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How Do You Factor a 4-Term Polynomial by Grouping?

Factor the polynomial 3x+7y-21-xy

Summary

  1. 7y and 21 have a common factor of 7
  2. 3x and xy have a common factor of x
  3. Factor an x out of the first set of parentheses
  4. Factor a 7 out of the second set of parentheses
  5. Factor a -1 out of the second set of parentheses
  6. Factor out a 3-y
  7. FOIL the binomials back together to check your answer

Notes

    1. 3x factors into 3•x
    2. 7y factors into 7•y
    3. So they don't have any common factors
    1. 3x factors into 3•x
    2. 21 factors into 3•7
    3. So they have a common factor of 3
    1. 3x factors into 3•x
    2. xy factors into x•y
    3. So they have a common factor of x
    1. 7y factors into 7•y
    2. 21 factors into 3•7
    3. So they have a common factor of 7
    1. 7y factors into 7•y
    2. xy factors into x•y
    3. So they have a common factor of y
    1. 7y and 21 have a common factor of 7
    2. 3x and xy have a common factor of x
    1. We could have grouped together 3x and 21, since they have a common factor of 3
    2. Or we could have grouped together 7y and -xy, since they have a common factor of y
    3. Either way, we'll get the same answer
    1. The associative property says that we can rearrange the order of terms in an expression without changing the value
    2. We just need to make sure each term keeps its sign when we move it
    3. Rearranging the terms will make the polynomial easier to factor
    1. Remember, we can move terms around without changing the value of the polynomial
    2. Since 3x and xy share a common factor of x, we put them next to each other
    3. Don't forget to bring the negative in front of the xy!
    4. 7y and 21 are grouped together because they have a common factor of 7
    5. Again, remember to keep the negative sign with the 21
    1. Now we can factor out the common factor from each set of parentheses
    1. Remember, 3x and xy share a common factor of x
    1. Bring the x out in front of the parentheses and divide it from each term
    2. So 3x/x leaves 3
    3. And -xy/x leaves -y
    1. Remember, 7y and -21 share a common factor of 7
    1. Bring the 7 out in front of the parentheses and divide it from each term
    2. So 7y/7 leaves y
    3. And -21/7 leaves 3
    1. We have a 3-y and a y-3
    2. Notice how these polynomials are almost the same, only their signs are different
    1. If we can get them to be the same, we can factor them out
    1. This will change the signs of both terms in the parentheses
    2. So bring the negative out front
    3. y becomes -y
    4. -3 becomes 3
    5. So y-3 becomes -y+3, which is the same as 3-y
    1. -y+3 is the same as 3-y
    2. So now both our parentheses are the same!
    1. 3-y is now a common factor
    1. Our first term is x(3-y)
    2. Our second term is -7(3-y)
    3. So 3-y is a common factor in these two terms
    1. 3-y is our common factor
    1. 3-y is our common factor
    1. We need to take the 3-y out of each of our terms
    1. Bring the 3-y out front
    2. Then put what's left in another set of parentheses: (x-7)
    1. Our polynomial factored into (3-y)(x-7)
    1. It's a useful method if your terms have common factors you can factor out
    1. First we found the common factors of 3x & xy, and 7y & 21
    2. Then we rearranged the terms and factored out an x and a 7
    3. Then we factored out a 3-y
    1. To check our answer, we can FOIL our two binomials back together
    2. Remember, FOIL is a method we use to multiply two binomials
    1. 3 and x are our FIRST terms
    1. 3 and -7 are our OUTER terms
    1. -y and x are our INNER terms
    1. -y and -7 are our LAST terms
    1. We got them in a different order, but we have the same monomials with the same signs