In mathematics, a quintic function is a function of the form
g ( x ) = a x 5 + b x 4 + c x 3 + d x 2 + e x + f , {\displaystyle g(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f,\,}
where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero. In other words, a quintic function is defined by a polynomial of degree five. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function. Setting g(x) = 0 and assuming a ≠ 0 produces a quintic equation of the form:
a x 5 + b x 4 + c x 3 + d x 2 + e x + f = 0. {\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0.\,}
Solving quintic equations in terms of radicals (nth roots) was a major problem in algebra from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem.
Finding roots of a quintic polynomial 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. However, there is no algebraic expression (that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the Abel–Ruffini theorem, first asserted in 1799 and completely proven in 1824. This result also holds for equations of higher degree. An example of a quintic whose roots cannot be expressed in terms of radicals is x5 − x + 1 = 0. Numerical approximations of quintics roots can be computed with root-finding algorithms for polynomials. Although some quintics may be solved in terms of radicals, the solution is generally too complicated to be used in practice.
Solvable quintics Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that is reducible, such as x5 − x4 − x + 1 = (x2 + 1)(x + 1)(x − 1)2. For example, it has been shown that
x 5 − x − r = 0 {\displaystyle x^{5}-x-r=0}
has solutions in radicals if and only if it has an integer solution or r is one of ±15, ±22440, or ±2759640, in which cases the polynomial is reducible. As solving reducible quintic equations reduces immediately to solving polynomials of lower degree, only irreducible quintic equations are considered in the remainder of this section, and the term "quintic" will refer only to irreducible quintics. A solvable quintic is thus an irreducible quintic polynomial whose roots may be expressed in terms of radicals. To characterize solvable quintics, and more generally solvable polynomials of higher degree, Évariste Galois developed techniques which gave rise to group theory and Galois theory. Applying these techniques, Arthur Cayley found a general criterion for determining whether any given quintic is solvable. This criterion is the following. Given the equation
a x 5 + b x 4 + c x 3 + d x 2 + e x + f = 0 , {\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0,}
the Tschirnhaus transformation x = y − b/5a, which depresses the quintic (that is, removes the term of degree four), gives the equation
y 5 + p y 3 + q y 2 + r y + s = 0 , {\displaystyle y^{5}+py^{3}+qy^{2}+ry+s=0,}
where
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