Resultants and algebraic numbers

Question

So if $\alpha\in\mathbb{R}$ and $\beta\in\mathbb{R}$ are algebraic numbers and their minimal polynomials are $f$ and $g$ in $\mathbb{Q}[x]$ respectively, then it is claimed that $h(z)=\text{Res}(f(x),g(z-x))$ is the minimal polynomial of $z=\alpha+\beta$. Now I understand why $\alpha+\beta$ is a root, but I don't understand why the resultant is irreducible. I know its an irreducible in the intermediates corresponding to the coefficients of $f$ and $g$ when they are arbitrary but I don't understand why we get irreducibility in the variable $z$, this was claimed but doesn't seem obvious or true in generality. Can someone clarify this for me please.