Inmathematics, azero (also sometimes called aroot) of areal-,complex-, or generallyvector-valued function${\displaystyle f}$, is a member${\displaystyle x}$ of thedomain of${\displaystyle f}$ such that${\displaystyle f(x)}$vanishes at${\displaystyle x}$; that is, the function${\displaystyle f}$ attains the value of 0 at${\displaystyle x}$, or equivalently,${\displaystyle x}$ is asolution to the equation${\displaystyle f(x)=0}$. A "zero" of a function is thus an input value that produces an output of 0.^[1]

A graph of the function${\displaystyle \cos (x)}$ for${\displaystyle x}$ in${\displaystyle \left[-2\pi ,2\pi \right]}$, withzeros at${\displaystyle -\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2}}$, and${\displaystyle \frac{3\pi }{2},}$ marked inred.

Aroot of apolynomial is a zero of the correspondingpolynomial function.^[2] Thefundamental theorem of algebra shows that any non-zeropolynomial has a number of roots at most equal to itsdegree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in analgebraically closed extension) counted with theirmultiplicities.^[3] For example, the polynomial${\displaystyle f}$ of degree two, defined by${\displaystyle f(x)=x^2-5x+6=(x-2)(x-3)}$ has the two roots (or zeros) that are2 and3.$${\displaystyle f(2)=2^2-5\times 2+6=0\text{ and }f(3)=3^2-5\times 3+6=0.}$$

If the function maps real numbers to real numbers, then its zeros are the${\displaystyle x}$-coordinates of the points where itsgraph meets thex-axis. An alternative name for such a point${\displaystyle (x,0)}$ in this context is an${\displaystyle x}$-intercept.

Everyequation in theunknown${\displaystyle x}$  may be rewritten as

${\displaystyle f(x)=0}$ 

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${\displaystyle f}$ . 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.

Every real polynomial of odddegree has an odd number of real roots (countingmultiplicities); 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 theintermediate value theorem: since polynomial functions arecontinuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).

The fundamental theorem of algebra states that every polynomial of degree${\displaystyle n}$  has${\displaystyle n}$  complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come inconjugate pairs.^[1]Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.

There are many methods for computing accurateapproximations of roots of functions, the best beingNewton's method, seeRoot-finding algorithm.

Forpolynomials, there are specialized algorithms that are more efficient and may provide all roots or all real roots; seePolynomial root-finding andReal-root isolation.

Some polynomial, including all those ofdegree no greater than 4, can have all their roots expressedalgebraically in terms of their coefficients; seeSolution in radicals.

In various areas of mathematics, thezero set of afunction is the set of all its zeros. More precisely, if${\displaystyle f:X\to \mathbb{R}}$  is areal-valued function (or, more generally, a function taking values in someadditive group), its zero set is${\displaystyle f^{-1}(0)}$ , theinverse image of${\displaystyle \{0\}}$  in${\displaystyle X}$ .

Under the same hypothesis on thecodomain of the function, alevel set of a function${\displaystyle f}$  is the zero set of the function${\displaystyle f-c}$  for some${\displaystyle c}$  in the codomain of${\displaystyle f.}$ 

The zero set of alinear map is also known as itskernel.

Thecozero set of the function${\displaystyle f:X\to \mathbb{R}}$  is thecomplement of the zero set of${\displaystyle f}$  (i.e., the subset of${\displaystyle X}$  on which${\displaystyle f}$  is nonzero).

Inalgebraic geometry, the first definition of analgebraic variety is through zero sets. Specifically, anaffine algebraic set is theintersection of the zero sets of several polynomials, in apolynomial ring${\displaystyle k\left[x_1,\dots ,x_n\right]}$  over afield. In this context, a zero set is sometimes called azero locus.

Inanalysis andgeometry, anyclosed subset of${\displaystyle {\mathbb{R}}^n}$  is the zero set of asmooth function defined on all of${\displaystyle {\mathbb{R}}^n}$ . This extends to anysmooth manifold as a corollary ofparacompactness.

Indifferential geometry, zero sets are frequently used to definemanifolds. An important special case is the case that${\displaystyle f}$  is asmooth function from${\displaystyle {\mathbb{R}}^p}$  to${\displaystyle {\mathbb{R}}^n}$ . If zero is aregular value of${\displaystyle f}$ , then the zero set of${\displaystyle f}$  is a smooth manifold of dimension${\displaystyle m=p-n}$  by theregular value theorem.

For example, the unit${\displaystyle m}$ -sphere in${\displaystyle {\mathbb{R}}^{m+1}}$  is the zero set of the real-valued function${\displaystyle f(x)=\Vert x{\Vert }^2-1}$ .

• Root-finding algorithm
• Bolzano's theorem, a continuous function that takes opposite signs at the end points of an interval has at least a zero in the interval.
• Gauss–Lucas theorem, the complex zeros of the derivative of a polynomial lie inside the convex hull of the roots of the polynomial.
• Marden's theorem, a refinement of Gauss–Lucas theorem for polynomials of degree three
• Sendov's conjecture, a conjectured refinement of Gauss-Lucas theorem
• zero at infinity
• Zero crossing, property of the graph of a function near a zero
• Zeros and poles of holomorphic functions

• Weisstein, Eric W."Root".MathWorld.