Editing Polynomial (section)

== Equations ==
{{Main|Algebraic equation}}
A ''polynomial equation'', also called an ''[[algebraic equation]]'', is an [[equation]] of the form<ref>{{Cite book |last=Proskuryakov |first=I.V. |chapter=Algebraic equation |editor=Hazewinkel, Michiel |editor-link=Michiel Hazewinkel |title=Encyclopaedia of Mathematics |volume=1 |publisher=Springer |year=1994 |isbn=978-1-55608-010-4 |chapter-url=https://books.google.com/books?id=PE1a-EIG22kC&pg=PA88}}</ref>
<math display="block">a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0 = 0.</math>
For example,
<math display="block"> 3x^2 + 4x - 5 = 0 </math>
is a polynomial equation.

When considering equations, the indeterminates (variables) of polynomials are also called [[variable (mathematics)|unknown]]s, and the ''solutions'' are the possible values of the unknowns for which the equality is true (in general more than one solution may exist). A polynomial equation stands in contrast to a ''polynomial [[identity (mathematics)|identity]]'' like {{math|(''x'' + ''y'')(''x'' − ''y'') {{=}} ''x''<sup>2</sup> − ''y''<sup>2</sup>}}, where both expressions represent the same polynomial in different forms, and as a consequence any evaluation of both members gives a valid equality.

In elementary [[algebra]], methods such as the [[quadratic formula]] are taught for solving all first degree and second degree polynomial equations in one variable. There are also formulas for the [[cubic equation|cubic]] and [[quartic equation]]s. For higher degrees, the [[Abel–Ruffini theorem]] asserts that there can not exist a general formula in radicals. However, [[root-finding algorithm]]s may be used to find [[numerical approximation]]s of the roots of a polynomial expression of any degree.

The number of solutions of a polynomial equation with real coefficients may not exceed the degree, and equals the degree when the [[complex number|complex]] solutions are counted with their [[multiplicity (mathematics)|multiplicity]]. This fact is called the [[fundamental theorem of algebra]].

=== Solving equations <span class="anchor" id="Solving polynomial equations"></span> ===
<!-- "Simple root (polynomial)" redirects here -->
{{main|Algebraic equation}}
{{See also|Root-finding of polynomials|Properties of polynomial roots}}

A ''root'' of a nonzero univariate polynomial {{math|''P''}} is a value {{mvar|a}} of {{mvar|x}} such that {{math|''P''(''a'') {{=}} 0}}. In other words, a root of {{mvar|P}} is a solution of the [[polynomial equation]] {{math|''P''(''x'') {{=}} 0}} or a [[zero of a function|zero]] of the polynomial function defined by {{math|''P''}}. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered.

A number {{math|''a''}} is a root of a polynomial {{math|''P''}} if and only if the [[#linear polynomial|linear polynomial]] {{math|''x'' − ''a''}} divides {{math|''P''}}, that is if there is another polynomial {{math|''Q''}} such that {{math|1=''P'' = (''x'' − ''a'') Q}}. It may happen that a power (greater than {{math|1}}) of {{math|''x'' − ''a''}} divides {{math|''P''}}; in this case, {{math|''a''}} is a ''multiple root'' of {{math|''P''}}, and otherwise {{math|''a''}} is a '''simple root''' of {{math|''P''}}. If {{math|''P''}} is a nonzero polynomial, there is a highest power {{math|''m''}} such that {{math|(''x'' − ''a'')<sup>''m''</sup>}} divides {{math|''P''}}, which is called the ''multiplicity'' of {{math|''a''}} as a root of {{math|''P''}}. The number of roots of a nonzero polynomial {{math|''P''}}, counted with their respective multiplicities, cannot exceed the degree of {{math|''P''}},<ref>{{cite book |last=Leung |first=Kam-tim |title=Polynomials and Equations |publisher=Hong Kong University Press |year=1992 |isbn=9789622092716 |page=134 |url=https://books.google.com/books?id=v5uXkwIUbC8C&pg=PA134|display-authors=etal}}</ref> and equals this degree if all [[complex number|complex]] roots are considered (this is a consequence of the [[fundamental theorem of algebra]]).
The coefficients of a polynomial and its roots are related by [[Vieta's formulas]].

