Last but not least, polynomial GCD algorithms and derived algorithms allow one to get useful information on the roots of a polynomial, without computing them.
22.
In this way, simply counting the sign changes in the leading coefficients in the Sturm chain readily gives the number of distinct real roots of a polynomial.
23.
The roots of a polynomial " f " are points on the affine line, which are the components of the algebraic set defined by the polynomial.
24.
A GCD computation allows to detect the existence of multiple roots, because the multiple roots of a polynomial are the roots of the GCD of the polynomial and its derivative.
25.
Descartes'work provided the basis for the calculus developed by rule of signs is also a commonly used method to determine the number of positive and negative roots of a polynomial.
26.
Since nonreal roots of a polynomial with real coefficients must occur in conjugate pairs, we can see that has exactly 2 nonreal roots and 1 real ( and positive ) root.
27.
A naive way to find the roots of a polynomial expression is to graph the equation and find the zeroes, where the graph crosses the horizontal ( x-) axis.
28.
It can be shown ( see Press, et al ., or Stoer and Bulirsch ) that the evaluation points are just the roots of a polynomial belonging to a class of orthogonal polynomials.
29.
For this computation, the representation involving the solving of only one univariate polynomial for each solution is preferable : computing the roots of a polynomial which has approximate coefficients is a highly unstable problem.
30.
Vincent's method was converted into its quotient " a i " as the lower bound, " lb ", on the values of the positive roots of a polynomial.
