Originally Posted by

**dannyboycurtis** Hi So I have to prove the following:

If $\displaystyle p(x)=a_{n}x^n+...+a_{1}x+a_{0}$ with $\displaystyle a_{0}<0$ and $\displaystyle a_{n}>0$ and n is an even positive integer, then prove that p has at least two distinct roots.

So my proof looks a little like this so far:

p is continuous on $\displaystyle \mathbb{R}$.

$\displaystyle \lim_{x\to +\infty}p(x)= +\infty$ and $\displaystyle \lim_{x\to -\infty}p(x)= -\infty$.

Hence $\displaystyle \exists \alpha, \beta \in \mathbb{R}$ such that:

$\displaystyle p(\alpha)<0<p(\beta)$.

...

At this point Im stuck. I found, via graphing example polynomials, that the negative $\displaystyle a_{0}$ values cause the polynomial to have two distinct roots, but how to state this mathematically to finish the proof?

Any suggestions would be appreciated thanks.