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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. – Dylan Moreland Jul 22 '12 at 19:09
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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"? – Code-Guru Jul 22 '12 at 19:10
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? – Dylan Moreland Jul 22 '12 at 19:13
If it's "precalculus", I doubt that he's working with finite fields. – Robert Israel Jul 22 '12 at 19:56
@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. – Dylan Moreland Jul 23 '12 at 1:02

1 Answer

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I am interpreting the question as follows. Let $P(x)$ be a polynomial such that the sum of any (non-empty) subset of the coefficients of $P(x)$ is non-zero. Is the same true of $(P(x))^n$?

The result need not hold. For example, let $P(x)=x-2$ and let $n=2$. Or let $P(x)=3x-1$ and let $n=3$. One can produce similar examples for any $n \gt 1$.

Remark: If the sum of all the coefficients of $P(x)$ is non-zero, then the sum of all the coefficients of $(P(x))^n$ cannot be $0$. For the sum of the coefficients of a polynomial $Q(x)$ is $0$ iff $Q(1)=0$. And if $(P(1))^n=0$, then $P(1)=0$.

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I see, so you think the question is about adding up the coefficients? That wasn't clear to me at all :-/ – Dylan Moreland Jul 22 '12 at 19:18
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@DylanMoreland: It is not clear to me either. Just looks like a plausible interpretation. – André Nicolas Jul 22 '12 at 19:20

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