Expressing a product in terms of the sum

Question

While solving a problem, I got to the expression $$(-a+b+c)(a-b+c)(a+b-c).$$ I would like to express it in terms of the sum $a+b+c$. Is there any possibility?

Answer 1

I don't think so. Here is an heuristic idea:

Call $s=a+b+c$. Then $$(s-2a)(s-2b)(s-2c) = s^3 -2(a+b+c)s^2 + 4(ab+bc+ac)s - abc = $$$$=-s^3 + 4(ab+bc+ac)s - abc$$

but now, $abc$ and $ab+bc+ac$ could be anything (they are somehow independent by $a+b+c$).

The reason is that the system $$\left\{ \begin{matrix} a+b+c = s \\ ab+ac+bc = t \\ abc=r \end{matrix}\right.$$ has always a solution $(a,b,c)$ for arbitrary values of $s,t,r$ (they are the roots of the polynomial $X^3-sX^2+tX-r$).