BUILT-IN MATHEMATICA SYMBOL

# Root

Root[f, k]
represents the exact k root of the polynomial equation .

Root[{f1, f2, ...}, {k1, k2, ...}]
represents the last coordinate of the exact vector such that is the root of the polynomial equation .

Root[{f, x0}]
represents the exact root of the general equation near .

Root[{f, x0, n}]
represents n roots of the equation near .

## Details and OptionsDetails and Options

• f must be a Function object such as .
• Root[f, k] is automatically reduced so that f has the smallest possible degree and smallest integer coefficients.
• The ordering used by Root[f, k] takes real roots to come before complex ones, and takes complex conjugate pairs of roots to be adjacent.
• The coefficients in the polynomial can involve symbolic parameters.
• For linear and quadratic polynomials , Root[f, k] is automatically reduced to explicit rational or radical form.
• For other polynomials, ToRadicals can be used to convert to explicit radicals.
• In Root[{f1, f2, ...}, {k1, k2, ...}], must be a Function object with i formal parameters, and should be a polynomial in x of degree at least .
• If for all i, is a polynomial in with rational number coefficients, then RootReduce can be used to represent Root[{f1, f2, ...}, {k1, k2, ...}] in the Root[f, k] form.
• Root[{f, x0}] represents an exact root of the general equation , which can be transcendental.
• In Root[{f, x0}], must be an approximate real or complex number such that exactly one root of lies within the numerical region defined by its precision.
• Root[{f, x0, n}] represents n roots, counting multiplicity, that lie within the numerical region defined by the precision of .
• 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 complex roots of a polynomial using validated numerical methods. SetOptions[Root, ExactRootIsolation->True] will make Root use symbolic methods that are usually much slower.

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

Solution to a quintic:

 Out[1]=

Numerical values:

 Out[2]=

Real solutions to an exp-log equation:

 Out[1]=

Real solution to a system of equations:

 Out[1]=