Question

How do you factor `m^4 - n ^4`?

Answer 1

`(m^2-n^2)(m^2+n^2)`

Explanation:

Remember the `a^2-b^2=(a-b)(a+b)`

Note that `m^4` is the same as `(m^2)^2` and the same for `n`. So we may write this as:
`color(white)("d")a^2color(white)("dd")-color(white)("d")b^2color(white)("dd") ->color(white)("dd")(a-b)(a+b)`

`(m^2)^2-(n^2)^2 ->(m^2-n^2)(m^2+n^2)`

Answer 2

`(m^2+n^2)(m+n)(m-n)`

Explanation:

What we have is a difference of squares. Recall that if we have an expression

`color(steelblue)a^2-color(steelblue)b^2`

This has an expansion of

`color(purple)((a+b))color(steelblue)((a-b))`

Let's rewrite our expression as

`color(steelblue)((m^2))^2-color(steelblue)((n^2))^2`

Since we have a difference of squares, we can rewrite this as

`color(purple)((m^2+n^2))color(steelblue)((m^2-n^2))`

What we have in purple is a sum of squares, which can't be factored with real numbers, but we have another difference of squares in blue.

Factoring this, we get

`bar( ul( | color(white)(2/2) color(purple)((m^2+n^2))color(steelblue)((m+n)(m-n)) color(white)(2/2)| ))`

Hope this helps!