Show that any polynomial of odd degree 2n+1: $f(x)=\sum_{k=0}^{2n+1} a_kx^k $, $a_{2n+1}\neq0$ has at least one real root.

Question

Show that any polynomial of odd degree 2n+1:

$$f(x)=\sum_{k=0}^{2n+1} a_kx^k $$

$a_{2n+1}\neq0$ has at least one real root.

I would like to prove this using IVT, how would I go about starting this?

Answer 1

Hint: Let $x\to+\infty$ and $x\to-\infty$. Then use "IVT".