Let $p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$ with $a_0,a_1,\dots,a_n \in \Bbb R$. Prove that if $a_0a_n < 0$, then $p$ has a positive root.

Question

Let $p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$ with $a_0,a_1,\dots,a_n \in \Bbb R$. Prove that if $a_0a_n < 0$, then $p$ has a positive root.

I was thinking of using the intermediate value theorem, but not quite sure how to formulate my proof and I also do not know how to show/why $p$ has a positive root.

Answer 1

$a_0 a_n < 0$ means that $a_0$ and $a_n$ have different signs and are non-zero. Now consider the value of the polynomial at $x=0$ and $x \to +\infty$. If $a_0 < 0$ then $p(0) < 0$, but since $a_0 a_n < 0$ we state that $a_n > 0$ thus $p(x \to +\infty) \to +\infty$. Polynomials are continuous functions thus at least one zero on the interval $(0;+\infty)$ should exist. Case $a_0 > 0$ can be considered similarly.