Zero of a function

A graph of the function cos(x) on the domain $\scriptstyle{[-2\pi,2\pi]}$, with x-intercepts indicated in red. The function has zeroes where x is $\scriptstyle\frac{-3\pi}{2}$, $\scriptstyle\frac{-\pi}{2}$, $\scriptstyle\frac{\pi}{2}$ and $\scriptstyle\frac{3\pi}{2}$.

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 is an input value that produces an output of zero ("0").

The fundamental theorem of algebra shows that any polynomial function must have a number of roots equal to the function's degree (though some of these may be repeated or complex values). For example, the function f defined by the formula

$f(x)=x^2-5x+6$

has zeroes at x = 2 and x = 3, since

$f(2) = 2^2 - 5 \cdot 2 + 6 = 0 \textstyle{\ and \ } f(3) = 3^2 - 5 \cdot 3 + 6 = 0.$

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.

The concept of complex numbers was developed to describe 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 roots 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.

Polynomial roots

Main article: Properties of polynomial roots

Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because one is the smallest odd whole number), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero in the process of changing from negative to positive or vice-versa.

Fundamental theorem of algebra

Main article: Fundamental theorem of algebra

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.

Computing roots

Main article: Root-finding algorithm

Main article: Equation solving

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

See also

• Zero (complex analysis)
• Pole (complex analysis)
• Fundamental theorem of algebra
• Newton's method
• Sendov's conjecture
• Marden's theorem

Further Reading

• Weisstein, Eric W., "Root", MathWorld.