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How Do You Factor the Greatest Common Factor out of a Polynomial?

Find and factor out the greatest common factor from 6x2y+14xy2-42xy-2x2y2

Summary

  1. GCF stands for "greatest common factor"
  2. Write out the factors of each monomial
  3. 2 is a common factor
  4. x is a common factor
  5. y is a common factor
  6. Multiply the common factors together to get a GCF of 2xy
  7. Use the distributive property in reverse to factor out the GCF
  8. The factors that are left after taking out the GCF make the terms inside the parentheses

Notes

    1. We need to factor each monomial individually
    1. We need to write 6x2y as a product of its factors
    1. We need to write 14xy2 as a product of its factors
    1. We need to write 42xy as a product of its factors
    1. We need to write 2x2y2 as a product of its factors
    1. Now that all our monomials are completely factored, we can pick out the factors they have in common
    1. If the factor is not in the first term, then it's not common to all of the terms because it's missing from the first one!
    1. If the factor is not in the first term, then it's not common to all of the terms because it's missing from the first one!
    1. Each of the monomials has a factor of 2
    2. So 2 is a common factor
    1. There is no 3 in 14xy2 or 2x2y2
    2. In order to be a common factor, 3 would need to be in ALL the terms
    3. Since it's not, 3 is not a common factor
    1. Each of the monomials has a factor of x
    2. So x is a common factor
    1. Our first term, 6x2y has 2 x's in its factorization
    2. But not all the monomials have 2 x's: 14xy2 and 42xy each only have 1
    1. So we only have one x as a common factor, not two
    2. We would only have two x's as common factors if EVERY term had two x's
    1. Each of the monomials has a factor of y
    2. So y is a common factor
    1. Now that we have identified the common factors of 2, x, and y, we need to multiply them to find the greatest common factor
    1. Now that we have our GCF of 2xy, we can factor it out of each of our terms
    1. When we distribute, we multiply something into parentheses
    2. Here, we are going to pull our factor of 2xy OUT of the parentheses
    1. So then if we wanted to redistribute the GCF back into the parentheses, we'd get our original polynomial
    1. 3x is what is left when we factor out a 2xy from 6x2y
    1. 7y is what is left when we factor out a 2xy from 14xy2
    1. -21 is what is left when we factor out a 2xy from -42xy
    1. -xy is what is left when we factor out a 2xy from -2x2y2
    1. Since our GCF, 2xy, is positive, factoring it out doesn't change the signs of our terms
    1. That wasn't so bad, right?
    1. Here's a hint that might make factoring a little easier
    1. So we already have each term's factors written out
    1. The factors we circled were the factors of the GCF, 2, x, and y
    2. Once you take those out, you're left with a 3 and an x
    3. These are the factors that were NOT common to all the terms
    4. So that's what's left after you factor out the GCF
    1. Pretty handy little trick, right?