Polynomial Factorisation

Question

Consider that we have a polynomial like $$x^3- (a + b +c ) x^2+abx-abc+s$$ Which is multiplication of $$(x-a)(x-b)(x-c)+s$$ Is it possible to reach value= $abc$ knowing the Coefficients and exponents of the upper polynomial or factorizing of the polynomial is difficult like integer factorization ?

Answer 1

So you said you are given a polynomial, and you wish to find $abc$ and $s$ which satisfies $$x^3 - (a+b+c)x^2 + abx - abc + s = (x-a)(x-b)(x-c)+s$$

By Vieta's formula, when we compare the coefficient of $x$, we know $$\begin{align}ab =& ab+bc+ca\\0=&c(a+b) \end{align}$$

If $a+b = 0$, the coefficient of $x^2$ gives $-c$. Multiply this with the negative of the coefficient of $x$ to get $abc$. Otherwise, if $c = 0$, we get $abc = 0$ definitely.

Therefore there can be two solutions for $abc$ from such polynomial $x^3 + Ax^2 + Bx + C$: either $abc = -AB$, or $abc = 0$.