Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically).
A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).
In particular, it includes the study of equations that involve nth roots and, more generally, algebraic expressions.
This makes the term algebraic equation ambiguous outside the context of the old problem.
The study of algebraic equations is probably as old as mathematics: the Babylonian mathematicians, as early as 2000 BC could solve some kinds of quadratic equations (displayed on Old Babylonian clay tablets).
Ancient mathematicians wanted the solutions in the form of radical expressions, like
The ancient Egyptians knew how to solve equations of degree 2 in this manner.
In the 9th century Muhammad ibn Musa al-Khwarizmi and other Islamic mathematicians derived the quadratic formula, the general solution of equations of degree 2, and recognized the importance of the discriminant.
During the Renaissance in 1545, Gerolamo Cardano published the solution of Scipione del Ferro and Niccolò Fontana Tartaglia to equations of degree 3 and that of Lodovico Ferrari for equations of degree 4.
Finally Niels Henrik Abel proved, in 1824, that equations of degree 5 and higher do not have general solutions using radicals.
Galois theory, named after Évariste Galois, showed that some equations of at least degree 5 do not even have an idiosyncratic solution in radicals, and gave criteria for deciding if an equation is in fact solvable using radicals.
A polynomial equation over the rationals can always be converted to an equivalent one in which the coefficients are integers.
For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the previously mentioned polynomial equation
The fundamental theorem of algebra states that the field of the complex numbers is closed algebraically, that is, all polynomial equations with complex coefficients and degree at least one have a solution.
It follows that all polynomial equations of degree 1 or more with real coefficients have a complex solution.
However, a monic polynomial of odd degree must necessarily have a real root.
By the intermediate value theorem, it must therefore assume the value zero at some real x, which is then a solution of the polynomial equation.
There exist formulas giving the solutions of real or complex polynomials of degree less than or equal to four as a function of their coefficients.
Abel showed that it is not possible to find such a formula in general (using only the four arithmetic operations and taking roots) for equations of degree five or higher.
Galois theory provides a criterion which allows one to determine whether the solution to a given polynomial equation can be expressed using radicals.
The explicit solution of a real or complex equation of degree 1 is trivial.
Solving an equation of higher degree n reduces to factoring the associated polynomial, that is, rewriting (E) in the form where the solutions are then the
This approach applies more generally if the coefficients and solutions belong to an integral domain.
To solve an equation of degree n, a common preliminary step is to eliminate the degree-n - 1 term: by setting
If the polynomial has real coefficients, it has: The best-known method for solving cubic equations, by writing roots in terms of radicals, is Cardano's formula.
may be reduced to a quadratic equation by a change of variable provided it is either biquadratic (b = d = 0) or quasi-palindromic (e = a, d = b).
Some cubic and quartic equations can be solved using trigonometry or hyperbolic functions.
Évariste Galois and Niels Henrik Abel showed independently that in general a polynomial of degree 5 or higher is not solvable using radicals.
Charles Hermite, on the other hand, showed that polynomials of degree 5 are solvable using elliptical functions.
Otherwise, one may find numerical approximations to the roots using root-finding algorithms, such as Newton's method.