New definitions for right, left and doubly coprime factorizations for nonlinear, input-affine state-space systems are introduced. These definitions are based on the state-to-output stability introduced by Baramov and Kimura (1996) and the chain-scattering formalism. Sufficient conditions for the existence of these factorizations as well as local state-space formulas for factors are given. Finally, these results are applied to obtain a parametrized set of stabilizing controllers to a fairly broad class of plants, for transforming the original feedback control configuration into the open loop model matching configuration and for thus extending the classical Youla-Kucera parametrization to nonlinear (local) cases.