Proof about an odd degree polynomial

[chaotixmonjuish]

How could one sufficiently prove that a polynomial of odd degree in R[x] with no multiple roots must have an odd number of real roots?

My book just refers back to a Corollary that states every polynomial of odd degree in R[x] has a root in R. However it doesn't say, all roots are in R.

 

[Cantab Morgan]

However it doesn't say, all roots are in R.

That's right. You're not trying to prove that all the roots are in R, merely that an odd number of them are. (You actually can't prove that all the roots are real. It's not hard to come up with a cubic equation with exactly one real root.)

One thing to ask yourself is whether complex roots have to come in pairs.

 

[Jamma]

Well I'm guessing its because all the complex roots will come as conjugate pairs, i.e. there are an even number of complex roots, so there will be an odd number of real roots left over.