>>11010170That's not quite true. Think of dividing two numbers by hand, if you divide 1/4 the result should be 0.25 (we aren't talking euclidean division, but division as it's taught on primary school). Given that the algorithm of dividing polynomials and numbers is essentially the same, it's not surprising that you can keep going
It's just not going to ever stop (it becomes a series) because a constant and a polynomial are "incommensurable" in the sense their ratio isn't on the ring of polynomials (e.g. 1/x is a rational function, not a polynomial).
To give an example of the kind of division I'm talking about consider 1/(1-x), the simplest example.
The first term of our quotient q should be such that 1-q(1-x) = qx, so q = 1 (if you put 1 = q(1-x)+qx you have the n = qm+r division formula)
For the second term we want q2 s.t. x-q2(1-x) = q2x, so we put q2 = x. So far we have 1/(1-x) = 1 + x + R, where R is the remainder x^2/(1-x). Of course, 1 + x + x^2/(1-x) = 1/(1-x).
Just keep going now.
