Sturm's Theorem for Polynomials

Let be the number of real roots of an algebraic equation with real coefficients whose real roots are simple over an interval and are not or . Then , the difference between the number of sign changes of the Sturm chain evaluated at and at .
By subdividing an interval until every subinterval contains at most one root, one can locate subintervals containing all the real roots in the original interval.

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The Sturm chain of a polynomial is the sequence of polynomials:
,
where
Here and are the polynomial quotient and remainder of . The chain ends when the polynomial is a constant.
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