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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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Product with what? –  Marek Feb 11 '13 at 7:48
    
Is it clear now? –  Turbo Feb 11 '13 at 7:51
    

2 Answers 2

up vote 3 down vote accepted

$(x^2 - x + 1) (x+1) = x^3 + 1$

If you want strictly positive:

$(4x^2 - x + 1) (2x+1) = 8 x^3 + 2x^2 + x + 1$

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thats trueI think I am missing something –  Turbo Feb 11 '13 at 7:53

Answered here: by Gerry Myerson http://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 5:42
    
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

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