Zero of a function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function formula_1, is a member formula_2 of the domain of formula_1 such that formula_4 "vanishes" at formula_2; that is, the function formula_1 attains the value of 0 at formula_2, or equivalently, formula_2 is the solution to the equation formula_9. A "zero" of a function is thus an input value that produces an output of 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 formula_1 of degree two, defined by formula_11 has the two roots formula_12 and formula_13, since formula_14 If the function maps real numbers to real numbers, then its zeros are the formula_2-coordinates of the points where its graph meets the "x"-axis. An alternative name for such a point formula_16 in this context is an  formula_2-intercept. Solution of an equation. Every equation in the unknown formula_2 may be rewritten as by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function formula_1. In other words, a "zero of a function" is precisely 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. 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 the smallest odd whole number is 1), 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 (which always happens for odd functions). Fundamental theorem of algebra. The fundamental theorem of algebra states that every polynomial of degree formula_21 has formula_21 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. Computing roots of functions, for example polynomial functions, frequently requires the use of specialised or approximation techniques (e.g., 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 (for more, see algebraic solution). Zero set. In various areas of mathematics, the zero set of a function  is the set of all its zeros. More precisely, if formula_23 is a real-valued function (or, more generally, a function taking values in some additive group), its zero set is formula_24, the inverse image of formula_25 in formula_26. The term "zero set" is generally used when there are infinitely many zeros, and they have some non-trivial topological properties. For example, a level set of a function formula_1 is the zero set of formula_28. The cozero set of formula_1 is the complement of the zero set of formula_1 (i.e., the subset of formula_26 on which formula_1 is nonzero). Applications . In algebraic geometry, the first definition of an algebraic variety is through zero sets. Specifically, an affine algebraic set is the intersection of the zero sets of several polynomials, in a polynomial ring formula_33 over a field. In this context, a zero set is sometimes called a "zero locus". In analysis and geometry, any closed subset of formula_34 is the zero set of a smooth function defined on all of formula_34. This extends to any smooth manifold as a corollary of paracompactness. In differential geometry, zero sets are frequently used to define manifolds. An important special case is the case  that formula_1 is a smooth function from formula_37 to formula_34. If zero is a regular value of formula_1, then the zero set of formula_1 is a smooth manifold of dimension formula_41 by the regular value theorem. For example, the unit formula_42-sphere in formula_43 is the zero set of the real-valued function formula_44.