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.

Consider the cases separately for odd and even degree polynomial functions and talk about their end behaviour. Perhaps draw a diagram to convince yourself, first. — Andrew Chin, Sep 01 '19 at 18:10
For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — Martin R, Sep 01 '19 at 18:10

score 4 · Accepted Answer · answered Sep 01 '19 at 18:22

$a_0 a_n < 0$ 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.

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.

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