Editing Closed-form expression (section)

== Example: roots of polynomials ==

The [[quadratic formula]]

<math display="block">x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}</math>

is a ''closed form'' of the solutions to the general [[quadratic equation]] <math>ax^2+bx+c=0.</math>

More generally, in the context of [[polynomial equation]]s, a closed form of a solution is a [[solution in radicals]]; that is, a closed-form expression for which the allowed functions are only {{mvar|n}}th-roots and field operations <math>(+, -, \times ,/).</math> In fact, [[field theory (mathematics)|field theory]] allows showing that if a solution of a polynomial equation has a closed form involving exponentials, logarithms or trigonometric functions, then it has also a closed form that does not involve these functions.{{cn|date=August 2023}}

There are expressions in radicals for all solutions of [[cubic equation]]s (degree 3) and [[quartic equation]]s (degree 4). The size of these expressions increases significantly with the degree, limiting their usefulness.

In higher degrees, the [[Abel–Ruffini theorem]] states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. A simple example is the equation <math>x^5-x-1=0.</math> [[Galois theory]] provides an [[algorithmic method]] for deciding whether a particular polynomial equation can be solved in radicals.