Some polynomials, such as {{math|''x''<sup>2</sup> + 1}}, do not have any roots among the [[real number]]s. If, however, the set of accepted solutions is expanded to the [[complex number]]s, every non-constant polynomial has at least one root; this is the [[fundamental theorem of algebra]]. By successively dividing out factors {{math|''x'' − ''a''}}, one sees that any polynomial with complex coefficients can be written as a constant (its leading coefficient) times a product of such polynomial factors of degree&nbsp;1; as a consequence, the number of (complex) roots counted with their multiplicities is exactly equal to the degree of the polynomial.

There may be several meanings of [[Equation solving|"solving an equation"]]. One may want to express the solutions as explicit numbers; for example, the unique solution of {{math|1=2''x'' − 1 = 0}} is {{math|1/2}}. This is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as [[algebraic expression]]s; for example, the [[golden ratio]] <math>(1+\sqrt 5)/2</math> is the unique positive solution of <math>x^2-x-1=0.</math> In the ancient times, they succeeded only for degrees one and two. For [[quadratic equation]]s, the [[quadratic formula]] provides such expressions of the solutions. Since the 16th century, similar formulas (using cube roots in addition to square roots), although much more complicated, are known for equations of degree three and four (see [[cubic equation]] and [[quartic equation]]). But formulas for degree 5 and higher eluded researchers for several centuries. In 1824, [[Niels Henrik Abel]] proved the striking result that there are equations of degree 5 whose solutions cannot be expressed by a (finite) formula, involving only arithmetic operations and radicals (see [[Abel–Ruffini theorem]]). In 1830, [[Évariste Galois]] proved that most equations of degree higher than four cannot be solved by radicals, and showed that for each equation, one may decide whether it is solvable by radicals, and, if it is, solve it. This result marked the start of [[Galois theory]] and [[group theory]], two important branches of modern [[algebra]]. Galois himself noted that the computations implied by his method were impracticable. Nevertheless, formulas for solvable equations of degrees 5 and 6 have been published (see [[quintic function]] and [[sextic equation]]).

When there is no algebraic expression for the roots, and when such an algebraic expression exists but is too complicated to be useful, the unique way of solving it is to compute [[numerical approximation]]s of the solutions.<ref>{{cite book |last=McNamee |first=J.M. |title=Numerical Methods for Roots of Polynomials, Part 1 |publisher=Elsevier |year=2007 |isbn=978-0-08-048947-6 |url=https://books.google.com/books?id=4PMqxwG-eqQC}}</ref> There are many methods for that; some are restricted to polynomials and others may apply to any [[continuous function]]. The most efficient [[algorithm]]s allow solving easily (on a [[computer]]) polynomial equations of degree higher than 1,000 (see ''[[Root-finding algorithm]]'').

For polynomials with more than one indeterminate, the combinations of values for the variables for which the polynomial function takes the value zero are generally called ''zeros'' instead of "roots". The study of the sets of zeros of polynomials is the object of [[algebraic geometry]]. For a set of polynomial equations with several unknowns, there are [[algorithm]]s to decide whether they have a finite number of [[complex number|complex]] solutions, and, if this number is finite, for computing the solutions. See [[System of polynomial equations]].

The special case where all the polynomials are of degree one is called a [[system of linear equations]], for which another range of different [[system of linear equations#Solving a linear system|solution methods]] exist, including the classical [[Gaussian elimination]].

A polynomial equation for which one is interested only in the solutions which are [[integer]]s is called a [[Diophantine equation]]. Solving Diophantine equations is generally a very hard task. It has been proved that there cannot be any general [[algorithm]] for solving them, or even for deciding whether the set of solutions is empty (see [[Hilbert's tenth problem]]). Some of the most famous problems that have been solved during the last fifty years are related to Diophantine equations, such as [[Fermat's Last Theorem]].