Zero of a function

A zero of a function is an input value where the function’s output is zero. It is also called a root and solves the equation formed by setting the function equal to zero.

Factor theorem: A polynomial has a factor x − a exactly when its value at a is zero. It turns a known zero into a factor, often reducing the equation’s degree.
Function (mathematics): A rule or correspondence assigning each input in a domain exactly one output. A zero is defined by comparing a function’s output with zero.
Fundamental theorem of algebra: Every nonconstant polynomial with complex coefficients has a complex root. It guarantees that complex zeros exist for every nonconstant polynomial.
Pole (complex analysis): An isolated singularity where a complex function grows without bound in a specific algebraic manner. A zero makes a function vanish, while a pole is a point of unbounded behavior.
Intermediate value theorem: A continuous function takes every value between its values at two points. Opposite signs at two endpoints guarantee at least one zero between them.
Domain of a function: The set of inputs for which a function is defined. Only domain elements can qualify as zeros, even if an equation has other formal solutions.
Eigenvalue: A scalar that changes a linear transformation into scalar multiplication along some nonzero vector. Eigenvalues are zeros of the characteristic polynomial of a matrix.
Critical point (mathematics): A point where a function’s derivative is zero or undefined, subject to the function’s domain. A zero concerns the function’s value; a critical point concerns its derivative.
Newton's method: An iterative numerical method that approximates a root using tangent lines. It can rapidly refine an approximate zero when suitable derivatives and starting values are available.
Range (mathematics): The set of outputs a function actually attains over its domain. A function has a zero precisely when zero belongs to its range.

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