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Difference between revisions of "Two-term equation"

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An algebraic equation of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094630/t0946301.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094630/t0946302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094630/t0946303.png" /> are complex numbers, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094630/t0946304.png" />. Two-term equations have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094630/t0946305.png" /> distinct complex roots
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An algebraic equation of the form $ax^n+b=0$, where $a$ and $b$ are [[complex number]]s, with $ab\neq0$. Two-term equations have $n$ distinct complex roots
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094630/t0946306.png" /></td> </tr></table>
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$$x_k=\left|\frac ba\right|^{1/n}\exp\left(\frac{2\pi k+\phi}{n}i\right),$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094630/t0946307.png" /></td> </tr></table>
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$$k=0,\ldots,n-1,\quad\phi=\arg\left(-\frac ba\right).$$
  
The roots of a two-term equation in the complex plane are located on the circle with radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094630/t0946308.png" /> and centre at the coordinate origin, at the vertices of the inscribed regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094630/t0946309.png" />-gon (cf. [[Regular polygons|Regular polygons]]).
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The roots of a two-term equation in the complex plane are located on the circle with radius $|b/a|^{1/n}$ and centre at the coordinate origin, at the vertices of the inscribed regular $n$-gon (cf. [[Regular polygons]]).

Latest revision as of 17:30, 23 December 2014

An algebraic equation of the form $ax^n+b=0$, where $a$ and $b$ are complex numbers, with $ab\neq0$. Two-term equations have $n$ distinct complex roots

$$x_k=\left|\frac ba\right|^{1/n}\exp\left(\frac{2\pi k+\phi}{n}i\right),$$

$$k=0,\ldots,n-1,\quad\phi=\arg\left(-\frac ba\right).$$

The roots of a two-term equation in the complex plane are located on the circle with radius $|b/a|^{1/n}$ and centre at the coordinate origin, at the vertices of the inscribed regular $n$-gon (cf. Regular polygons).

How to Cite This Entry:
Two-term equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Two-term_equation&oldid=13774
This article was adapted from an original article by A.I. Galochkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article