In the biased guest-host photorefractive polymer, the Manakov equations can be used to describe the optical soliton propagation and interaction. Hereby for such equations, via the Hirota method and symbolic computation, analytic soliton solutions in the bright-dark and dark-dark forms are obtained. Based on the choice of photorefractive polymer parameter and incident-optical-beam parameter, the bright-dark and dark-dark solitons as well as their interaction can occur in the polymer when the total intensity is much lower than the background illumination, and our analysis indicates that the incident light with different polarization directions influence little on the soliton propagation. γ, representing the soliton intensity far away from the soliton center, determines the appearance of bright or dark soliton under the background illumination. Through the graphic and asymptotic analysis on the two-soliton solutions along with the different γ, we find that there exist the elastic and inelastic interactions between the bright-dark solitons, while the interactions between the dark-dark solitons are always elastic.