<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] \begin{align}
a (b^2-c^2)+b (c^2-a^2)+c (a^2-b^2)
&=ab^2-ac^2+bc^2-ba^2+ca^2-cb^2\\
&=a^2c-a^2b+ab^2-ac^2+bc^2-cb^2\\
&=a^2 (c-b)-a (c^2-b^2)+bc (c-b)\\
&=(c-b)(a^2-ac-ab+bc)\\
&=(c-b)(bc-ab-ac-a^2)\\
&=(c-b)(b (c-a)-a (c-a))\\
&=(c-b)(b-a)(c-a)\\
&=(a-b)(b-c)(c-a)
\end{align}
</s>