Amit

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Name Amit
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Seen Mar 3 at 11:18
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comment When adding a constant makes a multivariate polynomial reducible?
About your first question - there need not be any simple criteria to say when such $m$ does not exist. Take e.g polynomial $x^2 + ny + m$. As long as $n \neq 0$, no value of $m$ will make this factorisable. Examples of this kind suggests that a simple criteria in terms of coefficients of the polynomial may not exist.