polynomial factorization when exponent is not given

Question

How can I factorise this equation, given i already know some of its factors which are: $(a-b)(b-c)(c-a).$ Equation is : $$a^nb^{n-1} + a^{n-1}c^n - a^nc^{n-1} - a^{n-1}b^n - b^{n-1}c^n + b^nc^{n-1}$$

Answer 1

Denote the above polynomial by $f(a,b,c)$. Then $f(a,b,c)=0$ for $a=b$ or $a=c$ or $b=c$. Hence we can write $$ f(a,b,c)=g(a,b,c)(a-b)(a-c)(b-c). $$ For $n=2$ we have $g(a,b,c)=1$. For $n\ge 3$ the homogeneous polynomial $g$ is given by $$ g(a,b,c)=a^{n-2}b^{n-2}+a^{n-2}b^{n-3}c+a^{n-2}b^{n-4}c^2+\ldots +b^{n-2}c^{n-2}, $$ where all monomials $a^ib^jc^k$ appear with $i+j+k=2n-4$. For example, for $n=3$ we have $g(a,b,c)=ab+ac+bc$, and for $n=4$ we have $$g(a,b,c)=a^2b^2+a^2bc+a^2c^2+ab^2c+abc^2+b^2c^2.$$

However, in general the polynomial $g(a,b,c)$ has no further nontrivial factors.