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:
[math]d^{6}+4dm^{9}-10dv^{6}m^{6}+15dv^{12}m^{3}-8dv^{18}+27v^{3}m^{3}[/math]to say d can only take rational integer values of factors of [math]27v^3m^3[/math] when [math]d,v,m[/math] are known integers?
What if I have constraint equations such that they covary? Such as [math]b=\frac{v^3m^6}{d^\frac{1}2}[/math] where b is another variable also known to be an integer, or more constraint equations.
For example, can I use the rational root theorem with:
[math]d^{6}+4dm^{9}-10dv^{6}m^{6}+15dv^{12}m^{3}-8dv^{18}+27v^{3}m^{3}[/math]to say d can only take rational integer values of factors of [math]27v^3m^3[/math] when [math]d,v,m[/math] are known integers?
What if I have constraint equations such that they covary? Such as [math]b=\frac{v^3m^6}{d^\frac{1}2}[/math] where b is another variable also known to be an integer, or more constraint equations.