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In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is Graph of a polynomial function of degree 4, with its 4 roots and 3 critical points.
In algebra, a quartic function is a function of the form = + + + +, α. where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form
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.
If a quartic polynomial P(x) is reducible in k[x], then it is the product of two quadratic polynomials or the product of a linear polynomial by a cubic polynomial. This second possibility occurs if and only if P(x) has a root in k.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
This method works well for cubic and quartic equations, but Lagrange did not succeed in applying it to a quintic equation, because it requires solving a resolvent polynomial of degree at least six. [ 37 ] [ 38 ] [ 39 ] Apart from the fact that nobody had previously succeeded, this was the first indication of the non-existence of an algebraic ...
Finding roots −2, −1 (repeated root), and −1/3 of the quartic 3x 4 +13x 3 +19x 2 +11x+2 using Lill's method. Black segments are labeled with their lengths (coefficients in the equation), while each colored line with initial slope m and the same endpoint corresponds to a real root.
Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods). In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography.