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 [latex]5xy + 10xz[/latex] where the GCF is the monomial [latex]5x[/latex], so you would have [latex]5x(y + 2z)[/latex]. 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 [latex]3ax - 7bx[/latex].

By observation, one can see that both have [latex]x[/latex] in common.

This means that [latex]3ax - 7bx = x(3a - 7b)[/latex].

Example 7.2.2

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

Both have [latex](2a + 5b)[/latex] as a common factor.

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

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

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 [latex]10ab + 15b^2 + 4a + 6b[/latex].

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

[latex]10ab + 15b^2 + 4a + 6b[/latex] becomes [latex]10ab + 15b^2[/latex] and [latex]4a + 6b[/latex].

Now find the common factor from each binomial.

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

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

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

[latex]5b(2a + 3b) + 2(2a + 3b)[/latex] can be factored as [latex](2a + 3b)(5b + 2)[/latex].

Questions

Factor the following polynomials.

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

Answer Key 7.2

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