Zero of a function

A graph of the function cos(x) on the domain , with x-intercepts indicated in red. The function has zeroes where x is , , and .

A graph of the function cos(x) on the domain $\scriptstyle{[-2\pi,2\pi]}$ , with x-intercepts 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 vector-valued 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 non-zero 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^2-5x+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 x-coordinates of the points where its graph meets the x-axis. An alternative name for such a point (x,0) in this context is an x-intercept.

Solution of an equation

Every equation in the unknown $x$ may easily be rewritten as

f (x) = 0

by regrouping all 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, "zero of a function" is a phrase denoting 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 article: Properties of 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

Main article: 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.^[1] Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

Computing roots

Main articles: Root-finding algorithm and Equation solving

Computing roots of functions, for example polynomial functions, frequently requires the use of specialised or approximation techniques (for example, 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. (See algebraic solution.)

Zero set

Main article: Zero set

In topology and other areas of mathematics, the zero set of a real-valued 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 zero-sets. For instance, for each set S of polynomials in k[x₁, ..., x_n], one defines the zero-locus Z(S) to be the set of points in Aⁿ 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 Aⁿ 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.

References

1 2 Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 535. ISBN 0-13-165711-9.