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In algebra, 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. If ''a'' is zero but one of the coefficients ''b'', ''c'', ''d'', or ''e'' is non-zero, the function is classified as either a quartic function, cubic function, quadratic function or linear function. Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess an additional local maximum and local minimum each. 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 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 (Abel–Ruffini theorem). ==Finding roots of a quintic equation== Finding the roots 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 explicit formula for general quintic equations over the rationals in terms of radicals; this statement is known as the Abel–Ruffini theorem, first published in 1824, which was the main motivation of the introduction of group theory by Évariste Galois a few years later. This result also holds for equations of higher degrees. An example of a quintic whose roots cannot be expressed in terms of radicals is This quintic is in Bring–Jerrard normal form. Some quintics may be solved in terms of radicals. However, the solution is generally too complex to be used in practice. Therefore, one commonly uses numerical approximations of the solutions, which can be provided by any root-finding algorithm, and in particular by any root-finding algorithm for polynomials. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「quintic function」の詳細全文を読む スポンサード リンク
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