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Extracting Factors from Polynomials

Extracting Factors from Polynomials. Learn to extract the greatest common factor from a polynomial. Extracting Factors. To factor a polynomial, we first begin by determining if the polynomial has a monomial factor other than 1.

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Extracting Factors from Polynomials

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

  2. 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. Polynomials

  3. 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). Polynomials

  4. 45 75 25 5 3 9 3 5 3 5 Prime Factorization To review how to find the prime factorization of a number, let’s look at a couple of examples. 1. 2. Prime Factorization of 75 is 3·5·5 Prime Factorization of 45 is 3·3·5 Polynomials

  5. Expanding a Monomial • To expanda 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 Polynomials

  6. 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+ 45x2 15x= 3 · 5 · x 45x2= 3 · 3 · 5 · x · x GCF=3 · 5 · x=15x Polynomials

  7. 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. 45x2= 3 · 3 · 5 · x · x 15x= 3 · 5 · x The GCF was 15x 1 3x 15x ( + ) 15x+ 45x2 = Polynomials

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

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