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Inmathematics, analgebraic equation orpolynomial equation is anequation of the form, whereP is apolynomial, usually withrational numbers forcoefficients.
For example, is an algebraic equation withinteger coefficients and
is amultivariate polynomial equation over the rationals.For many authors, the termalgebraic equation refers only to theunivariate case, that is polynomial equations that involve only onevariable. On the other hand, a polynomial equation may involve several variables (themultivariate case), in which case the termpolynomial equation is usually preferred.
Some but not all polynomial equations withrational coefficients have a solution that is analgebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can besolved algebraically). This can be done for all such equations ofdegree one, two, three, or four; but for degree five or more it can only be done for some equations,not all. A large amount of research has been devoted to compute efficiently accurate approximations of thereal orcomplex solutions of a univariate algebraic equation (seeRoot-finding algorithm) and of the common solutions of several multivariate polynomial equations (seeSystem of polynomial equations).
The term "algebraic equation" dates from the time when the main problem ofalgebra was to solve univariate polynomial equations. This problem was completely solved during the 19th century; seeFundamental theorem of algebra,Abel–Ruffini theorem andGalois theory.
Since then, the scope of algebra has been dramatically enlarged. In particular, it includes the study of equations that involventh roots and, more generally,algebraic expressions. This makes the termalgebraic equation ambiguous outside the context of the old problem. So the termpolynomial equation is generally preferred when this ambiguity may occur, specially when considering multivariate equations.
The study of algebraic equations is probably as old as mathematics: theBabylonian mathematicians, as early as 2000 BC could solve some kinds ofquadratic equations (displayed onOld Babylonianclay tablets).
Univariate algebraic equations over the rationals (i.e., withrational coefficients) have a very long history. Ancient mathematicians wanted the solutions in the form ofradical expressions, like for the positive solution of. The ancient Egyptians knew how to solve equations of degree 2 in this manner. The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. In the 9th centuryMuhammad ibn Musa al-Khwarizmi and other Islamic mathematicians derived thequadratic formula, the general solution of equations of degree 2, and recognized the importance of thediscriminant. During the Renaissance in 1545,Gerolamo Cardano published the solution ofScipione del Ferro andNiccolò Fontana Tartaglia toequations of degree 3 and that ofLodovico Ferrari forequations of degree 4. FinallyNiels Henrik Abel proved, in 1824, thatequations of degree 5 and higher do not have general solutions using radicals.Galois theory, named afterÉvariste 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.
The algebraic equations are the basis of a number of areas of modern mathematics:Algebraic number theory is the study of (univariate) algebraic equations over the rationals (that is, withrational coefficients).Galois theory was introduced byÉvariste Galois to specify criteria for deciding if an algebraic equation may be solved in terms of radicals. Infield theory, analgebraic extension is an extension such that every element is a root of an algebraic equation over the base field.Transcendental number theory is the study of the real numbers which are not solutions to an algebraic equation over the rationals. ADiophantine equation is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions.Algebraic geometry is the study of the solutions in analgebraically closed field of multivariate polynomial equations.
Two equations are equivalent if they have the same set ofsolutions. In particular the equation is equivalent to. It follows that the study of algebraic equations is equivalent to the study of polynomials.
A polynomial equation over the rationals can always be converted to an equivalent one in which thecoefficients areintegers. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the previously mentioned polynomial equation becomes
Becausesine,exponentiation, and 1/T are not polynomial functions,
isnot a polynomial equation in the four variablesx,y,z, andT over the rational numbers. However, it is a polynomial equation in the three variablesx,y, andz over the field of theelementary functions in the variableT.
Given an equation in unknownx
with coefficients in afieldK, one can equivalently say that the solutions of (E) inK are the roots inK of the polynomial
It can be shown that a polynomial of degreen in a field has at mostn roots. The equation (E) therefore has at mostn solutions.
IfK' is afield extension ofK, one may consider (E) to be an equation with coefficients inK and the solutions of (E) inK are also solutions inK' (the converse does not hold in general). It is always possible to find a field extension ofK known as therupture field of the polynomialP, in which (E) has at least one solution.
Thefundamental theorem of algebra states that thefield of thecomplex 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 acomplex solution. On the other hand, an equation such as does not have a solution in (the solutions are theimaginary unitsi and−i).
While the real solutions of real equations are intuitive (they are thex-coordinates of the points where the curvey =P(x) intersects thex-axis), the existence of complex solutions to real equations can be surprising and less easy to visualize.
However, amonic polynomial ofodd degree must necessarily have a real root. The associatedpolynomial function inx is continuous, and it approaches asx approaches and asx approaches. By theintermediate value theorem, it must therefore assume the value zero at some realx, 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 degreen reduces to factoring the associated polynomial, that is, rewriting (E) in the form
where the solutions are then the. The problem is then to express the in terms of the.
This approach applies more generally if the coefficients and solutions belong to anintegral domain.
If an equationP(x) = 0 of degreen has arational rootα, the associated polynomial can be factored to give the formP(X) = (X − α)Q(X) (bydividingP(X) byX − α or by writingP(X) −P(α) as alinear combination of terms of the formXk − αk, and factoring outX − α. SolvingP(x) = 0 thus reduces to solving the degreen − 1 equationQ(x) = 0. See for example thecasen = 3.
To solve an equation of degreen,
a common preliminary step is to eliminate the degree-n - 1 term: by setting, equation (E) becomes
Leonhard Euler developed this technique forthe casen = 3 but it is also applicable tothe casen = 4, for example.
To solve a quadratic equation of the form one calculates thediscriminant Δ defined by.
If the polynomial has real coefficients, it has:
The best-known method for solving cubic equations, by writing roots in terms of radicals, isCardano's formula.
For detailed discussions of some solution methods see:
A quartic equation with may be reduced to a quadratic equation by a change of variable provided it is eitherbiquadratic (b = d = 0) orquasi-palindromic (e = a,d = b).
Some cubic and quartic equations can be solved usingtrigonometry orhyperbolic functions.
Évariste Galois andNiels Henrik Abel showed independently that in general a polynomial of degree 5 or higher is not solvable using radicals. Some particular equations do have solutions, such as those associated with thecyclotomic polynomials of degrees 5 and 17.
Charles Hermite, on the other hand, showed that polynomials of degree 5 are solvable usingelliptical functions.
Otherwise, one may findnumerical approximations to the roots usingroot-finding algorithms, such asNewton's method.
