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

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Answer 1 · 2016-07-14T11:36:42

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

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. $(x^2)^2 = x^4$

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

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

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

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

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