Sextic equation
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a
where a ≠ 0 and the coefficients a, b, c, d, e, f, g may be
.A sextic function is a
Since a sextic function is defined by a polynomial with even degree, it has the same infinite limit when the argument goes to positive or negative
Solvable sextics
Some sixth degree equations, such as ax6 + dx3 + g = 0, can be solved by factorizing into radicals, but other sextics cannot. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory.[1]
It follows from Galois theory that a sextic equation is solvable in terms of radicals if and only if its
There are formulas to test either case, and, if the equation is solvable, compute the roots in term of radicals.[2]
The general sextic equation can be solved by the two-variable
Examples
Watt's curve, which arose in the context of early work on the steam engine, is a sextic in two variables.
One method of solving the cubic equation involves transforming variables to obtain a sextic equation having terms only of degrees 6, 3, and 0, which can be solved as a quadratic equation in the cube of the variable.
Etymology
The describer "sextic" comes from the Latin stem for 6 or 6th ("sex-t-"), and the Greek suffix meaning "pertaining to" ("-ic"). The much less common "hexic" uses Greek for both its stem (hex- 6) and its suffix (-ik-). In both cases, the prefix refers to the degree of the function. Often, these type of functions will simply be referred to as "6th degree functions".
See also
References
- ^ a b c Mathworld - Sextic Equation
- ^ T. R. Hagedorn, General formulas for solving solvable sextic equations, J. Algebra 233 (2000), 704-757