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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a zero (also sometimes called a root) of a real-,
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
-, or generally vector-valued function $f$, is a member $x$ of the domain of $f$ such that $f\left(x\right)$ ''vanishes'' at $x$; that is, the function $f$ attains the value of 0 at $x$, or equivalently, $x$ is the
solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solut ...
to the equation $f\left(x\right) = 0$. A "zero" of a function is thus an input value that produces an output of 0. A root of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...
is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...
has a number of roots at most equal to its 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 multiplicities. For example, the polynomial $f$ of degree two, defined by $f\left(x\right)=x^2-5x+6$ has the two roots (or zeros) that are 2 and 3. $f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0.$ If the function maps real numbers to real numbers, then its zeros are the $x$-coordinates of the points where its graph meets the ''x''-axis. An alternative name for such a point $\left(x,0\right)$ in this context is an $x$-intercept.

# Solution of an equation

Every
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
in the
unknown Unknown or The Unknown may refer to: Film * ''The Unknown'' (1915 comedy film), a silent boxing film * ''The Unknown'' (1915 drama film) * ''The Unknown'' (1927 film), a silent horror film starring Lon Chaney * ''The Unknown'' (1936 film), a ...
$x$ may be rewritten as :$f\left(x\right)=0$ 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 $f$. 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

Every real polynomial of odd degree has an odd number of real roots (counting 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, 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

The fundamental theorem of algebra states that every polynomial of degree $n$ has $n$ complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs.
Vieta's formulas In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta"). Basic formulas ...
relate the coefficients of a polynomial to sums and products of its roots.

# Computing roots

Computing roots of functions, for example polynomial functions, frequently requires the use of specialised or
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
techniques (e.g., Newton's method). However, some polynomial functions, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients (for more, see algebraic solution).

# Zero set

In various areas of mathematics, the zero set of a function is the set of all its zeros. More precisely, if $f:X\to\mathbb$ is a
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...
(or, more generally, a function taking values in some additive group), its zero set is $f^\left(0\right)$, the
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...
of $\$ in $X$. The term ''zero set'' is generally used when there are infinitely many zeros, and they have some non-trivial
topological properties In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...
. For example, a level set of a function $f$ is the zero set of $f-c$. The cozero set of $f$ is the complement of the zero set of $f$ (i.e., the subset of $X$ on which $f$ is nonzero). The zero set of a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
is also called kernel.

## Applications

In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the first definition of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
is through zero sets. Specifically, an affine algebraic set is the intersection of the zero sets of several polynomials, in a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, any closed subset of $\mathbb^n$ is the zero set of a
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
defined on all of $\mathbb^n$. This extends to any smooth manifold as a corollary of paracompactness. In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multil ...
, zero sets are frequently used to define
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s. An important special case is the case that $f$ is a
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
from $\mathbb^p$ to $\mathbb^n$. If zero is a regular value of $f$, then the zero set of $f$ is a smooth manifold of dimension $m=p-n$ by the regular value theorem. For example, the unit $m$-
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
in $\mathbb^$ is the zero set of the real-valued function $f\left(x\right)=\Vert x \Vert^2-1$.