algebraic equation

TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, an algebraic equation or polynomial equation is an
equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

of the form :$P = 0$ where ''P'' is a
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

with
coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s in some
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, often the field of the
rational number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s. For many authors, the term ''algebraic equation'' refers only to ''univariate equations'', that is polynomial equations that involves only one variable. On the other hand, a polynomial equation may involve several variables. In the case of several variables (''multivariate'' case), the term ''polynomial equation'' is usually preferred to ''algebraic equation''. For example, :$x^5-3x+1=0$ is an algebraic equation with integer coefficients and :$y^4+\frac-\frac+xy^2+y^2+\frac=0$ is a multivariate polynomial equation over the rationals. Some but not all polynomial equations with rational coefficients have a solution that is an
algebraic expressionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
one, two, three, or four; but for degree five or more it can only be done for some equations, not for all. A large amount of research has been devoted to compute efficiently accurate approximations of the
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

solutions of a univariate algebraic equation (see
Root-finding algorithm In mathematics and computing, a root-finding algorithm is an algorithm for finding Zero of a function, zeroes, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to ...
) and of the common solutions of several multivariate polynomial equations (see
System of polynomial equations A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for ...
).

# Terminology

The term "algebraic equation" dates from the time when the main problem of
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

was to solve
univariateIn mathematics, a univariate object is an expression, equation In mathematics, an equation is a statement that asserts the equality (mathematics), equality of two Expression (mathematics), expressions, which are connected by the equals sign "=". ...
polynomial equations. This problem was completely solved during the 19th century; see
Fundamental theorem of algebra The fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
,
Abel–Ruffini theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general algebraic equation, polynomial equations of quintic equation, degree five or higher with arbitrary coef ...
and
Galois theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Since then, the scope of algebra has been dramatically enlarged. In particular, it includes the study of equations that involve th roots and, more generally,
algebraic expressionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s. This makes the term ''algebraic equation'' ambiguous outside the context of the old problem. So the term ''polynomial equation'' is generally preferred when this ambiguity may occur, specially when considering multivariate equations.

# History

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 equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

s (displayed on
Old Babylonian Old Babylonian may refer to: *the period of the First Babylonian dynasty (20th to 16th centuries BC) *the historical stage of the Akkadian language of that time See also

*Old Assyrian (disambiguation) {{disambig ...
clay tablet In the Ancient Near East The ancient Near East was the home of early civilization A civilization (or civilisation) is any complex society that is characterized by urban development, social stratification, a form of government, and symb ...
s). Univariate algebraic equations over the rationals (i.e., with
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
coefficients) have a very long history. Ancient mathematicians wanted the solutions in the form of
radical expression In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, like $x=\frac$ for the positive solution of $x^2-x-1=0$. 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 century
Muhammad ibn Musa al-Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( fa, محمد بن موسی خوارزمی, Moḥammad ben Musā Khwārazmi; ), Arabized as al-Khwarizmi and formerly Latinized as ''Algorithmi'', was a Persian polymath A polymath ( el, πολυμα ...

and other Islamic mathematicians derived the
quadratic formula In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas ar ...

, the general solution of equations of degree 2, and recognized the importance of the
discriminant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. During the Renaissance in 1545,
Gerolamo Cardano Gerolamo (also Girolamo or Geronimo) Cardano (; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501 (O. S.)– 21 September 1576 (O. S.)) was an Italian polymath A polymath ( el, πολυμαθής, ', "having learn ...

published the solution of
Scipione del Ferro Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topic ...
and
Niccolò Fontana Tartaglia Niccolò Fontana Tartaglia (; 1499/1500 – 13 December 1557) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
to and that of
Lodovico Ferrari Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italians, Italian mathematician. Biography Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of Milan to Bologna. Lodovico settled in Bologna, and he be ...
for . Finally
Niels Henrik Abel Niels Henrik Abel (; ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

proved, in 1824, that equations of degree 5 and higher do not have general solutions using radicals.
Galois theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, named after
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
, 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.

# Areas of study

The algebraic equations are the basis of a number of areas of modern mathematics:
Algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, ...
is the study of (univariate) algebraic equations over the rationals (that is, with
rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογι ...
coefficients).
Galois theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
was introduced by
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
to specify criteria for deciding if an algebraic equation may be solved in terms of radicals. In field theory, an
algebraic extension In abstract algebra, a field extension ''L''/''K'' is called algebraic if every element of ''L'' is algebraic over ''K'', i.e. if every element of ''L'' is a root In vascular plants, the roots are the plant organ, organs of a plant that are ...
is an extension such that every element is a root of an algebraic equation over the base field.
Transcendental number theory Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...
is the study of the real numbers which are not solutions to an algebraic equation over the rationals. A
Diophantine equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
is a (usually multivariate) polynomial equation with integer coefficients for which one is interested in the integer solutions.
Algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

is the study of the solutions in an
algebraically closed field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of multivariate polynomial equations. Two equations are equivalent if they have the same set of
solutions Image:SaltInWaterSolutionLiquid.jpg, Making a saline water solution by dissolving Salt, table salt (sodium chloride, NaCl) in water. The salt is the solute and the water the solvent. In chemistry, a solution is a special type of Homogeneous and ...

