This is documentation for Mathematica 4, which was
based on an earlier version of the Wolfram Language.

Root

Root[f, k] represents the k root of the polynomial equation f[x] == 0.

f must be a Function object such as (#^5 - 2 # + 1)&.

Root[f, k] is automatically reduced so that f has the smallest possible degree and smallest integer coefficients.

The ordering used by Root takes real roots to come before complex ones, and takes complex conjugate pairs of roots to be adjacent.

The coefficients in the polynomial f[x] can involve symbolic parameters.

For linear and quadratic polynomials f[x], Root[f, k] is automatically reduced to explicit rational or radical form.

N finds the approximate numerical value of a Root object.

Operations such as Abs, Re, Round and Less can be used on Root objects.

Root[f, k] is treated as a numeric quantity if f contains no symbolic parameters.

Root by default isolates the roots of a polynomial using approximate numerical methods. No cases are known where this approach fails. SetOptions[Root, ExactRootIsolation->True] will however make Root use much slower but fully rigorous methods.

See The Mathematica Book: Section 1.5.7 and Section 3.4.2.