Explain the Intermediate Value Theorem and provide an example of how it can be applied to a polynomial function.

The Intermediate Value Theorem states that if a continuous function takes on two different values at two points, it must take on every value in between those points. For example, consider the polynomial function \( f(x) = x^3 - 3x + 2 \). If we evaluate \( f(0) \) and \( f(2) \), we find that \( f(0) = 2 \) and \( f(2) = 4 \), both positive. Evaluating \( f(1) \) gives \( f(1) = 0 \), showing a sign change. By the IVT, there must be a root between \( 0 \) and \( 2 \), and indeed, \( c = 1 \) is a root.