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→Example: roots of polynomials: Separating with commas the list of field operations. |
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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 (+, -, /, *). 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). However, they are rarely written explicitly because they are too complicated to be useful.
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