Calculate Power Series Coefficients
Mathematica, 24 23 bytes
Saved 1 byte thanks to Greg Martin
D[1/#2,{x,#}]/#!/.x-&gt;0&amp;

Pure function with two arguments # and #2. Assumes the polynomial #2 satisfies PolynomialQ[#2,x]. Of course there's a built-in for this:
SeriesCoefficient[1/#2,{x,0,#}]&amp;