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Concept
Quintic function
Summary
In mathematics, a quintic function is a function of the form
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:
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 the roots (zeros) of a given polynomial has been a prominent mathematical problem.
Solving linear, quadratic, cubic and quartic equations by factorization into radicals can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulae 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 proved 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 x^5 − x + 1 = 0.
Some quintics may be solved in terms of radicals. However, the solution is generally too complicated to be used in practice. Instead, numerical approximations are calculated using a root-finding algorithm for polynomials.
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.
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