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So, if the three non-monic coefficients of the depressed quartic equation, + + + =, in terms of the five coefficients of the general quartic equation are given as follows: =, = + and = +, then the criteria to identify a priori each case of quartic equations with multiple roots and their respective solutions are exposed below.
Each coordinate of the intersection points of two conic sections is a solution of a quartic equation. The same is true for the intersection of a line and a torus.It follows that quartic equations often arise in computational geometry and all related fields such as computer graphics, computer-aided design, computer-aided manufacturing and optics.
The values of trigonometric functions of angles related to / satisfy cubic equations. Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic. The solution of the general quartic equation relies on the solution of its resolvent cubic.
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.. Solving linear, quadratic, cubic and quartic equations in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions.
An algebraic solution to the problem was finally found first in 1965 by Jack M. Elkin (an actuary), by means of a quartic polynomial. [8] Other solutions were rediscovered later: in 1989, by Harald Riede; [9] in 1990 (submitted in 1988), by Miller and Vegh; [10] and in 1992, by John D. Smith [3] and also by Jörg Waldvogel. [11]
In mathematics, the term quartic describes something that pertains to the "fourth order", such as the function . It may refer to one of the following: Quartic function, a polynomial function of degree 4; Quartic equation, a polynomial equation of degree 4; Quartic curve, an algebraic curve of degree 4
An affine quartic surface is the solution set of an equation of the form ... The Fermat quartic, given by x 4 + y 4 + z 4 + w 4 =0 (an example of a K3 surface).
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