<s>[INST] Factorization of cyclic polynomial 
Factorize $$a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)$$ 

Since this is a cyclic polynomial, factors are also cyclic 
$$f(a) = a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)$$
$$f(b) = b(b^2-c^2)+b(c^2-b^2)+c(b^2-b^2) = 0 \Rightarrow a-b$$
is a factor of the given expression. Therefore, other factors are $(b-c)$ and $(c-a)$. The given expression may have a coefficient a constant factor which is nonzero. Let it be $m$.
$$\therefore a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2) = m(a-b)(b-c)(c-a)$$
Please guide further on how to find this coefficient.
 [/INST] From your last line $$a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)=m(a-b)(b-c)(c-a);$$
putting $a=0$, $b=1$, $c=2$ we get
$$
0(1^2-2^2)+1(2^2-0^2)+2(0^2-1^2)=m(0-1)(1-2)(2-0)\\
\implies 0+4-2=2m \implies 2m=2 \implies m=1.
$$
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