I am not sure that I understood this question, but I decided to list four aspects or ways of categorizing polynomials...I hope this helps.
1) Number of terms in the polynomial: Monomial, binomial, trinomial, etc.
2) Degree of the polynomial: The value of the highest power in the polynomial
3) Classification by degree: Linear if it has degree 1, quadratic if it has degree 2, cubic if it has degree three, etc.
4) Coefficients: The values of the multipliers of the variables in the polynomial. For example, in the quadratic binomial 2x2 – (1/3)x = 2x2 + (–1/3)x, the leading coefficient (the coefficient of the degree term) is 2 and the coefficient of the linear term is –1/3.
5) Completeness: A complete polynomial has terms of every power up to the degree. For example, a complete cubic polynomial will contain terms with third, second, first and zero power: x3 + 2x2 – (1/2)x + 5 is complete, but x3 – 2x + 3 is not complete because it doesn't contain an x2 term.