View source for Duhem–Margules equation

[[Image:Polynomialdeg5.png|thumb|right|300px|Polynomial of degree 5:<br>{{nowrap|''&fnof;''(''x'') {{=}} 1/20 (''x'' + 4)(''x'' + 2)&middot;}} {{nowrap|&middot;(''x'' + 1)(''x'' &minus; 1)(''x'' &minus; 3) + 2}}]]

[[Graph of a function|Graphical]] [[List of graphical methods|method]]s provide a means of determining or approximating the [[root of a function|roots]] of a [[polynomial]]—the values that make the polynomial equal to zero.<ref>{{citation| title=Graphical Representation of Complex Roots |author=James A. Ward |journal=National Mathematics Magazine |volume=11 |date=April 1937 |pages=297–303 |issue=7| doi=10.2307/3028785}}</ref> Practical tools for performing these include [[graph paper]], [[graphical calculator]]s and [[computer graphics]].<ref>{{citation |title=Determining Roots of Complex Functions with Computer Graphics |author=Robert Kowalski, Helen Skala |journal=Coll. Micro. |year=1990 |volume=VIII |pages=51–54 |issue=1}}</ref>

The [[fundamental theorem of algebra]] states that a ''n''th-degree polynomial with complex coefficients (including real coefficients) has ''n'' complex roots (not necessarily real even if the coefficients are real), although its roots may not all be different from each other.  If the polynomial has real coefficients, its roots are either real, or else occur as [[complex conjugate]]s.  Suppose a polynomial ''P''(''x'') is graphed as ''y'' =&nbsp;''P''(''x'').  At a real root, the graph of the polynomial crosses the ''x''-axis.  Thus, the real roots of a polynomial can be demonstrated graphically.<ref>{{citation |title=A graphical approach to understanding the fundamental theorem of algebra |author=Sudhir Kumar Goel, Denise T. Reid. |journal=Mathematics Teacher |date=December 2001 |volume=94 |issue=9 |page=749}}</ref>

For some kinds of polynomials, all the roots, including the complex roots, can  be found graphically. Polynomial equations up to the fifth degree may be solved graphically.<ref>Henriquez, Garcia, "The graphical interpretation of the complex roots of cubic equations," ''[[American Mathematical Monthly]]'' 42(6), June-July 1935, 383-384.</ref><ref name=Yanosik>{{citation| title=Graphical Solutions for Complex Roots of Quadratics, Cubics and Quartics |author=George A. Yanosik |journal=National Mathematics Magazine |volume=17 |date=January 1943 |pages=147–150 |issue=4| doi=10.2307/3028335}}</ref><ref>{{citation |journal=The Quarterly Journal of Mathematics |year=1934 |doi=10.1093/qmath/os-5.1.10 |publisher=Oxford University Press |title=On the number of real roots of a quintic equation |author=T.W.Chaundy |volume=os-5 |pages=10–22}}</ref>

The [[geometry|geometrical]] methods of [[compass and straightedge constructions|ruler and compass]] may be used to solve any [[linear equation|linear]] or [[quadratic equation]].  [[Descartes]] showed that the constructions of [[Euclid]] were equivalent to the [[algebra|algebraic]] solution of quadratics.<ref>{{citation |pages=120 et seq. |title= Geometry: Euclid and beyond |author=Robin Hartshorne |url=http://books.google.com/books?id=EJCSL9S6la0C |isbn= 978-0-387-98650-0 |year=2000}}</ref>

[[Cubic equations]] may be solved by [[solid geometry]].  [[Archimedes]]' work ''On the Sphere and the Cylinder'' provided solutions of some cubics and [[Omar Khayyam]] systematised this to provide geometrical solutions of all quadratics and cubics.<ref>{{citation |url=http://books.google.co.uk/books?id=4miP1_qN7aoC |chapter=ch. III Latency |title=The emergence of number |author=John N. Crossley |isbn=978-9971-5-0414-4 |year=1987}}</ref> 

== Complex roots of quadratic polynomials ==
For polynomials with real coefficients, a local minimum point above the ''x''-axis or a local maximum point below the ''x''-axis indicates the existence of two non-real [[complex number|complex]] roots, which are each other's complex conjugates.<ref>{{citation|title=From Cardano's Great Art to Lagrange's Reflections: Filling a Gap in the History of Algebra|series=Heritage of European mathematics|first=Jacqueline A.|last=Stedall|publisher=European Mathematical Society|year=2011|isbn=9783037190920|page=97|url=http://books.google.com/books?id=VQ-eI3EHBD4C&pg=PA97}}.</ref> The converse, however, is not true; for example, the cubic polynomial ''x''<sup>3</sup>&nbsp;+&nbsp;''x'' has two complex roots, but its graph has no local minima or maxima.

The simplest such case involves parabolas.

[[Image:Visual.complex.root.finding.png|300px|right|thumb|A second-degree polynomial function whose ''x'' intercepts are not real: {{nowrap|''y'' {{=}} (''x'' &minus; 5)<sup>2</sup> + 9}}. The "3" is the imaginary part of the ''x''-intercept. The real part is the ''x''-coordinate of the vertex. Thus the roots are {{nowrap|5 ± 3''i''}}.]] 

If a parabola has a global minimum point above the ''x''-axis, or a global maximum point below the ''x''-axis, then its ''x'' intercepts are not real.  For

: <math> y = a(x - h)^2 + k, \, </math>

if ''a'' and ''k'' are positive, then the roots are non-real complex numbers.  The number&nbsp;''k'' is then the height of the vertex above the ''x''-axis.  In the example in the illustration, we have&nbsp;''k''&nbsp;=&nbsp;9.  Suppose one goes ''k''&nbsp;units in the opposite direction from the vertex, i.e. away from the ''x''-axis, then horizontally as far as it takes to reach the curve (in the example, that distance is&nbsp;3.  The horizontal distance from that point to the curve is the absolute value of the imaginary part of the root.<ref name = "Norton1984">{{citation |title=Complex Roots Made Visible |author=Alec Norton, Benjamin Lotto |journal=The College Mathematics Journal |volume=15 |date=June 1984 |pages=248–249 |issue=3 |doi=10.2307/2686333}}</ref>  The ''x''-coordinate of the vertex is the real part.<ref name = "Norton1984"/> Thus, in the example, the roots are<ref name = "Norton1984"/>
: <math> 5 \pm 3i. \, </math>

This method is specific to quadratics and does not generalise to higher-degree polynomial equations.

==References==
{{reflist}}

==External links==
* [http://www.math.hmc.edu/funfacts/ffiles/10005.1.shtml ''Complex roots made visible'' at Mudd Math Fun Facts]
* [http://www.nctm.org/eresources/view_article.asp?article_id=6168 ''Connecting complex roots to a parabola's graph'' at ON-Math]
* [http://mathforum.org/library/drmath/view/53808.html ''Complex Roots'' at Dr. Math]. With comments by [[John Horton Conway|John Conway]]

{{DEFAULTSORT:Graphical Methods Of Finding Polynomial Roots}}
[[Category:Elementary algebra]]
[[Category:Root-finding algorithms]]