For a general polynomial f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_{2}x^2+a_{1}x+a_{0}, where a_{n}!=0, the degree is n, the terms are the terms you see for those a's that are nonzero, and the coefficients are a_{n},a_{n-1},a_{n-2},\ldots,a_{2},a_{1},a_{0}. Whether you decide to include those a's that are zero in your list of coefficients is, in a sense, dependent on what you are doing with the polynomial.
For example, if you are graphing it, then you don't need to worry about the a's that are zero. On the other hand, if you are using synthetic division to confirm a root of the polynomial (or to divide it by x-c), then the a's that are zero are important to include in your calculation.
