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](https://en.wikipedia.org/wiki/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?



History of Sylvester's resultant?
The [resultant](https://en.wikipedia.org/wiki/Resultant) of the [Sylvester matrix](https://en.wikipedia.org/wiki/Sylvester_matrix) first appeared in J. J. Sylvester, Philos. Magazine **16**, 132–135 (1840): [A method of determining by mere inspection the derivatives from two equations of any degree](https://zenodo.org/record/1931489#.X5nTCC9h3OQ).


![](https://ilorentz.org/beenakker/MO/Sylvester_resultant.png)
The last line is the resultant of the two equations at the top, which vanishes when two roots coincide.  

 I like the "ought to, and is" phrasing...




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The Sylvester matrix also appeared in an 1840 publication by Cauchy, who acknowledged the priority of Sylvester:


![](https://ilorentz.org/beenakker/MO/Cauchy_Sylvester.png)
[1876 reprint](http://www.numdam.org/article/NAM_1876_2_15__433_0.pdf) of Cauchy's 1840 article "Mémoire sur l'élimination d'une variable entre deux équations algébriques".


The name "resultant" originates from


E. Bézout (1764): *Recherches sur le degré des équations **résultantes** de l'évanouissement des inconnues et sur les moyens qu'on doit employer pour trouver ces équations.*  

Research on the degree of equations resulting from the vanishing of unknowns and on the methods one should use to obtain these equations.


