Editing Zero of a function

{{Short description|Point where function's value is zero}}
{{redirect|Root of a function|a half iterate of a function|Functional square root}}

{{Css Image Crop |Image = X-intercepts.svg |bSize = 300 |cWidth = 300 |cHeight = 110 |oLeft = 0 |oTop = 100 |Location = right |Description = A graph of the function <math>\cos(x)</math> for <math>x</math> in <math>\left[-2\pi,2\pi\right]</math>, with '''zeros''' at <math>-\tfrac{3\pi}{2},\;-\tfrac{\pi}{2},\;\tfrac{\pi}{2}</math>, and <math>\tfrac{3\pi}{2},</math> marked in <span style="color:red">red</span>.}} 

In [[mathematics]], a '''zero''' (also sometimes called a '''root''') of a [[Real number|real]]-, [[Complex number|complex]]-, or generally [[vector-valued function]] <math>f</math>, is a member <math>x</math> of the [[Domain of a function|domain]] of <math>f</math> such that <math>f(x)</math> ''vanishes'' at <math>x</math>; that is, the function <math>f</math> attains the value of 0 at <math>x</math>, or equivalently, <math>x</math> is a [[Solution (mathematics)|solution]] to the equation <math>f(x) = 0</math>. A "zero" of a function is thus an input value that produces an output of 0.<ref name="Foerster">{{cite book | last = Foerster | first = Paul A. | title = Algebra and Trigonometry: Functions and Applications, Teacher's Edition | edition = Classics | year = 2006 | page = [https://archive.org/details/algebratrigonome00paul_0/page/535 535] | publisher = [[Prentice Hall]] | location = Upper Saddle River, NJ | url = https://archive.org/details/algebratrigonome00paul_0/page/535 | isbn = 0-13-165711-9 }}</ref>

A '''root''' of a [[polynomial]] is a zero of the corresponding [[polynomial function]].<ref name=":0">{{Cite web|url=http://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx | title=Algebra - Zeroes/Roots of Polynomials |website=tutorial.math.lamar.edu| access-date=2019-12-15}}</ref> The [[fundamental theorem of algebra]] shows that any non-zero [[polynomial]] has a number of roots at most equal to its [[Degree of a polynomial|degree]], and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an [[algebraically closed extension]]) counted with their [[multiplicity (mathematics)|multiplicities]].<ref>{{Cite web|url=https://www.mathplanet.com/education/algebra-2/polynomial-functions/roots-and-zeros|title=Roots and zeros (Algebra 2, Polynomial functions)| website=Mathplanet |language=en|access-date=2019-12-15}}</ref> For example, the polynomial <math>f</math> of degree two, defined by <math>f(x)=x^2-5x+6=(x-2)(x-3)</math> has the two roots (or zeros) that are '''2''' and '''3'''.
<math display="block">f(2)=2^2-5\times 2+6= 0\text{ and }f(3)=3^2-5\times 3+6=0.</math>

If the function maps real numbers to real numbers, then its zeros are the <math>x</math>-coordinates of the points where its [[Graph of a function|graph]] meets the [[x-axis|''x''-axis]]. An alternative name for such a point <math>(x,0)</math> in this context is an '''<math>x</math>-intercept'''.

==Solution of an equation==
Every [[equation]] in the [[unknown (mathematics)|unknown]] <math>x</math> may be rewritten as

:<math>f(x)=0</math>

by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function <math>f</math>. In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.

== Polynomial roots ==
{{main|Properties of polynomial roots}}
Every real polynomial of odd [[Degree of a polynomial|degree]] has an odd number of real roots (counting [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicities]]); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the [[intermediate value theorem]]: since polynomial functions are [[Continuous function|continuous]], the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).

===Fundamental theorem of algebra===
{{main|Fundamental theorem of algebra}}
The fundamental theorem of algebra states that every polynomial of degree <math>n</math> has <math>n</math> complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in [[complex conjugate|conjugate]] pairs.<ref name="Foerster" /> [[Vieta's formulas]] relate the coefficients of a polynomial to sums and products of its roots.

