Example

To determine the quotient q and the remainder r when dividing a = 2X⁴ + X by b = X² + 1 in Q[X] we need the following steps.

• Compare the leading terms of a and b. Subtract 2X² · b from a in order to cancel the leading term of a:
  a - 2X² · b = 2X⁴ + X - 2X² · (X² + 1) = -2X² + X.

  From this step we conclude that 2X² will be a term of the quotient q. We now have

  a = 2X² · b + (-2X² + X).

  Since the degree of -2X² + X is not less than that of b we need a further step.

• Compare the leading terms of -2X² + X and b and subtract -2 · b from -2X² + X:
  -2X² + X + 2(X² + 1) = X + 2.

  The resulting polynomial has degree less than deg(b), so the division stops here. We conclude that

  q = 2X² - 2 and r = X + 2.

  You easily check the identity a = qb + r, i.e.,

  2X⁴ + X = (2X² - 2)(X² + 1) + (X + 2).