Rational Root Theorem for Multivariable Polynomial with Variable Coefficients

safwane

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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:

d6+4dm910dv6m6+15dv12m38dv18+27v3m3d^{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 27v3m327v^3m^3 when d,v,md,v,m are known integers?

What if I have constraint equations such that they covary? Such as b=v3m6d12b=\frac{v^3m^6}{d^\frac{1}2} where b is another variable also known to be an integer, or more constraint equations.
 
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.
 
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