Using Intermediate Value Theorem to prove # of polynomial roots

[KingNothing]

I've heard there's a proof out there of this, basically that (I think) you can use the intermediate value theorem to prove that an Nth-degree polynomial has no more than N roots.

I'm not in school anymore, just an interested engineer. Does anyone know where I can find this proof or any really strong hints on how to do it myself? I've been out of it for a while and I'm rusty.

 

[Deveno]

well the crux of any such proof is going to be that the derivative of an Nth-degree polynomial is going to be an (N-1)-th degree polynomial.

you don't even need limits for this, you can just define a derivative formally using the power rule.

why derivatives? because, in general, all one can say at the outset, is that a polynomial of odd degree has at least one root. to bring the IVT to bear, one has to find out how many times a polynomial p(x) can "change direction", that is, when its slope changes sign.

for an partial example of how this becomes an induction proof:

suppose p(x) is of degree 2, so that p(x) = ax² + bx + c. then p'(x) = 2ax + b, (and by assumption, a ≠ 0), so p'(x) has one root (-b/(2a), in fact).

this means that p(x) changes direction exactly once (for higher degrees it becomes "at most n-1 times" because some of the "humps" might not exist (f(x) = x³ doesn't have any, for example), so it can cross the x-axis at most twice (once going up, once going down).

you should be able to generalize, and make this a bit more rigorous, from this little bit.