Sections 4.5 and 4.6 due 10/1

Theorem 4.29 says that if a polynomial in the reals is irreducible then it is either of degree one or two. In the proof of it, it says that this is not necessarily the case--that higher degrees have roots in the complex numbers. Because of the way it is stated in the theorem, though, it makes me wonder if I am understanding it correctly. Are there really only irreducible polynomials of degrees 1 and 2?

There are many theorems in 4.5 about primes, coefficients, etc. Some of them seem obvious while others seem brilliant. All of them make me wonder if they are applicable. (I suppose they are because they are in the textbook.)