<s>[INST] Factoring $x^4-11x^2y^2+y^4$ I am brushing up on my precalculus and was wondering how to factor the expression
$$
x^4-11x^2y^2+y^4
$$
Thanks for any help!
 [/INST] We notice this looks a bit like $(x^2-y^2)^2$, so we write
\begin{align*}
x^4 - 11x^2y^2 + y^4
&= (x^4 - 2x^2y^2+y^4) - 9x^2y^2 \\
&= (x^2-y^2)^2 - (3xy)^2 \\
&= (x^2 + 3xy - y^2)(x^2 - 3xy - y^2).
\end{align*}
It factors further, but no longer over the integers (there will be square roots involved).
We also could have used $(x^2+y^2)^2$ instead of $(x^2-y^2)^2$, to get
\begin{align*}
x^4 - 11x^2y^2 + y^4
&= (x^4 + 2x^2y^2+y^4) - 13x^2y^2 \\
&= (x^2+y^2)^2 - (\sqrt{13}\;xy)^2 \\
&= (x^2 + \sqrt{13}\;xy + y^2)(x^2 - \sqrt{13}\; xy + y^2).
\end{align*}
The two factorizations may look different, but they are the same four linear factors combined in two different ways.
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