Two questions:
1) Why are numbers with variables in the radical and the denominator not considered polynomials?
2) What is the purpose behind knowing the degrees to a polynomial. What function does this serve ?
1) Same reason that numbers divisible by two are not considered odd. It's a definition.
2)There are contraints on the degree when polynomials are multiplied or composed. These constraints can be used to order polynomials as larger or smaller. This in turn allows a definition of division for polynomials even when the coefficients are not elements of Z, Q or R.