(Polynomial) Long Division
Travis Scrimshaw January 6, 2014
Division is the inverse operation to multiplication, so formally we want to find some x such that βx = α for
some given values α (the dividend) and β ̸ = 0 (the divisor) in some R, which can be either Z, Z[x], Q[x], R[x],
or C[x]. (For the curious reader, R could also be Q, R, or C, but these are too “big” because they all have a
well defined β −1.) The question becomes how do we do that, and the answer is (polynomial) long division.
Long division is an algorithm to find some q (the quotient) such that α = qβ + r where r (the remainder)
satisfies 0 ≤ d(r)