4.4, due on October 5

I struggled a little bit to understand some of the proofs for the theorems in this section even though the theorems for the most part were things that I am familiar with regarding polynomial functions. I particularly struggled with the proof of corollary 4.16 (especially because proof by induction is one of my weak points). I got lost when the book stated 0F=f(c)=(c-a)g(c). And couldn't follow after that. In the proof of corollary 4.18, I felt like I must have missed something from before because I didn't know that -c^-1d in Fwas a root of cx+d in F[x]. Or maybe I do know that but am just confused about what ring we are in. I also didn't understand the proof of the final corollary. Why would every element of F be a root of the polynomial f(x)-g(x)?

I was so happy to finally talk about functions. Since we had made such a strong distinction between the ways we use the symbol "x", I was wondering how functions were ever going to be tied to polynomials we were using in F[x]. It seems like such a simple description for who we define the polynomial functions that I don't know why I hadn't learned it earlier. What I wonder though, is how the continued separation between the polynomial and polynomial functions will continue to be relevant. In cases where I have applied mathematics, I have never seen a time when it would be useful to think of the polynomial as not a polynomial function. Then again, I probably don't have much experience so I'm sure there are plenty of situations where the ring of polynomials is useful.