In this paper, a method to design Infinite Impulse Response (IIR) multi-Notch filters based on the design of the denominator of the squared frequency response magnitude is demonstrated. This denominator is expressed as a positive polynomial of x = cos(ω), which is designed by a shape factor and the local maxima positions of the frequency response magnitude. The shape factor can be computed to comply, at any filter gain, with cut-off frequencies or bandwidths specifications for any selection of the local maxima positions. The designed filter is proven to be stable, ripple-free, and to have a realization based on the realization of an all-pass filter; thus, computationally efficient structures can be exploited. The optimal selection of the shape factor and the local maxima positions by using Particle Swarm Optimization (PSO) is also discussed. Finally, the effectiveness and performance of the polynomial design method is illustrated and compared with previous techniques through design examples.