root

If a function f(x) has the value 0 for x = a, a is a root of the equation f(x) = 0. The fundamental theorem of algebra states that any algebraic equation of the form a₀xⁿ + a₁xⁿ⁻¹ + ··· a_n−1x + a_n = 0, where the a_k's are real numbers, has at least one root. From this it follows readily that such an equation has roots, real or complex, in number equal to the index (here n) of the highest power of x.

Furthermore, if a + ib (where i = Ö−1) is a complex root of the given equation, so is a − ib, the conjugate of a + ib. Equations of degrees up to four may be solved algebraically. This statement means that the roots may be expressed as functions of the coefficients, the functions involving the elementary arithmetical processes of addition, multiplication, raising a number to a power, or extracting the root of a certain order of a given number. It was proved by H. Abel and by E. Galois that it is not possible to solve algebraically the general algebraic equation of degree higher than four. However, it is possible to determine the real roots of an algebraic equation to any desired degree of approximation. See also Calculus; Equations, theory of; Numerical analysis.

(1) The top level of a hierarchy. See tree, root directory and root domain.

(2) A person with unlimited access privileges who can perform any and all operations on the computer. Also called "superuser."

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1. the part of a human tooth covered by cementum. n 2. a nerve root; the part of a nerve adjacent to the center with which it is connected; in spinal and cranial nerves the part of the nerve between the cells of origin or termination and the ganglion.

This article is about the zeros of a function, which should not be confused with the value at zero. You may also want information on the Nth roots of numbers instead.

ƒ(x)=cosx on the interval [-2π,2π] with the x values of the red points being the roots (the x-intercepts).

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

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

This function 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. The x-value of such a point is called x-intercept. Therefore in this situation a root can be called an x-intercept.

The word root can also refer to the nth root of a number, a, as in . The square root of a number, a, is .

A substantial amount of mathematics was developed in order to find roots of various functions, especially polynomials. One wide-ranging concept, complex numbers, was developed to handle the roots of quadratic or cubic equations with negative discriminant (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

• zero (complex analysis)
• pole (complex analysis)
• nth root

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