Zero of a function

<h2 id="solution-of-an-equation">Solution of an equation</h2>
<p>Every <a href="/facts/Equation/pxFzLAVR">equation</a> in the <a href="/facts/Unknown_(mathematics)/pxFzLAVR">unknown</a> 
  
    
      
        x
      
    
    {\displaystyle x}
  
 may be rewritten as
</p>

f
        (
        x
        )
        =
        0
      
    
    {\displaystyle f(x)=0}

<p>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.
</p>
<h2 id="polynomial-roots">Polynomial roots</h2>
<p class="note">Main article: <a href="/facts/Properties_of_polynomial_roots/YKNYVKcj">Properties of polynomial roots</a></p>
<p>Every real polynomial of odd <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> has an odd number of real roots (counting <a href="/facts/Multiplicity_(mathematics)/dndUyetA">multiplicities</a>); 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 <a href="/facts/Intermediate_value_theorem/SaFdYE7W">intermediate value theorem</a>: since polynomial functions are <a href="/facts/Continuous_function/sKbl02pB">continuous</a>, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).
</p>
<h3>Fundamental theorem of algebra</h3>
<p class="note">Main article: <a href="/facts/Fundamental_theorem_of_algebra/vspflSsM">Fundamental theorem of algebra</a></p>
<p>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 <a href="/facts/Complex_conjugate/7TEaoQDe">conjugate</a> pairs.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> <a href="/facts/Vieta%2527s_formulas/u3eQJWkA">Vieta's formulas</a> relate the coefficients of a polynomial to sums and products of its roots.
</p>
<h2 id="computing-roots">Computing roots</h2>
<p class="note">See also: <a href="/facts/Equation_solving/2RKaXrDM">Equation solving</a></p>
<p>There are many methods for computing accurate <a href="/facts/Approximation/LvYHsXzq">approximations</a> of roots of functions, the best being <a href="/facts/Newton%2527s_method/VlRhI2FL">Newton's method</a>, see <a href="/facts/Root-finding_algorithm/IxTDkhBq">Root-finding algorithm</a>. 
</p><p>For <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>, there are specialized algorithms that are more efficient and may provide all roots or all real roots; see <a href="/facts/Polynomial_root-finding/akrqmLmu">Polynomial root-finding</a> and <a href="/facts/Real-root_isolation/yQ8cIVDR">Real-root isolation</a>.
</p><p>Some polynomial, including all those of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> no greater than 4, can have all their roots expressed <a href="/facts/Algebraic_function/yT7EKmwF">algebraically</a> in terms of their coefficients; see <a href="/facts/Solution_in_radicals/yVqArGpw">Solution in radicals</a>.
</p>
<h2 id="zero-set">Zero set</h2>
<p class="note">"Zero set" redirects here. For the musical album, see <a href="/facts/Zero_Set/39YKuyyk">Zero Set</a>.</p>
<p>In various areas of mathematics, the zero set of a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> is the set of all its zeros. More precisely, if 
  
    
      
        f
        :
        X
        →
        
          R
        
      
    
    {\displaystyle f:X\to \mathbb {R} }
  
 is a <a href="/facts/Real-valued_function/jetqlJpK">real-valued function</a> (or, more generally, a function taking values in some <a href="/facts/Abelian_group/FfWwmx4R">additive group</a>), its zero set is 
  
    
      
        
          f
          
            −
            1
          
        
        (
        0
        )
      
    
    {\displaystyle f^{-1}(0)}
  
, the <a href="/facts/Inverse_image/KANefMAl">inverse image</a> of 
  
    
      
        {
        0
        }
      
    
    {\displaystyle \{0\}}
  
 in 
  
    
      
        X
      
    
    {\displaystyle X}
  
. 
</p><p>Under the same hypothesis on the <a href="/facts/Codomain/MGc43nwQ">codomain</a> of the function, a <a href="/facts/Level_set/JUmNbdNh">level set</a> of a function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is the zero set of the function 
  
    
      
        f
        −
        c
      
    
    {\displaystyle f-c}
  
 for some 
  
    
      
        c
      
    
    {\displaystyle c}
  
 in the codomain of 
  
    
      
        f
        .
      
