Reference request: multivariable (homogeneous) polynomial division algorithm
I am looking for a reference for a result that shows that a (homogeneous) polynomial in an even number of variables (or any number of variables) can be divided by linear polynomials corresponding to roots.
Context:
The specific application I have in mind is rigorously proving the following facts related to resultants:
1. Show that $\det (Syl (f,g)) = \lambda\ \Pi_{i=1,j=1}^{i=n,j=m} (r_i - s_j)$, where $Syl(f,g)$ is the Sylvester matrix of the degree $n$ polynomial $f$ and the degree $m$ polynomial $g$.
2. Given: $\det (Syl (f,g)) = \lambda\ \Pi_{i=1,j=1}^{i=n,j=m} (r_i - s_j)$, show that $\lambda = 1$.
Basically I want to show that since the points $(r_i : s_j)$ are obviously zeros of the determinant of the Sylvester matrix, the corresponding linear polynomials divide $\det(Syl(f,g))$ (for the first part). This could be done I think if I could cite the existence of a suitable multivariable polynomial division algorithm. For the second part, I'll figure it out, but now I have no idea besides Vieta's formulas.
Related questions whose answers don't answer my question (I think):
Algorithms for factoring multivariate polynomials
Division algorithm of multivariate polynomial
Division algorithm for multivariate polynomials?
How to show that a root of a bivariate homogeneous polynomial divides the polynomial?
