# Zero of a function

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f(x)=cosx on the interval [-2π,2π], with x-intercepts indicated in red (the roots highlighted are -3π/2, -π/2, π/2, 3π/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,

$x \text{ such that } f(x) = 0\,.$

In other words, a "zero" of a function f is a value for x that produces a result of zero ("0").[1] For example, consider the function f defined by the formula

$f(x)=x^2-6x+9 \,.$

f has a root at 3 since

$f(3) = 3^2 - 6 \cdot 3 + 9 = 0.\$

If the function maps real numbers to real numbers, its zeros 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.

The concept of complex numbers was developed to handle the roots of cubic equations with negative discriminants (that is, those leading to expressions involving the square root of negative numbers). Complex numbers also occur as zeros of quadratic equations with negative discriminants.

The Riemann hypothesis, one of the most important unsolved problems in mathematics, concerns the location of the zeros of the Riemann zeta function.

## Computing roots

Computing roots of certain functions, especially polynomial functions, frequently requires the use of specialised or approximation techniques (for example, Newton's method).

## Polynomial roots

Every real polynomial of odd degree has at least one real number as a root. Many real polynomials of even degree do not have a real root, but 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. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.