In this paper, we deal with two problems of input-affine polynomial dynamical systems. One aims to obtain a state feedback controller such that a prescribed algebraic set is invariant for the resulting closed-loop system. The other aims to obtain a state feedback controller such that the resulting closed-loop system has a prescribed vector field on a given algebraic set. It is shown that the two problems can be represented by a particular inclusion of polynomials, and the inclusion can be solved. As a result, all the state feedback controllers required in the problems can be exactly represented by using free polynomial parameters.