In
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, a sextic (or hexic) polynomial is a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
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 (mathematics), field, often the field of the rational numbers.
For example, x^5-3x+1=0 is a ...
of degree six—that is, an
equation
In mathematics, an equation is a mathematical 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 ...
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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
,
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 (for example,
The set of all ...
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s,
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s or, more generally, members of any
field.
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 (mathematics), 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 of a polynomial, degree four, called a quartic polynomial.
A ''qu ...
s when graphed, except they may possess an additional
local maximum
In mathematical analysis, the maximum and minimum of a function (mathematics), function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given Interval (ma ...
and
local minimum
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...
each. The
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of a sextic function is a
quintic function
In mathematics, 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 word ...
.
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 something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol.
From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
. If the
leading coefficient
In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a c ...
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 Nth root, ...
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 (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
.
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]
Examples
Watt's curve, which arose in the context of early work on the
steam engine
A steam engine is a heat engine that performs Work (physics), 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 (locomotive), cyl ...
, is a sextic in two variables.
One method of solving the
cubic equation
In algebra, a cubic equation in one variable is an equation of the form
ax^3+bx^2+cx+d=0
in which is not zero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
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 languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
stem
Stem or STEM most commonly refers to:
* Plant stem, a structural axis of a vascular plant
* Stem group
* Science, technology, engineering, and mathematics
Stem or STEM can also refer to:
Language and writing
* Word stem, part of a word respon ...
for 6 or 6th ("sex-t-"), and the
Greek
Greek may refer to:
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 of all kno ...
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 and adjectives, and verb endings, which form the conjugation of verbs. Suffixes can ca ...
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
In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d, that is, a polynomial function of degree three. In many texts, the ''coefficients'' , , , and are supposed to be real numbers, and the function is considered as ...
*
Septic equation
References
{{DEFAULTSORT:Sextic Equation
Equations
Galois theory
Polynomials