Coefficient of Determination for High Order Polynomials

Question

Because the Coefficient of Determination makes use of the mean of y, is the Coefficient of Determination meaningless for higher order polynomials (higher order than linear polynomials)? I'm assuming this because I think it's easy to visualise how the variation from the mean gives a representation for how much of the variance is not described by the best fit line for linear graphs, but not for, say, fifth or sixth order polynomials.