Suppose that we have two polynomials that split: $$\begin{align*} f(x)=\sum_{k=0}^d a_{d-k}x^k&=\prod_{i=1}^d (x-\lambda_i),\\ g(x)=\sum_{k=0}^e b_{e-k}x^k&=\prod_{j=1}^e (x-\mu_j).\\ \end{align*}$$ Then the following result is often attributed to James Joseph Sylvester: $$ \det\begin{pmatrix} a_0 & a_1 & \cdots & a_d && \\ & \ddots &\ddots &&\ddots & \\ && a_0 & a_1 & \cdots & a_d \\ b_0 & b_1 & \cdots & b_e && \\ & \ddots & \ddots && \ddots & \\ && b_0 & b_1 & \cdots & b_e \end{pmatrix} = \prod_{i,j} (\lambda_i-\mu_j). $$

Can someone help me track down the original reference?