Zeros

A zero of a function is a value of the input for which the function outputs zero. If f is defined on a set and f(x) = 0 for some x in that set, then x is a zero of f. The collection of all zeros is called the zero set of f. When the function is a polynomial, its zeros are often called roots.

Zeros can be real or complex. For polynomials, the Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex zeros counted with multiplicity. Real polynomials may have real zeros or nonreal complex zeros that come in complex conjugate pairs.

The multiplicity of a zero describes how many times that value occurs as a root. For example, if a polynomial factors as (x − a)^m g(x) with g(a) ≠ 0, then a is a zero of multiplicity m.

In graphing terms, a zero is a point where the graph intersects the x-axis. In calculus, a zero is distinct from a critical point: a zero is where the function’s value is zero, whereas a critical point relates to where the derivative is zero or undefined.

Zeros of analytic functions have additional structure: unless the function is identically zero on a region, zeros are isolated points, and their multiplicities relate to the vanishing order of the function at those points.

Common methods to determine zeros include exact algebraic solving for polynomials and numerical techniques such as bisection, Newton-Raphson, or root-finding algorithms for more general functions.

Examples: the zeros of sin x are integer multiples of π; the zeros of x^2 − 2 are ±√2. The zero set concept generalizes to higher dimensions as the solution set of f(x) = 0.