This paper examines the liquid crystalline blue phases of mixtures of a readily available chiral system (consisting of cholesteryl nonanoate and cholesteryl benzoate) with various proportions of a low molecular weight and low polydispersity polymer. By observing the textures of samples using polarizing optical microscopy as they are heated from the chiral nematic through the blue phase and into the isotropic liquid phase, the temperature range where the blue phase of these mixtures is stable is measured as a function of polymer volume fraction for several molecular weights of polystyrene and poly(methyl methacrylate). The mixtures with polystyrene showed a blue phase stability range of up to 12 K, one of the largest measured on heating, and this range varied smoothly as a function of polymer volume fraction. The dependence of the range on the polymer volume fraction and molecular weight could be captured well by a theoretical model based on the Flory-Huggins and Maier-Saupe theories.