Let $\mathbf{A}$ be a commutative ring with unity, $f(x),g(x)\in\mathbf{A}[x]$. Denote Res($f,g$) be the resultant of $f$ and $g$. Assume that the coefficients of $f$ and $g$ are indeterminates and let $A[f,g]$ be the ring generated by these coefficients. Show that $\operatorname{Res}\left(f,g\right)$ is irreducible over $A[f,g]$.
This is a useful fact to know, as it implies that any polynomial in $A\left[f,g\right]$ that vanishes whenever $f$ and $g$ have a common root will be a multiple of $\operatorname{Res}\left(f,g\right)$, at least when $\mathbf{A}$ is an algebraically closed field. But how is it proved?