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This article is about the zeros of a function, which should not be confused with the value at zero.
ƒ(x)=cosx on the interval [-2π,2π], with roots indicated in red

In mathematics, a root (or a zero) of a real-, complex- or generally vector-valued function ƒ is a member x of the domain of ƒ such that ƒ(x) vanishes at x, that is,

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

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

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

ƒ has a root at 3 because f(3) = 32 − 6(3) + 9 = 0.

If the function is mapping from real numbers to real numbers, its zeros are the points where its graph meets the x-axis. An alternative name for the root in this context is the x-intercept.

Finding roots of certain functions, especially polynomials, frequently necessitates the use of specialised techniques (for example, Newton's method). The concept of complex numbers was developed to handle the roots of quadratic or cubic equations with negative discriminants (that is, those leading to expressions involving the square root of negative numbers).

All real polynomials of odd degree have a 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.

One of the most important unsolved problems in mathematics concerns the location of the roots of the Riemann zeta function.

See also


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