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May 2, 2016 at 22:09 comment added Josh Lackman @NoahStein Yes of course, that's the standard proof, but it gives no connection of the result with the hairy ball theorem which is what the question is about.
May 2, 2016 at 18:20 comment added Noah Stein Note that in the real odd degree case the polynomial takes different signs for large positive and negative arguments, so you can get by with a topologically simpler connectedness and continuity argument.
May 2, 2016 at 18:05 history answered Josh Lackman CC BY-SA 3.0