A method is presented which, in many cases, appears to be an improvement over the standard approach to the polynomial regression problem. This improvement is achieved by focusing attention on the deviation of the polynomial representation from the true underlying function. By fully utilizing the nature of this deviation, a model is constructed in which its properties are represented in terms of a Bayesian prior distribution. The model is analyzed to give parameter estimates and predictions of further observations. Comparisons are made with standard least squares procedures when the true underlying model is (a) quadratic and (b) linear and quadratic with a superimposed sine wave.