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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 $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.

I got good answers for the above question. The answer includes the phrase "...the last few coefficients of $g(x)h(x)$ are large enough...". Is there a way to overcome this?

The given polynomials $g(x)$ and $h(x)$ have constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$, $g(a)h(a)$ a prime for some $a > 1$, say $a=2$.

Above also solved! My polynomials have one last constraint: $x^{i}$ term coefficient of $g(x)h(x)$ cannot exceed say $2^{n−i+1}$ where $n$ is degree of $g(x)h(x)$ ($X^{0}$ TERM IS STILL $1$). The construction of $h_{1}(x)$ has large lower degree coefficients which means $h_{1}(x)$ has large higher degree coefficients. Is a factorization of the desired kind (unequal signs in coefficients) possible when $g(x)h(x)$ has this additional constraint? Again say $a=2$. The current construction by Ilya blows up the leading terms.

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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 $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.

I got good answers for the above question. The answer includes the phrase "...the last few coefficients of $g(x)h(x)$ are large enough...". Is there a way to overcome this?

The given polynomials $g(x)$ and $h(x)$ have constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$, $g(a)h(a)$ a prime for some $a > 1$, say $a=2$.

Above also solved! My polynomials have one last constraint: $x^{i}$ term coefficient of $g(x)h(x)$ cannot exceed say $2^{n−i+1}$ where $n$ is degree of $g(x)h(x)$ ($X^{0}$ TERM IS STILL $1$). The construction of $h_{1}(x)$ has large lower degree coefficients which means $h_{1}(x)$ has large higher degree coefficients. Is a factorization of the desired kind (unequal signs in coefficients) possible when $g(x)h(x)$ has this additional constraint? Again say $a=2$. The current construction by Ilya blows up the leading terms.

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# On reducible polynomials with positive coefficientsand, $1$ as constant coefficient andcertainboundsoncoefficients

Given $a \in \mathbb{Z}$ with $a > 1$. Let $g(x) \in \mathbb{Z}[x]$ be a polynomial with $g(a)=\pm 1$. Let $h(x) \in \mathbb{Z}[x]$ be a polynomial with $h(a)= p$, a prime. Let $g(x)$ and $h(x)$ have not all coefficients negative nor positive. Can $g(x)h(x)$ have only positive or only negative coefficients? Is there a way to generate such polynomials? What if $a < 1$? The motivation is to test if a given polynomial has only positive coefficients and assumes a prime value at a positive integer larger than $1$, then it is irreducible.

I got good answers for the above question. The answer includes the phrase "...the last few coefficients of $g(x)h(x)$ are large enough...". Is there a way to overcome this?

The given polynomials $g(x)$ and $h(x)$ have constant coefficient $\pm 1$ with $g(x)h(x)$ having constant coefficient $1$, $g(a)h(a)$ a prime for some $a > 1$, say $a=2$.

Above also solved! My polynomials have one last constraint: $x^{i}$ term coefficient of $g(x)h(x)$ cannot exceed say $2^{n−i+1}$ where $n$ is degree of $g(x)h(x)$ ($X^{0}$ TERM IS STILL $1$). The construction of $h_{1}(x)$ has large lower degree coefficients which means $h_{1}(x)$ has large higher degree coefficients. Is a factorization of the desired kind (unequal signs in coefficients) possible when $g(x)h(x)$ has this additional constraint? Again say $a=2$.

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