Consider a $n\times n$ matrix $A$ whose elements are some polynomials in the indeterminates $x_1, x_2,\ldots,x_m$. To calculate the determinant of such a matrix, one of the usual ways is to treat the determinant as a polynomial in $x_1,\ldots,x_m$ and identify its factors. The usual idea being if $x = y$ makes the determinant vanish then $x -y$ is one of the factors. What I however do not understand is how to identify its order, that is to identify the exact $k$ such that $(x - y)^k$ is the factor.
If setting $x=y$ makes the rank go down by $k$, then $(x-y)^k$ is a factor. Harald Helfgott and I used this idea in evaluating a determinant http://www.combinatorics.org/Volume_6/Abstracts/v6i1r16.html); actually the determinant was evaluated earlier by this method by Zavrotsky. The reference we gave for the fact relating the rank of the matrix and the multiplicity of $x-y$ as a factor is R. A. Frazer, W. J. Duncan, and A. R. Collar, Elementary Matrices and Some Applications to Dynamics and Differential Equations, Cambridge University Press, 1947, page 17.