Originally Posted by
FancyMouse I highly suspect that the proof of your fundamental theorem of polynomials *uses* the fact that any odd degree polynomial has a root in R. At least the only proof I know to prove the fundamental theorem of algebra, which uses a combination of Galois theory and Sylow theory, indeed uses this fact. I would appreciate it if you could have a check and tell me whether it uses this fact or not.
And there's a tiny bit problem in the statement. A should be nonzero otherwise the statement is false.
If A is nonzero, then WLOG a>0. The goal is to find two points x_1,x_2 such that f(x_1)>0 but f(x_2)<0. By IVT, there exists some zero in between. x_1 is chosen as big as possible, and x_2 is chosen as small (negative and the abs is as big) as possible. You can set any bound you like, as long as you can prove f(x_1)>0,f(x_2)<0.