Rational Root Theorem for Multivariable Polynomial with Variable Coefficients

[safwane]

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.

 

[JeffM]

Perhaps I am missing the point here. How is the root of a multivariate function defined? How is the rational root of a multivariate function defined? In any case, the rational root theorem merely gives a list of potential roots. Once you define your terms, I suspect the list will often be impractically long.

If d, v, and m are known values, your expression is a constant, which does not have a root.