Irrational Roots Theorems for Polynomial Functions

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Is any Irrational Roots Theorem been developed for polynomial functions in the same way as Rational Roots Theorems for polynomial functions? We can choose several possible RATIONAL roots to test when we have polynomial functions; but if there are suspected IRRATIONAL roots, can they be found using a theorem with a clear procedure?
 

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Is any Irrational Roots Theorem been developed for polynomial functions in the same way as Rational Roots Theorems for polynomial functions? We can choose several possible RATIONAL roots to test when we have polynomial functions; but if there are suspected IRRATIONAL roots, can they be found using a theorem with a clear procedure?
The only thing that comes to mind is Descartes's Rule of Signs (https://en.wikipedia.org/wiki/Descartes'_rule_of_signs), which can be used to determine an upper bound on the number of positive or negative real roots of a polynomial equation.

There is no theorem that I'm aware of that gives information about irrational roots the way that the rational root theorem does, which says that if there are rational roots, they have to be among a limited number of rational values.
 
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