A graph of the function cos(x) on the domain \scriptstyle, with xintercepts indicated in red. The function has zeroes where x is \scriptstyle\frac{3\pi}{2}, \scriptstyle\frac{\pi}{2}, \scriptstyle\frac{\pi}{2} and \scriptstyle\frac{3\pi}{2}.
In mathematics, a zero, also sometimes called a root, of a real, complex or generally vectorvalued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation

f(x) = 0.
In other words, a "zero" of a function is an input value that produces an output of zero (0).^{[1]}
A root of a polynomial is a zero of the associated polynomial function. The fundamental theorem of algebra shows that any nonzero polynomial 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(x)=x^25x+6
has the two roots 2 and 3, since

f(2) = 2^2  5 \cdot 2 + 6 = 0 \quad \textstyle{\rm {and} }\quad f(3) = 3^2  5 \cdot 3 + 6 = 0.
If the function maps real numbers to real numbers, its zeroes are the xcoordinates of the points where its graph meets the xaxis. An alternative name for such a point (x,0) in this context is an xintercept.
Contents

Solution of an equation 1

Polynomial roots 2

Fundamental theorem of algebra 2.1

Computing roots 3

Zero set 4

See also 5

References 6

Further reading 7
Solution of an equation
Every equation in the unknown x may easily be rewritten as

f(x) = 0
by regrouping all terms in the lefthand side. It follows that the solutions of such an equation are exactly the zeros of the function f. In other words "zero of a function" is a short sentence to denote a "solution of the equation obtained by equating the function to 0", and the study of zero 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 one is the smallest odd whole number), 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.
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 nonreal roots of polynomials with real coefficients come in conjugate pairs.^{[1]} Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.
Computing roots
Computing roots of certain functions, especially polynomial functions, frequently requires the use of specialised or approximation techniques (for example, Newton's method).
Zero set
In topology and other areas of mathematics, the zero set of a realvalued function f : X → R (or more generally, a function taking values in some additive group) is the subset f^{1}(0) of X (the inverse image of {0}).
Zero sets are important in many areas of mathematics. One area of particular importance is algebraic geometry, where the first definition of an algebraic variety is through zerosets. For instance, for each set S of polynomials in k[x_{1}, ..., x_{n}], one defines the zerolocus Z(S) to be the set of points in A^{n} on which the functions in S simultaneously vanish, that is to say

Z(S) = \{x \in \mathbb A^n \mid f(x) = 0 \text{ for all } f\in S\}. Then a subset V of A^{n} is called an affine algebraic set if V = Z(S) for some S. These affine algebraic sets are the fundamental building blocks of algebraic geometry.
See also
References

^ ^{a} ^{b} Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ:
Further reading
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