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| [[Image:Sextic Graph.svg|thumb|right|233px|Graph of a sextic function, with 5 [[critical point (mathematics)|critical points]].]]
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| In [[mathematics]], a '''sextic polynomial''' is a [[polynomial]] of [[Degree of a polynomial|degree]] six.
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| A '''sextic equation''' or '''hexic equation''' is a [[polynomial equation]] of degree six—that is, an [[equation]] whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form:
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| :<math>ax^6+bx^5+cx^4+dx^3+ex^2+fx+g=0,\,</math>
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| where {{math|''a'' ≠ 0}} and the ''coefficients'' {{math| ''a'', ''b'', ''c'', ''d'', ''e'', ''f'', ''g''}} may either be [[integers]], [[rational number]]s, [[real number]]s, [[complex number]]s or, more generally, members of any [[field (mathematics)|field]].
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| A '''sextic function''' is a [[function (mathematics)|function]] defined by a sextic polynomial. Because they have an even degree, sextic functions appear similar to [[quartic function]]s when graphed, except they may possess an additional [[Maxima and minima|local maximum]] and local minimum each. The [[derivative]] of a sextic function is a [[quintic function]].
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| 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]]. If the [[leading coefficient]] {{math|''a''}} is positive, then the function increases to positive infinity at both sides; and thus the function has a global minimum. Likewise, if ''a'' is negative, the sextic function decreases to negative infinity and has a global maximum.
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| ==Solvable sextics==
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| Some sixth degree equations 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]].<ref name="Mathworld - Sextic Equation"/>
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| It follows from Galois theory that a sextic equation is solvable in term of radicals if and only if its [[Galois group]] is contained either in the group of order 48 which [[stabilizer (group theory)|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.
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| There are formulas to test either case, and, if the equation is solvable, compute the roots in term of radicals.<ref>T. R. Hagedorn, ''General formulas for solving solvable sextic equations'', J. Algebra '''233''' (2000), 704-757</ref>
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| The general sextic equation can be solved in terms of [[Kampé de Fériet function]]s.<ref name="Mathworld - Sextic Equation">[http://mathworld.wolfram.com/SexticEquation.html Mathworld - Sextic Equation]</ref> A more restricted class of sextics can be solved in terms of [[hypergeometric functions|generalised hypergeometric functions]] in one variable using [[Felix Klein]]'s approach to solving the [[quintic equation]].<ref name="Mathworld - Sextic Equation"/>
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| ==See also==
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| *[[Cubic function]]
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| *[[Septic equation]]
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| ==References==
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| <references/>
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| {{Polynomials}}
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| {{DEFAULTSORT:Sextic Equation}}
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| [[Category:Equations]]
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| [[Category:Galois theory]]
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| [[Category:Polynomials]]
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| {{algebra-stub}}
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