  The M\"obius polynomial is an invariant of ranked posets, closely related to
the M\"obius function. In this paper, we study the M\"obius polynomial of face
posets of convex polytopes. We present formulas for computing the M\"obius
polynomial of the face poset of a pyramid or a prism over an existing polytope,
or of the gluing of two or more polytopes in terms of the M\"obius polynomials
of the original polytopes. We also present general formulas for calculating
M\"obius polynomials of face posets of simplicial polytopes and of Eulerian
posets in terms of their $f$-vectors and some additional constraints.
