# How do I use the intermediate value theorem to determine whether a polynomial function has a solution over a given interval?

To answer this question, we need to know what the intermediate value theorem says.

The theorem basically sates that:
For a given continuous function $f \left(x\right)$ in a given interval $\left[a , b\right]$, for some $y$ between $f \left(a\right)$ and $f \left(b\right)$, there is a value $c$ in the interval to which $f \left(c\right) = y$.

It's application to determining whether there is a solution in an interval is to test it's upper and lower bound.

Let's say that our $f \left(x\right)$ is such that $f \left(x\right) = {x}^{2} - 6 \cdot x + 8$ and we want to know if there is a solution between $1$ and $3$ (in the $\left[1 , 3\right]$ interval).
$f \left(1\right) = 3$
$f \left(3\right) = - 1$
From the theorem (since all polynomials are continuous), we know that there is a $c$ in $\left[1 , 3\right]$ such that $f \left(c\right) = 0$ ($- 1 \le 0 \le 3$)//

Hope it helps.