== Computing roots ==
{{see also|Equation solving}}
There are many methods for computing accurate [[approximation]]s of roots of functions, the best being [[Newton's method]], see [[Root-finding algorithm]]. 

For [[polynomial]]s, there are specialized algorithms that are more efficient and may provide all roots or all real roots; see [[Polynomial root-finding]] and [[Real-root isolation]].

Some polynomial, including all those of [[degree of a polynomial|degree]] no greater than 4, can have all their roots expressed [[algebraic function|algebraically]] in terms of their coefficients; see [[Solution in radicals]].

==Zero set==
{{redirect|Zero set|the musical album|Zero Set}}
In various areas of mathematics, the '''zero set''' of a [[function (mathematics)|function]] is the set of all its zeros. More precisely, if <math>f:X\to\mathbb{R}</math> is a [[real-valued function]] (or, more generally, a function taking values in some [[Abelian group|additive group]]), its zero set is <math>f^{-1}(0)</math>, the [[inverse image]] of <math>\{0\}</math> in <math>X</math>. 

Under the same hypothesis on the [[codomain]] of the function, a [[level set]] of a function <math>f</math> is the zero set of the function <math>f-c</math> for some <math>c</math> in the codomain of <math>f.</math>

The zero set of a [[linear map]] is also known as its [[kernel (algebra)|kernel]].

The '''cozero set''' of the function <math>f:X\to\mathbb{R}</math> is the [[complement (set theory)|complement]] of the zero set of <math>f</math> (i.e., the subset of <math>X</math> on which <math>f</math> is nonzero). 

=== Applications ===
In [[algebraic geometry]], the first definition of an [[algebraic variety]] is through zero sets. Specifically, an [[affine algebraic set]] is the [[set intersection|intersection]] of the zero sets of several polynomials, in a [[polynomial ring]] <math>k\left[x_1,\ldots,x_n\right]</math> over a [[field (mathematics)|field]]. In this context, a zero set is sometimes called a ''zero locus''.

In [[Mathematical analysis|analysis]] and [[geometry]], any [[closed set|closed subset]] of <math>\mathbb{R}^n</math> is the zero set of a [[smooth function]] defined on all of <math>\mathbb{R}^n</math>. This extends to any [[smooth manifold]] as a corollary of [[paracompactness]]. <!-- There is obvious overlap between this and the next paragraph, but it takes someone more experienced to merge the two. -->

In [[differential geometry]], zero sets are frequently used to define [[manifold]]s. An important special case is the case that <math>f</math> is a [[smooth function]] from <math>\mathbb{R}^p</math> to <math>\mathbb{R}^n</math>. If zero is a [[regular value]] of <math>f</math>, then the zero set of <math>f</math> is a smooth manifold of dimension <math>m=p-n</math> by the [[Submersion_(mathematics)#Local_normal_form|regular value theorem]].

For example, the unit <math>m</math>-[[sphere]] in <math>\mathbb{R}^{m+1}</math> is the zero set of the real-valued function <math>f(x)=\Vert x \Vert^2-1</math>.

== See also ==
*[[Root-finding algorithm]]
*[[Intermediate value theorem|Bolzano's theorem]], a continuous function that takes opposite signs at the end points of an interval has at least a zero in the interval.
*[[Gauss–Lucas theorem]], the complex zeros of the derivative of a polynomial lie inside the convex hull of the roots of the polynomial.
*[[Marden's theorem]], a refinement of Gauss–Lucas theorem for polynomials of degree three
*[[Sendov's conjecture]], a conjectured refinement of Gauss-Lucas theorem
*[[Vanish at infinity|zero at infinity]]
*[[Zero crossing]], property of the graph of a function near a zero
*[[Zeros and poles]] of holomorphic functions

== References ==
{{reflist}}

==Further reading==
* {{MathWorld |title=Root |urlname=Root}}

[[Category:Elementary mathematics]]
[[Category:Functions and mappings]]
[[Category:0 (number)]]