We propose two different approaches for finding optimum FIR polynomial predictive filters. The first reveals an important property of these filters and can lead to a numerical solution that is of the size of the number of constraints imposed instead of the size of the filter. With the second method, using a different parametrization of the filter coefficients, it is possible to obtain closed form expressions for the optimum filter coefficients and for the minimum output noise power, in the general case. Previous approaches were successful in providing closed forms expressions only for special cases.