www.VirtualNerd.com

How Do You Factor a 6-Term Polynomial by Grouping?

Factor 2x3 + 3x2 - 5x + 8x2y + 12xy – 20y by grouping.

Summary

  1. 'x' is a common factor of the first three terms
  2. '4y' is a common factor of the second three terms
  3. When we group terms together with parentheses, we're using the Associative Property
  4. We can re-write each group by factoring out its greatest common factor, or GCF
  5. The trinomial 2x2+3x-5 is common to both parts, so we can factor it from each
  6. We can factor the trinomial into two binomials

Notes

    1. A monomial has only one term
    2. Even though x2 is common to the first two terms, it would be better if we could split this whole polynomial into two groups of three terms each
    1. The first three terms of the polynomial are 2x3, 3x2, and -5x
    2. 'x' is the greatest common monomial factor of these terms, so it can be factored out of each
    3. 'GCF' stands for greatest common factor
    1. The second three terms of the polynomial are 8x2y, 12xy, and -20y
    2. '4y' is the greatest common monomial factor of these terms, so it can be factored out of each
    3. 'GCF' stands for greatest common factor
    1. The Associative Property says that we can regroup terms however we want when we're adding and subtracting
    2. So we can put parentheses around the first three terms and the last three terms to group them together
    3. The first three terms, 2x3, 3x2, and -5x are one group
    4. The second three terms, 8x2y, 12xy, and -20y are another group
    1. We can factor a greatest common factor out of each group
    2. The GCF of the first group is x, so factoring that out gives us x(2x2+3x-5)
    3. The GCF of the second group is 4y, so factoring that out gives us 4y(2x2+3x-5)
    4. 'GCF' stands for 'greatest common factor'
    1. Since we're factoring by grouping, we need to check the rewritten expression for another common factor
    2. This time, the new common factor is a trinomial
    3. It has three terms
    4. Remember, 'poly' means 'many' so a trinomial is just a three-term polynomial
    1. The polynomial 2x2+3x-5 is the common factor
    2. Since we're multiplying the whole polynomial by both x and 4y, we can factor it out just like if it were a single number or variable
    3. Factoring the polynomial out of the first part leaves 'x' and out of the second part leaves '4y'
    4. So we have (x+4y) times the polynomial we factored out
    1. Since x+4y is a first degree polynomial with no more common factors, it can't be factored anymore
    2. But the trinomial we just factored out can be factored some more
    3. The six term polynomial we started with can be factored into three 1st degree binomials:
    4. (2x+5), (x-1), and (x+4y)