Quintic function

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.

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.

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.

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.

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.

[3] Given the equation the Tschirnhaus transformation x = y − ⁠b/5a⁠, which depresses the quintic (that is, removes the term of degree four), gives the equation where Both quintics are solvable by radicals if and only if either they are factorisable in equations of lower degrees with rational coefficients or the polynomial P2 − 1024 z Δ, named Cayley's resolvent, has a rational root in z, where and Cayley's result allows us to test if a quintic is solvable.

During the second half of the 19th century, John Stuart Glashan, George Paxton Young, and Carl Runge gave such a parameterization: an irreducible quintic with rational coefficients in Bring–Jerrard form is solvable if and only if either a = 0 or it may be written where μ and ν are rational.

In 1994, Blair Spearman and Kenneth S. Williams gave an alternative, The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression where a = ⁠5(4ν + 3)/ν2 + 1⁠.

The substitution c = ⁠−m/ℓ5⁠, e = ⁠1/ℓ⁠ in the Spearman–Williams parameterization allows one to not exclude the special case a = 0, giving the following result: If a and b are rational numbers, the equation x5 + ax + b = 0 is solvable by radicals if either its left-hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers ℓ and m such that A polynomial equation is solvable by radicals if its Galois group is a solvable group.

This can be easily generalized to construct a solvable septic and other odd degrees, not necessarily prime.

At around the same time, Leopold Kronecker, using group theory, developed a simpler way of deriving Hermite's result, as had Francesco Brioschi.

Later, Felix Klein came up with a method that relates the symmetries of the icosahedron, Galois theory, and the elliptic modular functions that are featured in Hermite's solution, giving an explanation for why they should appear at all, and developed his own solution in terms of generalized hypergeometric functions.

[8] Similar phenomena occur in degree 7 (septic equations) and 11, as studied by Klein and discussed in Icosahedral symmetry § Related geometries.

However, in 1858, Charles Hermite published the first known solution of this equation in terms of elliptic functions.

If the mass of the smaller object (ME) is much smaller than the mass of the larger object (MS), then the quintic equation can be greatly reduced and L1 and L2 are at approximately the radius of the Hill sphere, given by: That also yields r = 1.5 × 109 m for satellites at L1 and L2 in the Sun-Earth system.