Quintic function

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.