A new method for solving the Boltzmann equation is presented and shown to generalize the “classical” spherical harmonics method. This new method utilizes polynomials that are spatially, as well as angularly, dependent and allows for the exact representation of the angular flux under certain conditions. The ideas behind using different truncation procedures as a means of truncating the infinite set of exact spherical harmonics equations to a finite set of approximate equations and allowing this procedure to supply more transport information to these approximate equations are explored. Preliminary results are also presented that show the differences and similarities of these methods as they relate to the exact results.