Question

How do you factor $x^4-y^4$ ?

Answer 1

Calculate the factors of $x^4-y^4$:

Clearly $x=y$ satisfies the equation $x^4-y^4$

As $x^4-x^4=0$

So, $x-y$ is a factor of $x^4-y^4$ -Equation$\left(1\right)$

Also, $x=-y$ satisfies the equation $x^4-y^4$

As $x^4-{\left(-x\right)}^4=x^4-x^4$ equals $0$

So, $x+y$ is a factor of $x^4-y^4$ -Equation$\left(2\right)$

From Equation $\left(1\right)$ and Equation $\left(2\right)$, we can say that

$\left(x+y\right)\left(x-y\right)$ is a factor of $x^4-y^4$

That is $x^2-y^2$ is a factor of $x^4-y^4$

Divide $x^4-y^4$ by $x^2-y^2$

$x^2-y^2\left(x^2+y^2\right)x^4-y^4\underline{-\left(x^4-x^2{y}^2\right)}\left(x^2{y}^2-y^4\right)\underline{-\left(x^2{y}^2-y^4\right)}0$

So, $x^2+y^2$ is a factor of $x^4-y^4$

Hence, factors of $x^4-y^4$ are $(x^2+y^2)$, $(x+y)$ and $(x-y)$.