The cases a) doesn't really have much to it.
If and has degree 0 this means that
for some
If the degree of is larger than 0
Then
if the degree of is 0 then
Then
All of this "division" is taking place in the field or rational numbers.
The division algorithm has an analoguen in that asserts: given two polynomials not identically 0, there exists such that
and either
, or
.
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 .
Subcase (1)
Prove directly
Subcase (2)
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.
Thanks to anyone willing to walk me through this.
Note: The polynomial was discussed in the previous problem here ---> http://www.mathhelpforum.com/math-he...hm-170136.html