Intermediate Value theorem application

**Andreamet** · April 9th 2009, 02:16 PM

Let p(x) be an odd-degree polynomial function. Prove that p(x)=0 has at least one real solution

**nzmathman** · April 9th 2009, 02:36 PM

Not sure if there is a formal proof of this, but one can see that as $x \to \infty$ then $p(x) \to \infty$ and as $x \to -\infty$ then $p(x) \to -\infty$ . (assuming leading coefficient is positive, if negative its just the other way around)

By definition any polynomial is continuous on $\mathbb{R}$ and thus by the IVT we can say there exists at least one $x \in \mathbb{R}$ such that $p(x)=0$

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