Let $F$ be a field and $f(X), g(X) \in F[X]$ two coprime non constant polynomials.
Question: Is the polynomial $P(X,Y):= f(X) g(Y)- g(X) f(Y)$ irreducible in the ring $F[X,Y]$ of polynomials in two variables?
The claim is clearly wrong if $f$ and $g$ are not coprime, so it's neccessary assumption. A possible approach I used was to divide $P(X,Y)$ by $f(X)$ and think about the polynomial $g(Y)-S \cdot f(Y) \in F[S][Y]$ where $S:= g(X)/f(X)$ and because the extension of $F$ by $F(S)$ is transcendental we can regard $S$ as a variable. I know that $g(Y)-S \cdot f(Y) $ is irreducible in $F[S][Y]$ but not know how it helps to answer the question if $P(X,Y)$ is irreducible in $F[X,Y]$.
In the ring $F(X)[Y]/(X-Y)$, you have $$P(X,Y)\bmod (X-Y)= f(X) g(X)- g(X) f(X)=0,$$ hence $P(X,Y)$ is divisible by $X-Y$.