We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs. We present a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials $q_1,...,q_m$ so that the joint probability density function of $q_1(G),...,q_m(G)$ is close to being bounded. This says essentially that any abnormalities in the distribution of $p(G)$ can be explained by the way in which p decomposes into the $q_i$. We then present some applications of this result.