Editing Zero of a function (section)

{{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'''.