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In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function, is a member of the domain of such that () vanishes at ; that is, the function attains the value of 0 at , or equivalently, is a solution to the equation () =.
Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation 2x 2 + 4x − 4 = 0. Although the display shows only five significant figures of accuracy, the retrieved value of xc is 0.732050807569, accurate to twelve significant figures. A quadratic function without real root: y = (x − 5) 2 + 9.
The quantity = is known as the discriminant of the quadratic equation. If the coefficients , , and are real numbers then when >, the equation has two distinct real roots; when =, the equation has one repeated real root; and when <, the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other.
They lead to efficient algorithms for real-root isolation of polynomials, which ensure finding all real roots with a guaranteed accuracy. Bisection method. The simplest root-finding algorithm is the bisection method. Let f be a continuous function, for which one knows an interval [a, b] such that f(a) and f(b) have opposite signs (a bracket).
Graph of a cubic function with 3 real roots (where the curve crosses the horizontal axis at y = 0).The case shown has two critical points.Here the function is () = (+).. In algebra, a cubic equation in one variable is an equation of the form
True roots must occur on both lists, so list of rational root candidates has shrunk to just x = 2 and x = 2/3. If k ≥ 1 rational roots are found, Horner's method will also yield a polynomial of degree n − k whose roots, together with the rational roots, are exactly the roots of the original polynomial. If none of the candidates is a ...
Repeated real roots. If the characteristic equation has a root r 1 that is repeated k times, then it is clear that y p (x) = c 1 e r 1 x is at least one solution. However, this solution lacks linearly independent solutions from the other k − 1 roots. Since r 1 has multiplicity k, the differential equation can be factored into
They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see Real-root isolation. The method [ edit ] The method is applicable for numerically solving the equation f ( x ) = 0 for the real variable x , where f is a continuous function defined on an interval [ a , b ] and where f ( a ) and f ...