In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, a sextic (or hexic) polynomial is a
polynomial
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
of
degree six.
A sextic equation is a
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
of degree six—that is, an
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form:
:
where and the ''coefficients'' may be
integers
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
,
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s,
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
s,
complex numbers or, more generally, members of any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
A sextic function is a
function defined by a sextic polynomial. Because they have an even degree, sextic functions appear similar to
quartic function
In algebra, a quartic function is a function of the form
:f(x)=ax^4+bx^3+cx^2+dx+e,
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 deg ...
s when graphed, except they may possess an additional
local maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
and local minimum each. The
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of a sextic function is a
quintic function
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, ...
.
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
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
. If the
leading coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves va ...
is positive, then the function increases to positive infinity at both sides and thus the function has a global minimum. Likewise, if is negative, the sextic function decreases to negative infinity and has a global maximum.
Solvable sextics
Some sixth degree equations, such as , can be solved by factorizing into radicals, but other sextics cannot.
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
.
It follows from Galois theory that a sextic equation is solvable in terms of radicals if and only if its
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
is contained either in the group of order 48 which
stabilizes a partition of the set of the roots into three subsets of two roots or in the group of order 72 which stabilizes a partition of the set of the roots into two subsets of three roots.
There are formulas to test either case, and, if the equation is solvable, compute the roots in term of radicals.
[T. R. Hagedorn, ''General formulas for solving solvable sextic equations'', J. Algebra 233 (2000), 704-757]
The general sextic equation can be solved in terms of
Kampé de Fériet function
In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet.
The Kampé de Fériet function is given by
:
^F_\left(
\begin
a_1,\cdots,a_p\colon b ...
s.
[Mathworld - Sextic Equation](_blank)
/ref> A more restricted class of sextics can be solved in terms of generalised hypergeometric functions in one variable using Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
's approach to solving the quintic equation
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
.
Examples
Watt's curve
In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius ''b'' with centers distance 2''a'' apart (taken to be at (±''a'', 0)). A line segment of length 2''c'' attaches to a p ...
, which arose in the context of early work on the steam engine
A steam engine is a heat engine that performs mechanical work using steam as its working fluid. The steam engine uses the force produced by steam pressure to push a piston back and forth inside a cylinder. This pushing force can be trans ...
, 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
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
stem
Stem or STEM may refer to:
Plant structures
* Plant stem, a plant's aboveground axis, made of vascular tissue, off which leaves and flowers hang
* Stipe (botany), a stalk to support some other structure
* Stipe (mycology), the stem of a mushro ...
for 6 or 6th ("sex-t-"), and the Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
suffix
In linguistics, a suffix is an affix which is placed after the stem of a word. Common examples are case endings, which indicate the grammatical case of nouns, adjectives, and verb endings, which form the conjugation of verbs. Suffixes can carry g ...
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
* Cayley's sextic
* Cubic function
*Septic equation
In algebra, a septic equation is an equation of the form
:ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h=0,\,
where .
A septic function is a function of the form
:f(x)=ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h\,
where . In other words, it is a polynomial of ...
References
{{DEFAULTSORT:Sextic Equation
Equations
Galois theory
Polynomials