The problem of bounding the zeros of a polynomial of degree n greater than or equal to 2 with real or complex coefficients, is considered and an improved Cauchy bound is proposed. This new bound is applied to two examples and compared with existing approaches. The methodology is then extended to the case of polynomials of degree n greater than or equal to 2 with interval Coefficients. A counter-example to theorem 3.8 in Adjiman, Androulakis, Maranas and Floudas (1996), used in the alpha BB algorithm to compute a tight lower bound on the minimum real part of the zeros of an interval polynomial is also presented. An alternative approach that uses the improved Cauchy bound as a starting point is developed.