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. $56ab+14-49a-16b$