A new rational closure approximation for the Doi model of liquid-crystalline polymers (LCPs) is presented and compared with unapproximated results. The closure is based upon proposing a parametric form for the orientation distribution function, the Bingham distribution, which is appropriate for the spherical geometry of the configuration space. The closure enjoys the advantage over ad hoc moment closures of always yielding physical answers; moreover, it is designed to give the exact solution behavior in a number of limits. Its behavior for steady two-dimensional flows reveals a range of applicability that extends significantly beyond the intended Limits. The main deficiency, a failure to predict a flow-aligning transition in simple shear, is shown to be due to the inability to predict skewing of the orientation distribution. Comparisons are also made for a center-gated disk geometry, addressing the performance in three-dimensional unsteady flow typically seen in processing of LCPs. The Bingham closure shows excellent quantitative agreement with unapproximated solutions, only underestimating the time scale of some transient behavior. The approximation is compared with other closures in the same vein, and suggestions are made for improvements.