Black hole spacetimes, like the Kerr spacetime, admit both stable and plunging orbits, separated in parameter space by the separatrix. Determining the location of the separatrix is of fundamental interest in understanding black holes, and is of crucial importance for modeling extreme mass-ratio inspirals. Previous numerical approaches to locating the Kerr separatrix were not always efficient or stable across all of parameter space. In this paper we show that the Kerr separatrix is the zero set of a single polynomial in parameter space. This gives two main results. First, we thoroughly analyze special cases (extreme Kerr, polar orbits, etc.), finding strict bounds on the limits of roots, and unifying a number of results in the literature. Second, we pose a stable numerical method which is guaranteed to quickly and robustly converge to the separatrix. This new approach is implemented in the Black Hole Perturbation Toolkit, and results in a ~45x speedup over the prior robust approach.