The Budan-Fourier Theorem

Given a polynomial of degree , the sequence , , ..., is called the Budan–Fourier sequence of .
Let be the number of real roots of over an open interval (i.e. excluding and ). Then , where is the difference between the number of sign changes of the Budan–Fourier sequence evaluated at and at , and is a non-negative even integer. Thus the Budan–Fourier theorem states that the number of roots in the interval is equal to or is smaller by an even number.


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[1] Wikipedia. "Budan's Theorem." (Mar 20, 2017)'s_theorem.
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