I'm learning Computer Algebra and met an exercise asking me to prove that $ \operatorname{res}(fg,h)=\operatorname{res}(f,h)\cdot\operatorname{res}(g,h) $ where $f(x)$, $g(x)$ and $h(x)$ are polynomials and $\operatorname{res}$ stand for resultant.
I know if we use the following fact: $\operatorname{res}(f,g)=\operatorname{lc}(f)^{\operatorname{deg}(g)}\operatorname{lc}(g)^{\operatorname{deg}(f)}\prod_{(x,y):f(x)=0,g(y)=0} (x-y)$ the proof would become obvious.
However, in our book the resultant was defined as the determinant of the Sylvester matrix. So I just want to find a proof using this definition directly. (I don't want to prove the fact above first.)
(Supposing that $\deg f=m,\deg g= n, \deg h = p$)
I first tried the multiplication of matrices but found that the Sylvester matrix of $(f,h)$ is $(m+p)\times(m+p)$, the Sylvester matrix of $(g,h)$ is $(n+p)\times(n+p)$, thus they cannot be multiplied. I even tried to extend their Sylvester matrix to $(m+n+p)\times(m+n+p)$. But I still can't get any useful result.
Can you please help? Thank you!