How do I show that a real polynomial function of odd degree has at least one real zero? That is when the function is 0 and cuts the x-line.
This is what I have got so far:
f(x) = x^(2k+1)
Can I then put x to zero and that proves it? However, that doesn't feel like I'm proving it for all real polynomial functions of odd degree.
You cannot prove it with an example.
Let $p$ be any polynomial of odd degree. if the leading coefficient is positive then $p(x) \to \infty$ as $ x \to \infty$ and $p(x) \to -\infty$ as $ x \to -\infty$. By IVP of continuous function it follows that $p$ vanishes at some point. The proof is similar when the leading coefficient is negative.