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Given a polynomial of m terms where no combination of those terms add to zero. Can the expansion of that same polynomial, where exponent n = a positive integer, ever contain two or more terms that add to zero ?

My instincts say "of course not" but I have no idea how to go about proving it. A simple proof may be staring me right in the face and I just can't see it!

Any suggestions or comments steering me in the right direction will be very much appreciated.

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    Could you think of a title which isn't just the tag? I think that would draw more eyeballs.2012-07-22
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    Can you give a concrete example of "a polynomial of m terms where no combination of those terms add to zero" and the "expansion of that same polynomial, where exponent n = a positive integer"?2012-07-22
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    Reading the question, I don't really understand it. Where are you taking your coefficients? If you view a polynomial abstractly then $f(x) = \sum a_ix^i$ is zero if and only if each $a_i = 0$. I guess there's always this subtlety, in that, for example, over a finite field with $p$ elements the polynomial $x^p - x$ induces the zero function on that field, but I don't think that's what you're talking about. And are you asking about the expansion of $f(x)^n$ with this mention of an exponent?2012-07-22
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    If it's "precalculus", I doubt that he's working with finite fields.2012-07-22
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    @Robert I agree; I just rattled off thoughts in hopes of getting some response. If "term" is supposed to mean "coefficient" then I want to edit the question.2012-07-23

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