| Revision as of 01:15, 4 December 2004 editKusma (talk | contribs)Autopatrolled, Administrators59,515 edits fundamental theorem of algebra needs polynomial to be nonconstant← Previous edit | Latest revision as of 21:11, 14 January 2018 edit undoD.Lazard (talk | contribs)Extended confirmed users33,720 edits typoTag: Redirect target changed | ||
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| #Redirect ] | |||
| In ], a '''zero''' of a ] ''f'' is a ] ''a'' such that ''f''(''a'') = 0. | |||
| {{R from merge}} | |||
| ==Multiplicity of a zero== | |||
| A complex number ''a'' is a '''simple zero''' of ''f'', or a '''zero of multiplicity 1''' of ''f'', if ''a'' is ''not'' a zero of the holomorophic function ''g'' such that | |||
| :<math>f(z)=(z-a)g(z).</math> | |||
| Generally, the ''']''' of the zero of ''f'' at ''a'' is the positive integer ''n'' for which there is a holomorphic function ''g'' such that | |||
| :<math>f(z)=(z-a)^ng(z)\ \mbox{and}\ g(a)\neq 0.</math> | |||
| ==Existence of zeroes== | |||
| The so-called ] (something of a misnomer) says that every nonconstant polynomial function with complex coefficients has at least one zero in the complex plane. This is in contrast to the situation with ] zeroes: some polynomial functions with real coefficients have no real zeroes (but since real numbers are complex numbers, they still have complex zeroes). An example is ''f''(''x'') = ''x''<sup>2</sub> + 1. | |||
| {{math-stub}} | |||
| ] | |||
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