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In mathematics, and, more specifically in numerical analysis and computer algebra, real-root isolation of a polynomial consist of producing disjoint intervals of the real line, which contain each one (and only one) real root of the polynomial, and, together, contain all the real roots of the polynomial.
The fundamental theorem of algebra states that every polynomial of degree has complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs. [2] Vieta's formulas relate the coefficients of a polynomial to sums and products of its 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.
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.
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.
Every polynomial in one variable x with real coefficients can be uniquely written as the product of a constant, polynomials of the form x + a with a real, and polynomials of the form x 2 + ax + b with a and b real and a 2 − 4b < 0 (which is the same thing as saying that the polynomial x 2 + ax + b has no real roots).
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