Inmathematics, aquintic function is afunction of the form
wherea,b,c,d,e andf are members of afield, typically therational numbers, thereal numbers or thecomplex numbers, anda is nonzero. In other words, a quintic function is defined by apolynomial ofdegree five.
Because they have an odd degree, normal quintic functions appear similar to normalcubic functions when graphed, except they may possess one additionallocal maximum and one additional local minimum. Thederivative of a quintic function is aquartic function.
Settingg(x) = 0 and assuminga ≠ 0 produces aquintic equation of the form:
Solving quintic equations in terms ofradicals (nth roots) was a major problem in algebra from the 16th century, whencubic andquartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with theAbel–Ruffini theorem.
Finding theroots (zeros) of a given polynomial has been a prominent mathematical problem.
Solvinglinear,quadratic,cubic andquartic 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 noalgebraic expression (that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as theAbel–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 isx5 −x + 1 = 0.
Numerical approximations of quintics roots can be computed withroot-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.
Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that isreducible, such asx5 −x4 −x + 1 = (x2 + 1)(x + 1)(x − 1)2. For example, it has been shown[1] that
has solutions in radicalsif and only if it has aninteger solution orr 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. Asolvable 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 togroup theory andGalois theory. Applying these techniques,Arthur Cayley found a general criterion for determining whether any given quintic is solvable.[2] This criterion is the following.[3]
Given the equation
theTschirnhaus transformationx =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 polynomialP2 − 1024z Δ, namedCayley's resolvent, has a rational root inz, where
and
Cayley's result allows us to test if a quintic is solvable. If it is the case, finding its roots is a more difficult problem, which consists of expressing the roots in terms of radicals involving the coefficients of the quintic and the rational root of Cayley's resolvent.
In 1888,George Paxton Young described how to solve a solvable quintic equation, without providing an explicit formula;[4] in 2004,Daniel Lazard wrote out a three-page formula.[5]
There are several parametric representations of solvable quintics of the formx5 +ax +b = 0, called theBring–Jerrard form.
During the second half of the 19th century, John Stuart Glashan, George Paxton Young, andCarl Runge gave such a parameterization: anirreducible quintic with rational coefficients in Bring–Jerrard formis solvable if and only if eithera = 0 or it may be written
whereμ andν 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. Using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second.
The substitution in the Spearman–Williams parameterization allows one to not exclude the special casea = 0, giving the following result:
Ifa andb are rational numbers, the equationx5 +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 numbersℓ andm such that
A polynomial equation is solvable by radicals if itsGalois group is asolvable group. In the case of irreducible quintics, the Galois group is a subgroup of thesymmetric groupS5 of all permutations of a five element set, which is solvable if and only if it is a subgroup of the groupF5, of order20, generated by the cyclic permutations(1 2 3 4 5) and(1 2 4 3).
If the quintic is solvable, one of the solutions may be represented by analgebraic expression involving a fifth root and at most two square roots, generallynested. The other solutions may then be obtained either by changing the fifth root or by multiplying all the occurrences of the fifth root by the same power of aprimitive 5th root of unity, such as
In fact, all four primitive fifth roots of unity may be obtained by changing the signs of the square roots appropriately; namely, the expression
where, yields the four distinct primitive fifth roots of unity.
It follows that one may need four different square roots for writing all the roots of a solvable quintic. Even for the first root that involves at most two square roots, the expression of the solutions in terms of radicals is usually highly complicated. However, when no square root is needed, the form of the first solution may be rather simple, as for the equationx5 − 5x4 + 30x3 − 50x2 + 55x − 21 = 0, for which the only real solution is
An example of a more complicated (although small enough to be written here) solution is the unique real root ofx5 − 5x + 12 = 0. Leta =√2φ−1,b =√2φ, andc =4√5, whereφ =1+√5/2 is thegolden ratio. Then the only real solutionx = −1.84208... is given by
or, equivalently, by
where theyi are the four roots of thequartic equation
More generally, if an equationP(x) = 0 of prime degreep with rational coefficients is solvable in radicals, then one can define an auxiliary equationQ(y) = 0 of degreep − 1, also with rational coefficients, such that each root ofP is the sum ofp-th roots of the roots ofQ. Thesep-th roots were introduced byJoseph-Louis Lagrange, and their products byp are commonly calledLagrange resolvents. The computation ofQ and its roots can be used to solveP(x) = 0. However thesep-th roots may not be computed independently (this would providepp−1 roots instead ofp). Thus a correct solution needs to express all thesep-roots in term of one of them. Galois theory shows that this is always theoretically possible, even if the resulting formula may be too large to be of any use.
