I want to factorize $a^2(b − c)^3 + b^2(c − a)^3 + c^2(a − b)^3$ . By inspection , I can see that substituting $b$ for $a$ yields $0$ thus $(a-b)$ is a factor . Similarly $(c-a)$ and $(b-c)$ are factors . But I can't figure out other factors. Yes , I know that I can find it using long division , but there must be an easy way . Please help.
( Using wolfram alpha , I know that the factorization is $(a-b)(c-a)(b-c)(ab+ac+bc)$ , but I want to know how to figure that out . ( without long division ) )
Edit : see this page to understand what I mean - mathnerds - Is there any similar method for this question ?