In this paper, a third-degree transcendental polynomial is studied, and the distribution of its zeros is established. Then, the results are applied to study an SEIR model with a time delay. We show that, under certain conditions, as the time delay increases, a stable endemic equilibrium becomes unstable, and a periodic solution emerges through Hopf bifurcation. By finding the normal form of the system, the direction and stability of the periodic solution are established. Numerical simulations are performed to demonstrate the theoretical results.