Search results
Results From The WOW.Com Content Network
The discriminant of a polynomial is a function of its coefficients that is zero if and only if the polynomial has a multiple root, or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is square-free .
In mathematics, a cubic function is a function of the form () = + + +, that is, a polynomial function of degree three. In many texts, the coefficients a , b , c , and d are supposed to be real numbers , and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to ...
Suppose the zeroes z 1, z 2, and z 3 of a third-degree polynomial p(z) are non-collinear. There is a unique ellipse inscribed in the triangle with vertices z 1, z 2, z 3 and tangent to the sides at their midpoints: the Steiner inellipse. The foci of that ellipse are the zeroes of the derivative p'(z).
Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots.
If x is a simple root of the polynomial , then Laguerre's method converges cubically whenever the initial guess, , is close enough to the root . On the other hand, when x 1 {\displaystyle \ x_{1}\ } is a multiple root convergence is merely linear, with the penalty of calculating values for the polynomial and its first and second derivatives at ...
The resolvent cubic of an irreducible quartic polynomial P(x) can be used to determine its Galois group G; that is, the Galois group of the splitting field of P(x). Let m be the degree over k of the splitting field of the resolvent cubic (it can be either R 4 (y) or R 5 (y); they have the same splitting field).
A root of a polynomial is a zero of the corresponding polynomial function. [1] The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree , and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically ...
The largest zero of this polynomial which corresponds to the second largest zero of the original polynomial is found at 3 and is circled in red. The degree 5 polynomial is now divided by () to obtain = + + which is shown in yellow. The zero for this polynomial is found at 2 again using Newton's method and is circled in yellow.