<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] Note that (can be easily seen with rule of Sarrus)$$
        \begin{vmatrix}
        x & y & z \\
        z & x & y \\
        y & z & x \\
        \end{vmatrix}=x^3+y^3+z^3-3xyz
$$
On the other hand, it is equal to (if we add to the first row 2 other rows)
$$
        \begin{vmatrix}
        x+y+z & x+y+z & x+y+z \\
        z & x & y \\
        y & z & x \\
        \end{vmatrix}=(x+y+z)\begin{vmatrix}
        1 & 1 & 1 \\
        z & x & y \\
        y & z & x \\
        \end{vmatrix}=(x+y+z)(x^2+y^2+z^2-xy-xz-yz)
$$
 just as we wanted. The last equality follows from the expansion of the determinant by first row.
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