A simple model is introduced to describe liquid crystal transitions and phase separations in binary mixtures of a flexible polymer and a liquid crystal. By combining the McMillan theory for the smectic A phase of liquid crystals with the Flory-Huggins theory for the isotropic mixing of two components, we examine binodal and spinodal lines on the temperature-concentration plane. We predict the appearance of phase separations such as the smectic A-nematic, smectic A-isotropic, and smectic A-nematic-isotropic phase separations. We also find a tricritical point caused by the interference between a second-order liquid crystal transition and a phase separation.