Remarks

• If the polynomials a and b are elements of Z[X] (so that the coefficients are not from a field), then the polynomials q and r are not necessarily in Z[X]. Take for instance a = 1 and b = 2.
• By hand, the easiest way to compute the quotient and the remainder is by a long division. In fact, this procedure is hidden in the existence part of the proof of the theorem. In every step the leading terms of b and a minus the product of b and the `quotient so far' are compared.