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{{Css Image Crop |Image = X-intercepts.svg |bSize = 300px |cWidth = 300 |cHeight = 110 |oLeft = 0 |oTop = 100 |Location = right |Description = A graph of the function cos(''x'') on the domain <math>\scriptstyle{[-2\pi,2\pi]}</math>, with ''x''-intercepts indicated in red. The function has '''zeroes''' where ''x'' is <math>\scriptstyle\frac{-3\pi}{2}</math>, <math>\scriptstyle\frac{-\pi}{2}</math>, <math>\scriptstyle\frac{\pi}{2}</math> and <math>\scriptstyle\frac{3\pi}{2}</math>.}}


In [[mathematics]], a '''zero''', also sometimes called a '''root''', of a real-, complex- or generally [[vector-valued function]] ''f'' is a member ''x'' of the [[Domain of a function|domain]] of ''f'' such that ''f''(''x'') '''vanishes''' at ''x''; that is,
:<math>x \text{ such that } f(x) = 0\,.</math>
In other words, a "zero" of a function is an input value that produces an output of zero (0).<ref name="Foerster">{{cite book | last = Foerster | first = Paul A. | title = Algebra and Trigonometry: Functions and Applications, Teacher's Edition | edition = Classics | year = 2006 | page = 535 | publisher = [[Prentice Hall]] | location = Upper Saddle River, NJ | url = http://www.amazon.com/Algebra-Trigonometry-Functions-Applications-Prentice/dp/0131657100 | isbn = 0-13-165711-9}}</ref>


A '''root''' of a [[polynomial]] is a zero of the associated [[polynomial function]].
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The [[fundamental theorem of algebra]] shows that any non-zero [[polynomial]] has a number of roots at most equal to its [[Degree of a polynomial|degree]] and that the number of roots and the degree are equal when one considers the [[complex number|complex]] roots (or more generally the roots in an [[algebraically closed extension]]) counted with their [[multiplicity (mathematics)|multiplicities]]. For example, the polynomial ''f'' of degree two, defined by
:<math>f(x)=x^2-5x+6</math>
has the two roots 2 and 3, since
:<math>f(2) = 2^2 - 5 \cdot 2 + 6 = 0 \quad \textstyle{\rm {and} }\quad f(3) = 3^2 - 5 \cdot 3 + 6 = 0.</math>
 
If the function maps [[real number]]s to real numbers, its zeroes are the ''x''-coordinates of the points where its [[Graph of a function|graph]] meets the [[x-axis|''x''-axis]]. An alternative name for such a point (''x'',0) in this context is an '''''x''-intercept'''.
 
== Polynomial roots ==
{{main|Properties of polynomial roots}}
Every real polynomial of odd [[Degree of a polynomial|degree]] has an odd number of real roots (counting [[Multiplicity_(mathematics)#Multiplicity_of_a_root_of_a_polynomial|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 function|continuous]], the function value must cross zero in the process of changing from negative to positive or vice-versa.  
 
===Fundamental theorem of algebra===
{{main|Fundamental theorem of algebra}}
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 [[complex conjugate|conjugate]] pairs.<ref name="Foerster" /> [[Vieta's formulas]] relate the coefficients of a polynomial to sums and products of its roots.
 
== Computing roots ==
{{main|Root-finding algorithm}}
{{main|Equation solving}}
Computing roots of certain functions, especially [[polynomial function]]s, frequently requires the use of specialised or [[approximation]] techniques (for example, [[Newton's method]]).
 
==Zero set==
{{main|Zero set}}
 
In topology and other areas of mathematics, the '''zero set''' of a real-valued [[function (mathematics)|function]] ''f'' : ''X'' → '''R''' (or more generally, a function taking values in some [[Abelian group|additive group]]) is the [[subset]] <math>f^{-1}(0)</math> of ''X'' (the [[inverse image]] of {0}).
 
Zero sets are important in many areas of mathematics. One area of particular importance is algebraic geometry, where the first definition of an [[algebraic variety]] is through zero-sets. For instance, for each set ''S'' of polynomials in ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], one defines the zero-locus ''Z''(''S'') to be the set of points in '''A'''<sup>''n''</sup> on which the functions in ''S'' simultaneously vanish, that is to say
:<math>Z(S) = \{x \in \mathbb A^n \mid f(x) = 0 \text{ for all } f\in S\}.</math> Then a subset ''V'' of '''A'''<sup>''n''</sup> is called an '''affine algebraic set''' if ''V'' = ''Z''(''S'') for some ''S''. These affine algebraic sets are the fundamental building blocks of algebraic geometry.
 
== See also ==
* [[Zero (complex analysis)]]
* [[Pole (complex analysis)]]
* [[Fundamental theorem of algebra]]
* [[Newton's method]]
* [[Sendov's conjecture]]
* [[Marden's theorem]]
 
== References ==
{{reflist}}
 
==Further reading==
* {{MathWorld |title=Root |urlname=Root}}
 
[[Category:Elementary mathematics]]
[[Category:Functions and mappings]]
[[Category:Zero]]
 
[[ru:Нуль функции]]

Latest revision as of 09:57, 15 November 2014


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While some men and women seeking a real estate agent are promoting there are many people who are actually thinking of buying a house for the 1st time ever. Numerous first-time purchasers are beginning their own family, which individuals understand that buying a house is a huge part of this. There are a number of customers that believe that they don"t need extensive assistance as this is not their first house nevertheless they actually might.

Many people that are searching for a local agent are in reality seeking to offer a home and also aiding people sell is simply another facet of the task for a highly skilled agent. Although some men and women try to skip from a realtor the truth is that selling a home without you can end up being very difficult. For alternative viewpoints, consider checking out: las vegas nv real estate agent. Selling a home without one isn"t difficult yet it"s recommended to go with an agent.

A lot of people which are in the marketplace to get realize currently the loan qualification procedure is often likely to be the first step that they must experience. Loan qualification is a challenging procedure for many people and also this is why a superb realtor needs to be there to assist buyers with the procedure. While it"s refreshing to appreciate that one has been certified it is also just the beginning of a much larger procedure.

The sad fact about selling a home in today"s times is that many people are pressured into quick selling the house whether they wished to or not. Brief selling just signifies the training of selling a house at the most up-to-date market price inspite of the cost which was purchased it for. Realtors Vegas contains more concerning when to think over this activity. No one obviously wants to do such a thing however some individuals have no other option.

Simply because the bank is foreclosing doesn"t imply that short selling is not a solution which is still on the table for the resident of the house. The resident should act in a short time however simply because they should list their home and find someone that is willing to get it in a short time. The financial institution will end their notice of property foreclosure in some of the instances when they find out the home is short sold.

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8872 S Eastern Ave #200, Las Vegas, NV 89123
(702) 321-1763

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