Algebraic equations in C. Extraction of roots

Algebraic equations can have solutions (roots) that are complex numbers. For example, the equation $z^n = w$ (where $z$ and $w$ are complex numbers and $n$ is a positive integer) can be solved using polar form and De Moivre's Theorem. Every nonzero complex number $w$ has exactly $n$ distinct $n$-th roots in the complex plane.

Important properties

• A quadratic equation with real coefficients may have complex roots if the discriminant is negative.

• Any polynomial equation of degree $n$ has exactly $n$ complex roots (counting multiplicities), according to the Fundamental Theorem of Algebra.

• To extract $n$-th roots of a complex number $w = re^{i\theta}$, use $z_k = r^{1/n} e^{i(\theta + 2\pi k)/n}$ for $k = 0, 1, ..., n-1$.

• The $n$-th roots of unity are the solutions to $z^n = 1$ and are evenly spaced around the unit circle.