Given $a \in \mathbb{Z}$ with $|a| > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)=\pm p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive coefficients? Is there a way to generate such polynomials?
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1Product with what? – Marek Feb 11 '13 at 07:48
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Is it clear now? – Turbo Feb 11 '13 at 07:51
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Now posted to MO, http://mathoverflow.net/questions/121568/on-reducible-polynomials – Gerry Myerson Feb 12 '13 at 22:32
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$(x^2 - x + 1) (x+1) = x^3 + 1$
If you want strictly positive:
$(4x^2 - x + 1) (2x+1) = 8 x^3 + 2x^2 + x + 1$
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Answered here: by Gerry Myerson https://mathoverflow.net/questions/121568/on-reducible-polynomials
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Well, answered in part. OP would really like an example with $a\gt1$, while my example at MO has $a=-3$. And I didn't address the question of a systematic way to generate examples. – Gerry Myerson Feb 13 '13 at 05:42
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Ilya Bogdanov has now posted an example there with $a=2$ (and degree $16$), and some details on how to go about finding such things. – Gerry Myerson Feb 13 '13 at 22:04