# How do you factor the expression 2a ^5 - 32ab ^8?

Jan 29, 2016

First, extract a common factor.

#### Explanation:

2a(${a}^{4} - 16 {b}^{8}$)

Now you can factor the interior of the parentheses as a difference of squares.
.
2a(a^2 - 4b^4)(a^2 + 4b^4)

As you can see, we can do another difference of squares.

2a(a - 2b^2)(a + 2b^2)(a^2 + 4b^4)

This is fully factored, since we can simplify it no more.

Hopefully this helps!

Jan 29, 2016

$2 a \left(a - 2 {b}^{2}\right) \left(a + 2 {b}^{2}\right) \left({a}^{2} + 4 {b}^{4}\right)$

#### Explanation:

1. common factor in the 2 terms ? yes 2a

expression is :$2 a \left({a}^{4} - 16 {b}^{8}\right)$

Note: difference of squares ${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

now. ${a}^{4} - 16 {b}^{8} \textcolor{b l a c k}{\text{ is a difference of squares}}$

where $a = {a}^{2} , b = 4 {b}^{4}$

so ${a}^{4} - 16 {b}^{8} = \left({a}^{2} - 4 {b}^{4}\right) \left({a}^{2} + 4 {b}^{4}\right)$

Note now that $\left({a}^{2} - 4 {b}^{4}\right)$ is a difference of squares.

In the same way as above : ${a}^{2} - 4 {b}^{4} = \left(a - 2 {b}^{2}\right) \left(a + 2 {b}^{2}\right)$

'Pull this together ' to get:

$2 {a}^{5} - 32 a {b}^{8} = 2 a \left(a - 2 {b}^{2}\right) \left(a + 2 {b}^{2}\right) \left({a}^{2} + 4 {b}^{4}\right)$