The IVT is a very useful and straightforward theorem that states that if a function f(x) is continuous on a closed interval, [a,b] , the function will attain every y-value in between f(a) and f(b) (those are the "intermediate" values for which the theorem is named), and will do so somewhere on that interval [a,b].
The IVT is frequently invoked, as it is in this case, to prove the existence of a root (an x that solves an equation) on a given interval. The process is simple: Gather all the terms in the equation on one side, so that we have an equation = 0. The function = 0 needs to be continuous on the interval (this one is by virtue of being a polynomial). Then, plug in -1 for x and evaluate, then plug in 1 for x and evaluate again. One of those should give you a value < 0 and the other should give a value > 0. Since the function changes signs on the interval, the function has a zero and the equation has a root.
f(x) = x8 - 4x3 - x - 1. f(-1) = 5 , f(1) = -5 and since -5 < 0 < 5 , there is an x1 with -1 < x1 < 1 s.t. f(x1) = 0.
