Chapter 7: Factoring

7.2 Factoring by Grouping

First thing to do when factoring is to factor out the GCF. This GCF is often a monomial, like in the problem 5xy + 10xz where the GCF is the monomial 5x, so you would have 5x(y + 2z). However, a GCF does not have to be a monomial; it could be a binomial. Consider the following two examples.

Example 7.2.1

Find and factor out the GCF for 3ax - 7bx.

By observation, one can see that both have x in common.

This means that 3ax - 7bx =  x(3a - 7b).

Example 7.2.2

Find and factor out the GCF for 3a(2a + 5b) - 7b(2a + 5b).

Both have (2a + 5b) as a common factor.

This means that if you factor out (2a + 5b), you are left with 3a - 7b.

The factored polynomial is written as (2a + 5b)(3a - 7b).

In the same way as factoring out a GCF from a binomial, there is a process known as grouping to factor out common binomials from a polynomial containing four terms.

Find and factor out the GCF for 10ab + 15b^2 + 4a + 6b.

To do this, first split the polynomial into two binomials.

10ab + 15b^2 + 4a + 6b becomes 10ab + 15b^2 and 4a + 6b.

Now find the common factor from each binomial.

10ab + 15b^2 has a common factor of 5b and becomes 5b(2a + 3b).

4a + 6b has a common factor of 2 and becomes 2(2a + 3b).

This means that 10ab + 15b^2 + 4a + 6b = 5b(2a + 3b) + 2(2a + 3b).

5b(2a + 3b) + 2(2a + 3b) can be factored as (2a + 3b)(5b + 2).

Questions

Factor the following polynomials.

  1. 40r^3-8r^2-25r+5
  2. 35x^3-10x^2-56x+16
  3. 3n^3-2n^2-9n+6
  4. 14v^3+10v^2-7v-5
  5. 15b^3+21b^2-35b-49
  6. 6x^3-48x^2+5x-40
  7. 35x^3-28x^2-20x+16
  8. 7n^3+21n^2-5n-15
  9. 7xy-49x+5y-35
  10. 42r^3-49r^2+18r-21
  11. 16xy-56x+2y-7
  12. 3mn-8m+15n-40
  13. 2xy-8x^2+7y^3-28y^2x
  14. 5mn+2m-25n-10
  15. 40xy+35x-8y^2-7y
  16. 8xy+56x-y-7
  17. 10xy+30+25x+12y
  18. 24xy+25y^2-20x-30y^3
  19. 3uv+14u-6u^2-7v
  20. 56ab+14-49a-16b

<a class=”internal” href=”/intermediatealgebraberg/back-matter/answer-key-7-2/”>Answer Key 7.2

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