1
$\begingroup$

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.

  • 0
    @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

1 Answers 1

7

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 $P(1)=0$.

  • 2
    @DylanMoreland: It is not clear to me either. Just looks like a plausible interpretation.2012-07-22