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.

You can put arbitrary values in $f(a,b,c)$ and get the value of $m$
Let's put $a=1, b=2, c=0$ in  $$ a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2) = m(a-b)(b-c)(c-a)$$
we get, $m = 1$
