is 3·3·5 Prime Factorization of 75 is 3·5·5

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FHSPolynomials5 Expanding a Monomial To expand a monomial, we find the prime factorization of the coefficient, and write the variables without exponents. For example: 24x 2 y 3 = 15a 2 b = 8xyz = 2 · 2 · 2 · 3 · x · x · y · y · y 3 · 5 · a · a · b 2 · 2 · 2 · x · y · z

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FHSPolynomials6 Finding the GCF 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 = 3 · 5 · x 45x 2 = 3 · 3 · 5 · x · x 15x + 45x 2 GCF = 3 · 5 · x = 15x

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FHSPolynomials7 45x 2 = 3 · 3 · 5 · x · x 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 + 45x 2 = 15x ( + ) 3x 1 15x = 3 · 5 · x The GCF was 15x

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FHSPolynomials8 4n 4 = 2 · 2 · n · n · n · n 8n 2 = 2 · 2 · 2 · n · n Finding the GCF Let’s try another example 6n 3 = 2 · 3 · n · n · n 4n 4 + 6n 3 – 8n 2 2n 2 ( + – ) GCF = 2 · n · n = 2n 2 4n 4 + 6n 3 – 8n 2 = 2n 2 3n 4

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