On Fri, 30 Oct 2015, Jeremy Murphy wrote:
Division is interesting because it's not actually clear to me what the result should be - is it a polynomial (plus remainder) or is it a rational function (suitable reduced by the greatest common divisor).
Yes, I was initially troubled by this question but resolved, admittedly more through intuition than proof, that polynomial division is Euclidean (integer) division: the / operator gives you the quotient, and % gives you the remainder. Someone with a deeper understanding of abstract algebra could presumably validate or discredit this claim. However, if one accepts this, then everything falls neatly into place, for example the /= operator makes sense, which it obviously wouldn't otherwise.
This looks like a sensible choice. The situation is pretty similar to integers. 10 / 4 could return a rational type, but the choice was made to stay in the original type and use the Euclidean domain structure instead. You might want to provide a div-like function for people who want both the quotient and the remainder without duplicating too much computation. -- Marc Glisse