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This has been generalized by Budan's theorem (1807), into a similar result for the real roots in a half-open interval (a, b]: If f(x) is a polynomial, and v is the difference between of the numbers of sign variations of the sequences of the coefficients of f(x + a) and f(x + b), then v minus the number of real roots in the interval, counted ...
For polynomials with real coefficients, it is often useful to bound only the real roots. It suffices to bound the positive roots, as the negative roots of p(x) are the positive roots of p(–x). Clearly, every bound of all roots applies also for real roots. But in some contexts, tighter bounds of real roots are useful.
This is illustrated by Wilkinson's polynomial: the roots of this polynomial of degree 20 are the 20 first positive integers; changing the last bit of the 32-bit representation of one of its coefficient (equal to –210) produces a polynomial with only 10 real roots and 10 complex roots with imaginary parts larger than 0.6.
This can be verified by noting that p(x) can be factored as (x 2 − 1)(x 2 + x + 1), where the first factor has the roots −1 and 1, and second factor has no real roots. This last assertion results from the quadratic formula , and also from Sturm's theorem, which gives the sign sequences (+, –, –) at −∞ and (+, +, –) at +∞ .
For n = 5, 10, none of the non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sum z + z = 2 Re z of each root with its complex conjugate (also a 5th root of unity) is an element of the ring Z[ 1 + √ 5 / 2 ] (D = 5).
The polynomial x 2 + 1 = 0 has roots ± i. Any real square matrix of odd degree has at least one real eigenvalue. For example, if the matrix is orthogonal, then 1 or −1 is an eigenvalue. The polynomial + has roots , +,, and thus can be factored as
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[8]: 11, Lemma.4.7 Note that Vrahatis and Iordanidis [8] prove a lower bound on the number of evaluations, and not an upper bound. A fourth method uses an intermediate value theorem on simplices. [9] Again, no upper bound on the number of queries is given.