. In particular the equation $P = Q$ is equivalent to $P-Q = 0$. 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 the
coefficient In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s are
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
s. For example, multiplying through by 42 = 2·3·7 and grouping its terms in the first member, the previously mentioned polynomial equation $y^4+\frac=\frac-xy^2+y^2-\frac$ becomes :$42y^4+21xy-14x^3+42xy^2-42y^2+6=0.$ Because
sine In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

,
exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer An integer (from the Latin wikt: ...
, and 1/''T'' are not polynomial functions, :$e^T x^2+\fracxy+\sin\left(T\right)z -2 =0$ is ''not'' a polynomial equation in the four variables ''x'', ''y'', ''z'', and ''T'' over the rational numbers. However, it is a polynomial equation in the three variables ''x'', ''y'', and ''z'' over the field of the
elementary function In mathematics, an elementary function is a function (mathematics), function of a single variable (mathematics), variable (typically Function of a real variable, real or Complex analysis#Complex functions, complex) that is defined as taking addit ...
s in the variable ''T''.

# Theory

## Polynomials

Given an equation in unknown :$\left(\mathrm E\right) \qquad a_n x^n + a_ x^ + \dots + a_1 x + a_0 = 0$, with coefficients in a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, one can equivalently say that the solutions of (E) in are the roots in of the polynomial :
field extension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of , one may consider (E) to be an equation with coefficients in and the solutions of (E) in are also solutions in (the converse does not hold in general). It is always possible to find a field extension of known as the rupture field of the polynomial , in which (E) has at least one solution.

## Existence of solutions to real and complex equations

The
fundamental theorem of algebra The fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
states that the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
of the
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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. On the other hand, an equation such as $x^2 + 1 = 0$ does not have a solution in $\R$ (the solutions are the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation . Although there is no real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be ...
s and ). While the real solutions of real equations are intuitive (they are the -coordinates of the points where the curve intersects the -axis), the existence of complex solutions to real equations can be surprising and less easy to visualize. However, a
monic polynomial In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
of odd degree must necessarily have a real root. The associated
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
in is continuous, and it approaches $-\infty$ as approaches $-\infty$ and $+\infty$ as approaches $+\infty$. By the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if ''f'' is a continuous function whose domain contains the interval 'a'', ''b'' then it takes on any given value between ''f''(''a'') and ''f''(''b'') at some point ...

, it must therefore assume the value zero at some real , which is then a solution of the polynomial equation.

## Connection to Galois theory

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 Abel ''Hábel''; ar, هابيل, Hābīl is a Biblical figure in the Book of Genesis The Book of Genesis,, "''Bərēšīṯ''", "In hebeginning" the first book of the Hebrew Bible The Hebrew Bible or Tanakh (; Hebrew: , or ), is the ...

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
provides a criterion which allows one to determine whether the solution to a given polynomial equation can be expressed using radicals.

# Explicit solution of numerical equations

## Approach

The explicit solution of a real or complex equation of degree 1 is trivial. Solving an equation of higher degree reduces to factoring the associated polynomial, that is, rewriting (E) in the form :$a_n\left(x-z_1\right)\dots\left(x-z_n\right)=0$, where the solutions are then the $z_1, \dots, z_n$. The problem is then to express the $z_i$ in terms of the $a_i$. This approach applies more generally if the coefficients and solutions belong to an
integral domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
.

## General techniques

### Factoring

If an equation of degree has a rational root , the associated polynomial can be factored to give the form (by dividing by or by writing as a
linear combination In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of terms of the form , and factoring out . Solving thus reduces to solving the degree equation . See for example the .

### Elimination of the sub-dominant term

To solve an equation of degree , :$\left(\mathrm E\right) \qquad a_n x^n + a_ x^ + \dots + a_1 x + a_0 = 0$, a common preliminary step is to eliminate the degree- term: by setting $x = y-\frac$, equation (E) becomes :$a_ny^n + b_y^ + \dots +b_1 x +b_0 = 0$.
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal c ...

developed this technique for the case but it is also applicable to the case , for example.

To solve a quadratic equation of the form $ax^2 + bx + c = 0$ one calculates the ''
discriminant In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'' Δ defined by $\Delta = b^2 - 4ac$. If the polynomial has real coefficients, it has: * two distinct real roots if $\Delta > 0$ ; * one real double root if $\Delta = 0$ ; * no real root if $\Delta < 0$, but two complex conjugate roots.

## Cubic equations

The best-known method for solving cubic equations, by writing roots in terms of radicals, is Cubic equation#Cardano's formula, Cardano's formula.

## Quartic equations

For detailed discussions of some solution methods see: * Tschirnhaus transformation (general method, not guaranteed to succeed); * Bezout method (general method, not guaranteed to succeed); * Ferrari method (solutions for degree 4); * Euler method (solutions for degree 4); * Lagrange method (solutions for degree 4); * Descartes method (solutions for degree 2 or 4); A quartic equation $ax^4+bx^3+cx^2+dx+e=0$ with $a\ne0$ may be reduced to a quadratic equation by a change of variable provided it is either Quartic function#Biquadratic equation, biquadratic () or Quartic function#Quasi-palindromic equation, quasi-palindromic (). Some cubic and quartic equations can be solved using trigonometry or hyperbolic functions.

## Higher-degree equations

Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ...
and
Niels Henrik Abel Niels Henrik Abel (; ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

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 the cyclotomic polynomials of degrees 5 and 17. Charles Hermite, on the other hand, showed that polynomials of degree 5 are solvable using elliptical functions. Otherwise, one may find numerical analysis, numerical approximations to the roots using root-finding algorithms, such as Newton's method.