ordinaryroot

Ordinaryroot is a term used in some mathematical references to denote a root of an equation that is simple, meaning it has multiplicity one. While the concept applies to polynomials, it can also be described for analytic functions. An ordinaryroot r satisfies f(r) = 0 and, in the polynomial case, the derivative at r is nonzero.

In contrast, roots with higher multiplicity are called multiple or repeated roots. For a polynomial f, a root r is ordinary if and only if f(r) = 0 and f'(r) ≠ 0. If f'(r) = 0, the root has multiplicity greater than one and is not an ordinaryroot. This derivative criterion provides a practical test for distinguishing ordinaryroots from multiple roots.

Examples illustrate the distinction. Consider f(x) = x^2 − 3x + 2, which factors as (x − 1)(x − 2). The roots r = 1 and r = 2 satisfy f(r) = 0 and f'(r) ≠ 0, so both are ordinaryroots. By contrast, g(x) = (x − 1)^2 has a root at x = 1 with multiplicity two, and g'(1) = 0, so x = 1 is not an ordinaryroot.

In practice, the concept helps in numerical root-finding and local analysis, since simple zeros behave regularly under perturbations and satisfy standard linear approximations. Some texts use ordinaryroot synonymously with the more common term simple root. See also simple root, multiple root, and root multiplicity.