This paper is devoted to filtering, smoothing, and prediction of polynomial processes that are partially observed. These problems are known to allow for an explicit solution in the simpler case of linear Gaussian state space models. The key insight underlying the present piece of research is that in linear filtering applications polynomial processes and their discrete-time counterpart are indistinguishable from Gaussian processes sharing their first two moments. We describe the construction of these Gaussian equivalents of polynomial processes and explicitly compute optimal linear filters, predictors and smoothers for polynomial processes in discrete and continuous time. The consideration of Gaussian equivalents also opens the door to parameter estimation and linear-quadratic optimal control in the context of polynomial processes.