Questions about polynomials with degree $5$. There is no general algebraic solution to these equations as proven by the Abel-Ruffini theorem, although some quintics are solvable.
A quintic equation is an equation in the form:
$$ax^5 + bx^4 + cx^3 + dx^2+ex+f = 0 $$
where $a,b,c,d,e,f$ are members of a field, typically either the rational numbers, the real numbers, or the complex numbers, and $a \ne 0$.
According to the Fundamental Theorem of Algebra, quintic equations always have $5$ roots. This number includes complex roots, as well as repeated roots.
The Abel-Ruffini theorem states that there is no algebraic solution to a quintic equation with arbitrary coefficients. An algebraic solution is a solution which uses only addition, subtraction, multiplication, division, exponentiation, and $n$th root extraction. However, this theorem does not imply that all quintics do not have an algebraic solution (one counterexample is $(x-1)^5 = 0$), or that a specific quintic equation is not solvable using radicals. Sextic ($6$th degree) equations and polynomials of higher degrees also do not have a general algebraic solution under this theorem.
There are several methods to find the roots of solvable quintics. One method is to use the Tschrinhaus transformation $x = y - \frac{b}{5a}$, which depresses the quintic or removes the fourth-degree term. Then the original quintic has a solvable root if the transformed quintic is a product of lower-order polynomials with rational coefficients, or if Cayley's resolvent, the polynomial $P^2 - 1024z \Delta$ is solvable. Alternatively, quintics of the form $x^5+ax+b = 0$ can be represented parametrically using the Bring-Jerrard form.
The roots of quintics and other higher-order polynomials can be approximated by Newton's method, the secant method, the method of false position, Padé approximants, and other root-finding algorithms.
Quintic equations are relevant to problems in celestial mechanics. Solving for the locations of the Lagrangian points of an astronomical orbit, where the masses of both objects are non-negligible, requires solving a quintic. For example, finding a stable location for a satellite between the Sun and the Earth requires solving a quintic.
References (not for academic use):
https://en.wikipedia.org/wiki/Quintic_function
https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
https://mathworld.wolfram.com/QuinticEquation.html
Further reading:
