Question

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

Answer 1

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(x)` in a given interval `[a,b]`, for some `y` between `f(a)` and `f(b)`, there is a value `c` in the interval to which `f(c) = 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(x)` is such that `f(x) = x^2 - 6*x + 8` and we want to know if there is a solution between `1` and `3` (in the `[1,3]` interval).
`f(1) = 3`
`f(3) = -1`
From the theorem (since all polynomials are continuous), we know that there is a `c` in `[1,3]` such that `f(c) = 0` (`-1 <= 0 <= 3`)//

Hope it helps.