If i know the amount of of rational roots in a polynomial and the amount of real roots in a polynomial. Can subtract the amount of rational roots in from the amount of Real roots in the polynomial to get the amount of irrational roots in a polynomial?
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Yes, you can do that since real numbers contain only rational and irrational numbers. So you CAN subtract the number of rational roots from the number of real roots, to get the number of irrational roots. But there is no guarantee that they will be algebraic.
Aniket Gupta
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I mean, if your coefficients are transcendental then sure, there's no guarantee the roots will be algebraic. But surely otherwise they are algebraic by definition? :) – Izaak van Dongen Feb 22 '20 at 15:41
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Yes, exactly. In the question, the domain of the coefficients are not mentioned, so we need to consider all possible real coefficients, which include transcendental numbers as well. – Aniket Gupta Feb 22 '20 at 16:09