[Math] Proving that roots of a quadratic lie between two values

Question

To prove that one of the roots of a quadratic $f(x) = ax^2 + bx + c$ with real coefficients lies between two values $x_1, x_2$ is it enough to prove that:

$$f(x_1) < 0 < f(x_2) $$

Can this be generalized to a polynomial with an arbitrary degree?

Answer 1

Any polynomial, $f$, is a continuous function on $\mathbb{R}$. If there exists an $x_1$ such that $f(x_1)<0$ and an $x_2$ such that $f(x_2)>0$, (and wlog assume that $x_1<x_2$) then it follows by the intermediate value theorem that there exists an $x\in (x_1, x_2)$ such that

\begin{equation*} f(x)=0. \end{equation*}

Thus, not only is your statement true for any polynomial, it is true for any continuous function on $\mathbb{R}$.