Zeroes

In mathematics, a zero (plural zeros or zeroes) of a function f is an input value x such that f(x) = 0. Zeros are also called roots or solutions of the equation f(x) = 0, and the collection of all zeros is the zero set.

For polynomials, zeros are the roots of the polynomial. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex zeros counted with multiplicity. Real zeros lie on the real axis, while nonreal zeros occur as complex conjugate pairs when coefficients are real.

Multiplicity: If (x − r)^m divides the polynomial, r is a zero of multiplicity m. Multiplicity affects the graph near the zero: a simple zero (multiplicity 1) typically crosses the axis, whereas even multiplicities cause the graph to touch but not cross.

Beyond polynomials, zeros can refer to inputs where other functions vanish. Zeros are found by algebraic factorization, solving equations, or numerical root-finding methods such as the bisection method, Newton-Raphson, or the secant method, under appropriate conditions.

In trigonometric functions, zeros occur at regular intervals: sin x has zeros at x = kπ; cos x at x = π/2 + kπ. Zeros are important in analysis, signal processing, and numerical computation, as they identify where a function or system output is null.