Recall that the asymptotic behavior of a polynomial is determined by its highest degree term. Consider how the highest degree term dominates the growth of the polynomial as the input size increases.

For two polynomials, compare their highest degree terms. If the degrees are different, the polynomial with the higher degree will asymptotically dominate the other. If the degrees are the same, compare their leading coefficients.

To rigorously prove the asymptotic relationship, use the limit definition of Big-O notation. Specifically, compute the limit of the ratio of the two polynomials as the input size approaches infinity. The result will indicate the asymptotic relationship between them.