So the discriminant of a polynomial of degree $n$ in the form of determinant of the resultant matrix can be written as
$$\det(D)\det(A-BD^{-1}C)$$
where $A, B, C, D$ are block matrices of the resultant matrix. $A$ is $(n-1)\times (n-1)$, $B$ is $(n-1)\times n$ and so on.
Now is it possible to find the valuation of $\det(A-BD^{-1}C)$ without explicitly calculating the discriminant?