# How do you factor x^4 – y^4?

Jul 22, 2018

$\left(x + y\right) \left(x - y\right) \left({x}^{2} + {y}^{2}\right)$

#### Explanation:

Expression $= {x}^{4} - {y}^{4}$

Recall the factorization of the difference of two squares:

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

In our example, we will use this factorization twice.

Note: ${x}^{4} = {\left({x}^{2}\right)}^{2} \mathmr{and} {y}^{4} = {\left({y}^{2}\right)}^{2}$

Applying the factorization above:

Expression $= \left({x}^{2} + {y}^{2}\right) \left({x}^{2} - {y}^{2}\right)$

Now, the second factor above is also the difference of two squares.

Hence, Expression $= \left(x + y\right) \left(x - y\right) \left({x}^{2} + {y}^{2}\right)$

$\left(x - y\right) \left(x + y\right) \left({x}^{2} + {y}^{2}\right)$

#### Explanation:

Given that

${x}^{4} - {y}^{4}$

$= {\left({x}^{2}\right)}^{2} - {\left({y}^{2}\right)}^{2}$

$= \left({x}^{2} - {y}^{2}\right) \left({x}^{2} + {y}^{2}\right)$

$= \left(x - y\right) \left(x + y\right) \left({x}^{2} + {y}^{2}\right)$