Intermediate Value Theorem
If f is continuous on the closed interval [ a , b] and K is any number between f(a) and f(b), then there is at least one number c in the interval such that f(c) = K.
You can use this theorem to determine if a polynomial has at least one real root.
To begin using this we first need to know if
f(a) < 0 < f(b) or f(b) < 0 < f(a)
If either of those cases are true. We know that there is a c in the closed interval such that f(c) = 0.
As an example:
we set f(x) = x^2 - 2. We will now test the function for some closed interval. Lets say [1 , 2]. Then lets test these numbers 1 and 2.
f(1) = -1 < 0 and f(2) = 2 > 0.
Therefore we know by the theorem that there exists a number c in the interval [1 , 2] such that f(c) = 0.
Since f increases on [1 , 2], there is only one such number. This number we can see is the square root of 2.