M and N are positive integers with M>N. The division algorithm for integers tells us there exists integers Q and R such that M=QN+R with 0[tex]\leq[/tex]R<N. The division algorithm for real polynomials tells us that there exist real polynomials q and r such that xM - 1 = q(xN - 1) + r with r = 0 or deg r < N. Find q and r.
The Attempt at a Solution
I have rewritten it as xQN+R - 1 = q(xN - 1) + r, so I think q must be of degree QN+R-N = N(Q-1)+R
However, it is not then always (in fact more likely not) true that N(Q-1)+R<N so 1 must have more than 1 term. But I am struggling to know where to go from here.
Thanks for any help :)