Symbolic algebra manipulation is applied computationally to the mechanical system described by the inverted pendulum of two links, to yield analytic roots of the open-loop characteristic equation obtained from the linearized, symbolic dynamic equations of motion of the system. Detailed analysis of the characteristic polynomial form is presented, together with conditions relating polynomial root properties to physical parameter inequalities. Finally, features of the theoretical form of the characteristic polynomial for the generaln-link system are proposed, by simple explication of results provided by this low order system, and also those existing for the corresponding single-link case which may be obtained and checked directly.