Difficult:
It was difficult for me to follow the proof on the last two pages. For example I am not sure why they multiplied by a_(n) b_(m)^(-1) x(n-m). It did not make sense to me. I understand long division using polynomials, and I understand how the division algorithm applies to F[x]. But that was all as far as those couple of pages. The rest of the section I understood very well. It is difficult for me to visualize rings and fields and how they relate to this material, but I still understood most of it. Everything was very straightforward for this section. But probably the most difficult part, like I said earlier, was the Division Algorithm in F[x].
Reflective:
I have never thought of x being not any particular number but just being treated like a number. The analogy using pi help a lot with that explanation. It was very interesting in this section to try to understand the connection that R and R[x] have in polynomials. It was also interesting to note that the ring R is a sub-ring of the polynomial ring R[x] and yet, if R is an integral domain, then so is R[x]. Usually it seems like it would be the other way around on the latter statement.
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