<s>[INST] Factorize the polynomial $x^3+y^3+z^3-3xyz$ I want to factorize the polynomial $x^3+y^3+z^3-3xyz$. Using Mathematica I find that it equals $(x+y+z)(x^2+y^2+z^2-xy-yz-zx)$. But how can I factorize it by hand?
 [/INST] \begin{align}
x^3+y^3+z^3-3xyz\\
&= x^3+y^3+3x^2y+3xy^2+z^3-3xyz-3x^2y-3xy^2\\
&= (x+y)^3+z^3-3xy(x+y+z)\\
&= (x+y+z)((x+y)^2+z^2-(x+y)z)-3xy(x+y+z)\\
&= (x+y+z)(x^2+2xy+y^2+z^2-yz-xz-3xy)\\
&= (x+y+z)(x^2+y^2+z^2-xy-yz-zx)
\end{align}
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