Part I: What was the most difficult part of the material for you?
I am lost at the start of theorem 4.20 (Rational root test) because it discusses about the variables r and s, but I don't even understand where they got an s! I know that is comes from the root of f(x) but I guess it is worded strangely and I would need an example that I work on to see the connection.
Part II: Write something reflective about the reading.
Theorem 4.25 I think is really amazing because it seems impossible that EVERY non-constant polynomial in the C[x] has a root in C but after seeing the proof and thinking about it, it makes sense. But still, it is surprising. Also, many of the theorems after had similar ideas with finding things dealing with EVERY non-constant polynomials in a polynomial ring, which is really amazing.
Which topics and theorems do you think are the most important out of those we have studied?
The Euclidean Algorithm, the fact that we can write the gcd of f and g as a linear combination for some u and v, and the theorem that says "If F is a field, then for any f in that field polynomial and any a in F, the element a is a root of f iff (x-a) divides f."
What kinds of questions do you expect to see on the exam?
I expect there to be proofs, definitions, and many some small calculations that we may have done similar in the homework. But I think the proofs will be proven with the theorems and definitions we have memorized.
What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.
I am still trying to understand better how to prove if a function is surjective or not. Most of the time it just seems obvious, so I don't know if I am missing something or if I understand it so well that I am making it more complicated. One example is, problem 15 on 3.3 which says, Show that the complex conjugation function f:C to C (whose rule is f(a+bi)=a-bi) is a bijection.
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