How to deal problems in finding real roots of a polynomial?

After encountering many numerical problems related to finding real roots of a polynomial, I have fixed a simple path:

1. For an odd-degree polynomial: complex roots come in pairs. If a polynomial is of degree $d$ which is odd then the polynomial must have at least one real root.

2. For a even-degree polynomial I have always seen it has no real root!

I am looking for a counterexample.
please can anyone tell is there any general approach to solve this kind of problem? Often I do mistake in these.