<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] Use Newton's identities: 
$p_3=e_1 p_2 - e_2 p_1 + 3e_3$ and so $p_3-3e_3 =e_1 p_2 - e_2 p_1 = p_1(p_2-e_2)$ as required.
Here
$p_1= x+y+z = e_1$
$p_2= x^2+y^2+z^2$
$p_3= x^3+y^3+z^3$
$e_2 = xy + xz + yz$
$e_3 = xyz$
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