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Factoring by Grouping

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What if you had a polynomial expression like 3x^2 - 6x + 2x - 4 in which some of the terms shared a common factor but not all of them? How could you factor this expression? After completing this Concept, you'll be able to factor polynomials like this one by grouping.

Fisch Video: Factoring Polynomials by Grouping

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CK-12 Foundation: 0913S Factoring By Grouping

Guidance

Sometimes, we can factor a polynomial containing four or more terms by factoring common monomials from groups of terms. This method is called factor by grouping.

The next example illustrates how this process works.

Example A

Factor 2x+2y+ax+ay .

Solution

There is no factor common to all the terms. However, the first two terms have a common factor of 2 and the last two terms have a common factor of a . Factor 2 from the first two terms and factor a from the last two terms:

2x + 2y + ax + ay = 2(x + y) + a(x + y)

Now we notice that the binomial (x + y) is common to both terms. We factor the common binomial and get:

(x + y)(2 + a)

Example B

Factor 3x^2+6x+4x+8 .

Solution

We factor 3x from the first two terms and factor 4 from the last two terms:

3x(x+2)+4(x+2)

Now factor (x+2) from both terms: (x+2)(3x+4) .

Now the polynomial is factored completely.

Practice

Factor by grouping.

  1. 6x^2-9x+10x-15
  2. 5x^2-35x+x-7
  3. 9x^2-9x-x+1
  4. 4x^2+32x-5x-40
  5. 2a^2-6ab+3ab-9b^2
  6. 5x^2+15x-2xy-6y

Additional Practice Opportunities (including self-check)

  1. Braingenie: Factoring Polynomials by Grouping Terms

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