It is possible that some of the roots ofQ are rational (as in the first example of this section) or some are zero. In these cases, the formula for the roots is much simpler, as for the solvablede Moivre quintic
where the auxiliary equation has two zero roots and reduces, by factoring them out, to thequadratic equation
such that the five roots of the de Moivre quintic are given by
whereyi is any root of the auxiliary quadratic equation andω is any of the fourprimitive 5th roots of unity. This can be easily generalized to construct a solvableseptic and other odd degrees, not necessarily prime.
There are infinitely many solvable quintics in Bring–Jerrard form which have been parameterized in a preceding section.
Up to the scaling of the variable, there are exactly five solvable quintics of the shape, which are[6] (wheres is a scaling factor):
Paxton Young (1888) gave a number of examples of solvable quintics:
| Root: | |
An infinite sequence of solvable quintics may be constructed, whose roots are sums ofnthroots of unity, withn = 10k + 1 being aprime number:
| Roots: | ||
| Root: | ||
| Root: | ||
| Root: | ||
| Root: |
There are also two parameterized families of solvable quintics:The Kondo–Brumer quintic,
and the family depending on the parameters
where
Analogously tocubic equations, there are solvable quintics which have five real roots all of whose solutions in radicals involve roots of complex numbers. This iscasus irreducibilis for the quintic, which is discussed in Dummit.[7]: p.17 Indeed, if an irreducible quintic has all roots real, no root can be expressed purely in terms of real radicals (as is true for all polynomial degrees that are not powers of 2).
About 1835,Jerrard demonstrated that quintics can be solved by usingultraradicals (also known asBring radicals), the unique real root oft5 +t −a = 0 for real numbersa. In 1858,Charles Hermite showed that the Bring radical could be characterized in terms of the Jacobitheta functions and their associatedelliptic modular functions, using an approach similar to the more familiar approach of solvingcubic equations by means oftrigonometric functions. At around the same time,Leopold Kronecker, usinggroup theory, developed a simpler way of deriving Hermite's result, as hadFrancesco Brioschi. Later,Felix Klein came up with a method that relates the symmetries of theicosahedron,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 ofgeneralized hypergeometric functions.[8] Similar phenomena occur in degree7 (septic equations) and11, as studied by Klein and discussed inIcosahedral symmetry § Related geometries.
ATschirnhaus transformation, which may be computed by solving aquartic equation, reduces the general quintic equation of the form
to theBring–Jerrard normal formx5 −x +t = 0.
The roots of this equation cannot be expressed by radicals. However, in 1858,Charles Hermite published the first known solution of this equation in terms ofelliptic functions.[9]At around the same timeFrancesco Brioschi[10] andLeopold Kronecker[11]came upon equivalent solutions.
SeeBring radical for details on these solutions and some related ones.
Solving for the locations of theLagrangian points of an astronomical orbit in which the masses of both objects are non-negligible involves solving a quintic.
More precisely, the locations ofL2 andL1 are the solutions to the following equations, where the gravitational forces of two masses on a third (for example, Sun and Earth on satellites such asGaia and theJames Webb Space Telescope atL2 andSOHO atL1) provide the satellite's centripetal force necessary to be in a synchronous orbit with Earth around the Sun:
The ± sign corresponds toL2 andL1, respectively;G is thegravitational constant,ω theangular velocity,r the distance of the satellite to Earth,R the distance Sun to Earth (that is, thesemi-major axis of Earth's orbit), andm,ME, andMS are the respective masses of satellite,Earth, andSun.
Using Kepler's Third Law and rearranging all terms yields the quintic
with:
Solving these two quintics yieldsr = 1.501 × 109m forL2 andr = 1.491 × 109m forL1. TheSun–Earth Lagrangian pointsL2 andL1 are usually given as 1.5 million km from Earth.
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 theHill sphere, given by:
That also yieldsr = 1.5 × 109m for satellites at L1 and L2 in the Sun-Earth system.
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