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The rule states that if the nonzero terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign changes between consecutive (nonzero) coefficients, or is less than it by an even number.
The theory of continued fractions allows us to compute the best approximations of a real number: for the second definition, they are the convergents of its expression as a regular continued fraction. [ 3 ] [ 4 ] [ 5 ] For the first definition, one has to consider also the semiconvergents .
It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial (x – r) can be factored out of the polynomial using polynomial long division, resulting in a polynomial of lower degree whose roots are also roots of the original polynomial.
In his Essai sur la théorie des nombres (1798), Adrien-Marie Legendre derives a necessary and sufficient condition for a rational number to be a convergent of the simple continued fraction of a given real number. [4] A consequence of this criterion, often called Legendre's theorem within the study of continued fractions, is as follows: [5 ...
Budan's may provide a real-root-isolation algorithm for a square-free polynomial (a polynomial without multiple root): from the coefficients of polynomial, one may compute an upper bound M of the absolute values of the roots and a lower bound m on the absolute values of the differences of two roots (see Properties of polynomial roots).
Finding the real roots of a polynomial with real coefficients is a problem that has received much attention since the beginning of 19th century, and is still an active domain of research. Most root-finding algorithms can find some real roots, but cannot certify having found all the roots.
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Clearly, every bound of all roots applies also for real roots. But in some contexts, tighter bounds of real roots are useful. For example, the efficiency of the method of continued fractions for real-root isolation strongly depends on tightness of a bound of positive roots. This has led to establishing new bounds that are tighter than the ...
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