Monday, 23 September 2013

calculus - No complex roots of a polynomial implies its derivative has no complex roots



It's a well known result that if a degree n2 polynomial P(x) with real coefficients has n distinct real roots, then its derivative P(x) will have n1 distinct real roots. This is a consequence of Rolle's Theorem.



I would like to know if the following statement true:



If a degree n2 polynomial P(x) with real coefficients has n real roots (not necessarily distinct), may consist of multiplicities >1), then P(x) will have n1 real roots (not necessarily distinct).




I've spent a while looking for a proof or a counterexample to this online but had no luck in doing so. I've also tried working out a proof for myself but ran into trouble when considering multiplicities of roots greater than one.


Answer



This follows by Rolle's theorem by essentially the same argument. Let $a_1

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