# Budan's theorem

## Budan's theorem

[′bü‚dänz ‚thir·əm]
(mathematics)
The theorem that the number of roots of an n th-degree polynomial lying in an open interval equals the difference in the number of sign changes induced by n differentiations at the two ends of the interval.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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More specifically, the paper demonstrates the applicability of Descartes' Rule of Signs, Budan's Theorem, and Sturm's Theorem from the theory of equations and rules developed in the business literature by Teichroew, Robichek, and Montalbano (1965a, 1965b), Mao (1969), Jean (1968, 1969), and Pratt and Hammond (1979).
In addition, Descartes' Rule of Signs is actually a special case of Budan's Theorem, which will be discussed next.
Additional information concerning multiple roots can be gained by the application of Budan's Theorem. Unfortunately, the theorem requires a greater understanding of mathematics and might test the abilities of any practitioner unfamiliar with calculus.
Budan's Theorem states that in an nth degree polynomial where f(x) = 0, the number of real roots for a [less than or equal to] x [less than or equal to] b is at most S(a) - S(b), where S(a) and S(b) are the number of variations in signs in the sequence of f(x) and its derivatives when x = a and x = b (Skrapek et al., 1976: 40-41).
Sturm's Theorem is more awkward to apply and requires greater computational skill than Budan's Theorem. However, the application of Sturm's Theorem is preferable to Budan's Theorem because of the additional information gained.
As with Budan's Theorem, the user specifies the range to be evaluated.
By, in turn, using Descartes' Rule of Signs, Budan's Theorem, and Sturm's Theorem, an increasingly clearer picture is gained of the number of positive solutions to the pump problem.
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