The polynomial closure of a class of languages C is the set of languages that are finite unions of languages of the form L0a1L1... anLn, where the ai's are letters and the Li's are elements of C. The main result of this paper gives an equational description of the polynomial closure of C, given an equational description of C, when C is a lattice of regular languages closed under quotients. As an application of this result, we give a set of equations defining the polynomial closure of the languages of the form u* or A*, where u is a word, and we use these equations to show that this class is decidable.