How Euler did it: Roots of a polynomial

On the Mathematical Association of America there's a column entitled: "How Euler Did It." The link downloads a PDF file, by the way.

This month, Ed Sandifer writes on a method Euler developed to find a root of a polynomial. The method begins by computing the power series expansion of the reciprocal of the polynomial. Then, we do a partial fraction expansion of this quotient. From the expansion we can set up a recursion. As we iterate the recursion we can find the largest root.

The paper is worth reading.