Vanish (mathematics)

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

In other words, a "zero" of a function is an input value that produces an output of zero (0).

A root of a polynomial is a zero of the corresponding 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

has the two roots 2 and 3, since

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.

Every equation in the unknown x may be rewritten as

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.

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