We consider the plasmon resonances in multi-layer structures. We show the plasmon modes are equivalent to the eigenvalue problem of a matrix, whose order is the same to the number of layers. For any number of layers, the exact characteristic polynomial is derived by a conjecture, which is verified by using induction. It is shown that all the solutions to the characteristic polynomial are real and exist in the span [-1, 2]. Numerical examples are presented for finding all the plasmon modes.