How to obtain the number of real valued zeroes of a polynomial?

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While I know there's no analytical formula for the roots of a general polynomial of degree five and higher, I wonder whether there is at least something like a parabola's discriminant to determine how many of the roots are real-valued?

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Descartes rule of signs could help very well. The intermediate value theorem can be used as well.

Since a polynomial is continous at every point in the real-number system. You can use the intermediate value theorem on an interval,

[a, b] such that if f(a) < 0 and f(b) > 0 , there is a point x = c in (a, b) such that f(c) = 0.