Two-term equation

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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:
This article was adapted from an original article by A.I. Galochkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article