Examples

• 5 + X - 11X² in Q[X] has degree 2. Its leading term is -11X² and its leading coefficient is -11.
• 5 + X - 11X² considered as a polynomial in Z/11Z[X] has degree 1, leading term X and leading coefficient 1 (mod 11).
• (2X + 1)³ in Q[X] has degree 3, leading term 8X³ and leading coefficient 8. You find the leading term for instance by expanding this expression.
• The product of the two linear, i.e., degree 1, polynomials 2X + 1, 2X + 2 in Z/4Z[X] is not a polynomial of degree 2, because the product of the leading coefficients is 0. Expanding shows that (2X + 1)(2X + 2) = 2X + 2 is again a linear polynomial. This is the reason why we restrict the definition of degree to polynomial rings R[X] with R a field like Q, R, C or Z/pZ, with p prime.