The division algorithm has an analoguen in that asserts: given two polynomials not identically 0, there exists such that
Prove this. One route is to proceed by the following steps:
a) If or the degree of is 0, prove the assertion.
b) Proceed by induction on the degree of and assume results for all cases where is of degree less than . Let be of degree .
Form polynomial and use inductive hypothesis.
OK. If someone could just do the first part of this, a), then I think that MAYBE I can do the rest. Please don't just go and post an entire solution. I really want to try and do some of this. I just can't get started.