Proof of Roots of Odd Degree Polynomial

Date: 05/10/2000 at 00:08:37
From: Anonymous
Subject: Taylor Series (specific problem)

Hello,

I am a student in a Calculus II course and we're currently studying 
complex numbers (power series, Taylor series, sequences, etc.) I was 
given on a homework assignment the following question:

1) Prove that any real polynomial p: R->R of odd degree has a root.

I don't know how to begin this proof (or finish it for that matter). 
It is one of the simpler problems on the assignment so if you could do 
this and post it so that I may use it as a guide for all other proofs, 
that would be fantastic.

Please help! Thank you.

Date: 05/10/2000 at 13:23:21
From: Doctor Rob
Subject: Re: Taylor Series (specific problem)

Thanks for writing to Ask Dr. Math.

Assume that the leading coefficient is positive.

Observe that lim p(x) as x -> +infinity is +infinity, so that from 
some point on in the positive direction, p(x) > 0. (Eventually the 
leading term will dominate.)

Observe that lim p(x) as x -> -infinity is -infinity, so that from 
some point on in the negative direction, p(x) < 0.

Now use the fact that polynomial functions are continuous, and apply 
the Intermediate Value Theorem, to conclude that p(x) = 0 somewhere in 
between (maybe more than one place, but at least one.)

A similar argument works if the leading coefficient is negative.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/