A polynomial of degree k that normally has k + 1 terms can be considered as having n + 1 terms for any n >= k; just make the coefficients to be zero. The definition does not say that coefficients have to be non-zero. We are not talking about a degree here. Thus, when we need to add two polynomials, we can assume that they have the same number of terms.

Do you doubt that the sum of two polynomials with integer coefficients is another such polynomial? For example, what about ?