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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.

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    One could use the Mean Value Theorem, or more specifically Rolle's Theorem. But it is more work than the standard algebraic argument, and does not generalize readily to fields other than the reals. We can replace Rolle's Theorem by the Intermediate Value Theorem (applied to the derivative) at the cost of additional complication. But it can be done.2011-11-01
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    @AndréNicolas: this comment is spot on. Could you make it into an answer?2011-11-01

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