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.

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.