Is it valid to use the rational root theorem on a polynomial with variable integer coefficients? Does it change anything if you have constraints making the variables interdependent?
For example, can I use the rational root theorem with:
$$ d^{6}+4dm^{9}-10dv^{6}m^{6}+15dv^{12}m^{3}-8dv^{18}+27v^{3}m^{3} $$ to say $d$ can only take rational integer values of factors of $27v^3m^3$ when $d,v,m$ are known integers?
What if I have constraint equations such that they covary? Such as $b=\frac{v^3m^6}{d^\frac{1}2}$ where $b$ is another variable also known to be an integer, or more constraint equations.