Chapter 5.2: 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 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 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.

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

Answers to odd questions

1. $8r^2(5r-1)-5(5r-1)$
$(5r-1)(8r^2-5)$

3. $n^2(3n-2)-3(3n-2)$
$(3n-2)(n^2-3)$

5. $3b^2(5b+7)-7(5b+7)$
$(5b+7)(3b^2-7)$

7. $7x^2(5x-4)-4(5x-4)$
$(5x-4)(7x^2-4)$

9. $7x(y-7)+5(y-7)$
$(y-7)(7x+5)$

11. $8x(2y-7)+1(2y-7)$
$(2y-7)(8x+1)$

13. $2x(y-4x)+7y^2(y-4x)$
$(y-4x)(2x+7y^2)$

15. $5x(8y+7)-y(8y+7)$
$(8y+7)(5x-y)$

17. $12y+10xy+30+25x$
$2y(6+5x)+5(6+5x)$
$(6+5x)(2y+5)$

19. $-6u^2+3uv+14u-7v$
$-3u(2u-v)+7(2u-v)$
$(2u-v)(-3u+7)$

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