    
    {\displaystyle f.}

</p><p>The zero set of a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> is also known as its <a href="/facts/Kernel_(algebra)/GGSpUPMJ">kernel</a>.
</p><p>The cozero set of the function 
  
    
      
        f
        :
        X
        →
        
          R
        
      
    
    {\displaystyle f:X\to \mathbb {R} }
  
 is the <a href="/facts/Complement_(set_theory)/yWUePIYY">complement</a> of the zero set of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 (i.e., the subset of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 on which 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is nonzero). 
</p>
<h3>Applications</h3>
<p>In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, the first definition of an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a> is through zero sets. Specifically, an <a href="/facts/Affine_algebraic_set/WVynz2Kn">affine algebraic set</a> is the <a href="/facts/Set_intersection/RPfUNJh9">intersection</a> of the zero sets of several polynomials, in a <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> 
  
    
      
        k
        
          [
          
            
              x
              
                1
              
            
            ,
            …
            ,
            
              x
              
                n
              
            
          
          ]
        
      
    
    {\displaystyle k\left[x_{1},\ldots ,x_{n}\right]}
  
 over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. In this context, a zero set is sometimes called a <i>zero locus</i>.
</p><p>In <a href="/facts/Mathematical_analysis/XXeR26Ih">analysis</a> and <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, any <a href="/facts/Closed_set/upYnRTUj">closed subset</a> of 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 is the zero set of a <a href="/facts/Smooth_function/mffL4ch2">smooth function</a> defined on all of 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
. This extends to any <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> as a corollary of <a href="/facts/Paracompactness/rjTrvF7j">paracompactness</a>. 
</p><p>In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, zero sets are frequently used to define <a href="/facts/Manifold/rXPERJtu">manifolds</a>. An important special case is the case that 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a <a href="/facts/Smooth_function/mffL4ch2">smooth function</a> from 
  
    
      
        
          
            R
          
          
            p
          
        
      
    
    {\displaystyle \mathbb {R} ^{p}}
  
 to 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
. If zero is a <a href="/facts/Regular_value/ARWB4sfr">regular value</a> of 
  
    
      
        f
      
    
    {\displaystyle f}
  
, then the zero set of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a smooth manifold of dimension 
  
    
      
        m
        =
        p
        −
        n
      
    
    {\displaystyle m=p-n}
  
 by the <a href="/facts/Submersion_(mathematics)/ARWB4sfr">regular value theorem</a>.
</p><p>For example, the unit 
  
    
      
        m
      
    
    {\displaystyle m}
  
-<a href="/facts/Sphere/jywYLNM2">sphere</a> in 
  
    
      
        
          
            R
          
          
            m
            +
            1
          
        
      
    
    {\displaystyle \mathbb {R} ^{m+1}}
  
 is the zero set of the real-valued function 
  
    
      
        f
        (
        x
        )
        =
        ‖
        x
        
          ‖
          
            2
          
        
        −
        1
      
    
    {\displaystyle f(x)=\Vert x\Vert ^{2}-1}
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Root-finding_algorithm/IxTDkhBq">Root-finding algorithm</a></li>
<li><a href="/facts/Intermediate_value_theorem/SaFdYE7W">Bolzano's theorem</a>, a continuous function that takes opposite signs at the end points of an interval has at least a zero in the interval.</li>
<li><a href="/facts/Gauss%25E2%2580%2593Lucas_theorem/mnAct7WV">Gauss–Lucas theorem</a>, the complex zeros of the derivative of a polynomial lie inside the convex hull of the roots of the polynomial.</li>
<li><a href="/facts/Marden%2527s_theorem/NbR5HF7h">Marden's theorem</a>, a refinement of Gauss–Lucas theorem for polynomials of degree three</li>
<li><a href="/facts/Sendov%2527s_conjecture/7vPfgLUK">Sendov's conjecture</a>, a conjectured refinement of Gauss-Lucas theorem</li>
<li><a href="/facts/Vanish_at_infinity/nCAl6pYS">zero at infinity</a></li>
<li><a href="/facts/Zero_crossing/g70XI9o6">Zero crossing</a>, property of the graph of a function near a zero</li>
<li><a href="/facts/Zeros_and_poles/gnA3U8VP">Zeros and poles</a> of holomorphic functions</li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Root.html">"Root"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 535. ISBN 0-13-165711-9. <a href="0-13-165711-9" target="_blank">0-13-165711-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Algebra - Zeroes/Roots of Polynomials". tutorial.math.lamar.edu. Retrieved 2019-12-15. <a href="http://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx" target="_blank">http://tutorial.math.lamar.edu/Classes/Alg/ZeroesOfPolynomials.aspx</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Roots and zeros (Algebra 2, Polynomial functions)". Mathplanet. Retrieved 2019-12-15. <a href="https://www.mathplanet.com/education/algebra-2/polynomial-functions/roots-and-zeros" target="_blank">https://www.mathplanet.com/education/algebra-2/polynomial-functions/roots-and-zeros</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 535. ISBN 0-13-165711-9. <a href="0-13-165711-9" target="_blank">0-13-165711-9</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Zero of a function open-in-new

Zero of a function