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

Jun 24, 2018

$\left({m}^{2} - {n}^{2}\right) \left({m}^{2} + {n}^{2}\right)$

Explanation:

Remember the ${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

Note that ${m}^{4}$ is the same as ${\left({m}^{2}\right)}^{2}$ and the same for $n$. So we may write this as:
$\textcolor{w h i t e}{\text{d")a^2color(white)("dd")-color(white)("d")b^2color(white)("dd") ->color(white)("dd}} \left(a - b\right) \left(a + b\right)$

${\left({m}^{2}\right)}^{2} - {\left({n}^{2}\right)}^{2} \to \left({m}^{2} - {n}^{2}\right) \left({m}^{2} + {n}^{2}\right)$

Jun 24, 2018

$\left({m}^{2} + {n}^{2}\right) \left(m + n\right) \left(m - n\right)$

Explanation:

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

${\textcolor{s t e e l b l u e}{a}}^{2} - {\textcolor{s t e e l b l u e}{b}}^{2}$

This has an expansion of

$\textcolor{p u r p \le}{\left(a + b\right)} \textcolor{s t e e l b l u e}{\left(a - b\right)}$

Let's rewrite our expression as

${\textcolor{s t e e l b l u e}{\left({m}^{2}\right)}}^{2} - {\textcolor{s t e e l b l u e}{\left({n}^{2}\right)}}^{2}$

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

$\textcolor{p u r p \le}{\left({m}^{2} + {n}^{2}\right)} \textcolor{s t e e l b l u e}{\left({m}^{2} - {n}^{2}\right)}$

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

$\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{p u r p \le}{\left({m}^{2} + {n}^{2}\right)} \textcolor{s t e e l b l u e}{\left(m + n\right) \left(m - n\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}$

Hope this helps!