Question

How do you factor `(a+b)^4 - (a-b)^4`?

Answer 1

=`4ab[(a+b)^2+ (a-b)^2]`

Explanation:

This is easier than it seems at first glance.
DO NOT multiply out the brackets - that will only make things worse!

This expression is written in the form `x^2 - y^2` which is difference of 2 squares.

Let (a+b) be `x` and (a-b) be `y`, just to make it easier to work with.

`x^4 - y^4` is also the difference of squares. `(x^2)^2 = x^4`

=`(x^2 + y^2)(x^2-y^2)`

=`color(blue)((x^2 + y^2))color(red)((x+y))color(orange)((x-y))" replace " x and y`

=`color(blue)([(a+b)^2+ (a-b)^2])color(red)([(a+b) +(a-b)])color(orange)([(a+b)- (a-b)])`

`color(blue)([(a+b)^2+ (a-b)^2])color(red)([(2a)color(orange)((2b))])`
=`4ab[(a+b)^2+ (a-b)^2]`