Mathematicians are a strange bunch. They will toil away, fussing over mathematical equations, twisting them this way and that. Like a mad science experiment, but with fewer lab explosions and more chalkboard dust. Whenever they prove a new theorem, they give it a big, fancy name and go brag to their mathematician buddies, who say stuff like, "Sick! That is super gnarly, bro." Like we said, a strange bunch.
That's the way it was with the Intermediate Value Theorem. If a polynomial is below the x-axis at one value of x, and above the x-axis at another value of x, then it had to have been on the x-axis at some point in between.
Yep, that's the whole idea behind the Intermediate Value Theorem. Seems more like the Duh Theorem, right? Well, it looks obvious on a graph, but someone needed to actually prove it to be true in all cases for polynomials. So, thanks bro.
In practical terms, any time we see y change signs between two values of x, we know that there is at least one root between them.