Proof of a theorem about polynomial [duplicate]

Question

I have known a theorem from the link Derivation of the general forms of partial fractions

The theorem is that a polynomial of odd degree has a root.But I don't know its proof.Can anyone prove it with intuition.

Answer 1

Suppose the polynomial $P$ is monic, for this is irrelevant as the existence of a zero. Also, I presume, it's over the reals, for else this is just not true.

Then $\lim_{x\to-\infty}P(x)=-\infty$ and $\lim_{x\to\infty}P(x)=\infty$ because the polynomial is dominated by its leading term, which is an odd function -- it's $x^{2n+1}$. Since $P$ is a continuous function with both a positive and negative value, it must have a zero. Pictorially, if it has a value above and below the $x$-axis and the graph is continuous, then it has to cross the $x$-axis.

I am not entirely sure why you refer to partial fractions.