How can I show that an odd degreed polynomial with coefficients in the real space always has a root in $\mathbb{R}$? How can I show that every odd degreed polynomial with coefficients in the real space will have a rea...--prophetes.ai

How can I show that an odd degreed polynomial with coefficients in the real space always has a root in $\mathbb{R}$? How can I show that every odd degreed polynomial with coefficients in the real space will have a real root?

Hint: Let $f$ be an odd degree polynomial. What is $\lim\limits_{x \rightarrow \infty} f(x)?$ What is $\lim\limits_{x \rightarrow -\infty} f(x)?$

Can you now use Intermediate value theorem to conclude your result?