We can use the Intermediate Value Theorem (IVT) to show that certain
equations have solutions, or that certain polynomials have roots. For
instance, the polynomial f(x)=x4+x−3 is complicated, and finding its
roots is very complicated. However, it's easy to check that
f(−1)=−3 and f(2)=15. Since
−3<0<15, there
has to be a point c between −1 and 2 with f(c)=0. In other words,
f(x) has a root somewhere between −1 and 2. We don't know where,
but we know it exists.