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Let us have two univariate polynomials $P\left({x}\right)$ and $Q\left({x}\right)$, both with integer coefficients, such that

  1. the sum of the coefficients of $P\left({x}\right)$ is equal to 1.
  2. $P\left({x}\right)$ has at least one negative coefficient.

Could it be proved that $R\left({x}\right)=P\left({x}\right)Q\left({x}\right)$ has at least one non-positive coefficient, meaning with non-positive a coefficient which is not a positive integer, or otherwise could you give me some counterexample?

Thanks in advance!

Juan Moreno
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1 Answers1

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$(x^2-x+1)\cdot (2x^2+3x+2) = 2x^4+x^3+x^2+x+2$

Hope that answers your questions.

Alien
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  • Perfectly, thanks! – Juan Moreno Aug 25 '20 at 22:06
  • How did you find that polynomial? –  Aug 25 '20 at 22:15
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    @CitrusCornflakes this is a counter example, you can set polynomial as ax^2 + bx +c, then expand and apply the conditions (assuming this is what the OP did) – Anindya Prithvi Aug 25 '20 at 22:18
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    @CitrusCornflakes First, there is the well-known factorization $(x^2-x+1)(x^2+x+1) = x^4+x^2+1$, but this fails to account for coefficients equal to zero, so I just adjusted the second polynomial a bit to make the zero-coefficients become positive and so that the middle term stays positive. Sil has also given an intuitive counterexample in the comments on the post if that is not to your liking with the intuition from the fact that $x^2-x+1$ is the 6th cyclotomic polynomial. – Alien Aug 25 '20 at 22:19