The grey prediction model, as a time-series analysis tool, has been used in various fields only with partly known distribution information. The grey polynomial model is a novel method to solve the problem that the original sequence is in accord with a more general trend rather than the special homogeneous or non-homogeneous trend, but how to select the polynomial order still needs further study. In this paper the tuned background coefficient is introduced into the grey polynomial model and then the algorithmic framework for polynomial order selection, background coefficient search and parameter estimation is proposed. The quantitative relations between the affine transformation of accumulating sequence and the parameter estimates are deduced. The modeling performance proves to be independent of the affine transformation. The numerical example and application are carried out to assess the modeling efficiency in comparison with other conventional models.