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Zero of a function

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From Wikipedia, the free encyclopedia

(Redirected fromZero set)

Point where function's value is zero

"Root of a function" redirects here. For a half iterate of a function, seeFunctional square root.

A graph of the function

\cos(x)

for

x {\displaystyle x}

\left[-2\pi ,2\pi \right]

, withzeros at

-{\tfrac {3\pi }{2}},\;-{\tfrac {\pi }{2}},\;{\tfrac {\pi }{2}}

, and

{\tfrac {3\pi }{2}},

marked inred.

Inmathematics, azero (also sometimes called aroot) of areal-,complex-, or generallyvector-valued function $f {\displaystyle f}$ , is a member $x {\displaystyle x}$ of thedomain of $f {\displaystyle f}$ such that $f(x)$ vanishes at $x {\displaystyle x}$ ; that is, the function $f {\displaystyle f}$ attains the value of 0 at $x {\displaystyle x}$ , or equivalently, $x {\displaystyle x}$ is asolution to the equation $f(x)=0$ .^[1] A "zero" of a function is thus an input value that produces an output of 0.^[2]

Aroot of apolynomial is a zero of the correspondingpolynomial function.^[1] 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 $f {\displaystyle f}$ of degree two, defined by $f(x)=x^{2}-5x+6=(x-2)(x-3)$ has the two roots (or zeros) that are2 and3. $f(2)=2^{2}-5\times 2+6=0{\text{ and }}f(3)=3^{2}-5\times 3+6=0.$

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

Solution of an equation

[edit]

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

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 $f {\displaystyle f}$ . 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.

Polynomial roots

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Main article:Properties of polynomial roots

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).

Fundamental theorem of algebra

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Main article:Fundamental theorem of algebra

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 inconjugate pairs.^[2]Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.