In this paper the problem of density in the space C(X), for a compact set X⊂C, of polynomial modules of the type p+z^dq:p,q∈C[z] for integer d>1, as well as several related problems are studied. We obtain approximability criteria for Carathéodory compact sets using the concept of a d-Nevanlinna domain, which is a new special analytic characteristic of planar simply connected domains. In connection with this concept we study the problem of taking roots in the model spaces, that is, in the subspaces of the Hardy space H² which are invariant under the backward shift operator.