Zero of a function

In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f {displaystyle f} is a member x {displaystyle x} of the domain of f {displaystyle f} such that f ( x ) {displaystyle f(x)} vanishes at x {displaystyle x} ; that is, x {displaystyle x} is a solution of the equation f ( x ) = 0 {displaystyle f(x)=0} .In other words, a 'zero' of a function is an input value that produces an output of 0 {displaystyle 0} . In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f {displaystyle f} is a member x {displaystyle x} of the domain of f {displaystyle f} such that f ( x ) {displaystyle f(x)} vanishes at x {displaystyle x} ; that is, x {displaystyle x} is a solution of the equation f ( x ) = 0 {displaystyle f(x)=0} .In other words, a 'zero' of a function is an input value that produces an output of 0 {displaystyle 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 {displaystyle f} of degree two, defined by has the two roots 2 {displaystyle 2} and 3 {displaystyle 3} , since If the function maps real numbers to real numbers, its zeros are the x {displaystyle x} -coordinates of the points where its graph meets the x-axis. An alternative name for such a point ( x , 0 ) {displaystyle (x,0)} in this context is an x {displaystyle 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 {displaystyle 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. 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. The fundamental theorem of algebra states that every polynomial of degree n {displaystyle n} has n {displaystyle 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 of functions, for example polynomial functions, frequently requires the use of specialised or approximation techniques (for example, Newton's method). However, some polynomial functions, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients. (See algebraic solution.)