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represents the exact k^(th) root of the polynomial equation .
represents the exact root of the general equation near .
represents n roots of the equation near .
  • Root 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 can involve symbolic parameters.
  • For linear and quadratic polynomials , Root is automatically reduced to explicit rational or radical form.
  • For other polynomials, ToRadicals can be used to convert to explicit radicals.
  • Root represents an exact root of the general equation , which can be transcendental.
  • In Root, must be an approximate real or complex number such that exactly one root of lies within the numerical region defined by its precision.
  • Root 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.
  • Root 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.
Solution to a quintic:
Numerical values:
Real solutions to an exp-log equation:
Solution to a quintic:
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Numerical values:
Click for copyable input
Real solutions to an exp-log equation:
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Some exact values are generated automatically:
Evaluate to high precision:
Roots of a polynomial:
Real roots of an exp-log function:
Roots of an analytic function in a bounded region:
A triple root:
Roots of a polynomial with symbolic coefficients:
Roots of a quadratic with symbolic coefficients:
When a, b, c and the roots are real, the roots are always ordered by their values:
The "standard" formulas for the roots of a quadratic do not guarantee the ordering of roots:
Find the series with respect to a parameter:
Complex components of roots:
Exact comparisons:
Algebraic number coefficients are automatically lifted to integers:
Find Puiseux series at branch points:
The setting of is reflected in the third argument of a Root object:
Root isolation is performed the first time the numerical value of the root is needed:
The symbolic complex root isolation method is usually slower than the validated numeric one:
The root isolation method may affect the ordering of nonreal roots:
Solve polynomial equations of any degree in closed form in terms of Root:
Solve the characteristic equation of a Hilbert matrix:
Find the minimum of a parameterized polynomial:
Solve a constant coefficient differential equation of any degree:
Solve a constant coefficient difference equation of any degree:
Resolve a piecewise function:
Solve univariate exp-log equations and inequalities over the reals:
Solve univariate elementary function equations over bounded intervals and regions:
Solve univariate analytic equations over bounded intervals and regions:
Find real roots of high-degree sparse polynomials and algebraic functions:
Solve univariate transcendental optimization problems:
Integrate a piecewise function with an exp-log inequality condition:
Evaluate the hard hexagon entropy constant:
Solve Kepler's equation:
Compute the Laplace limit constant:
Plot a root as a function of a parameter:
Extract the polynomial from a Root object:
Series expansions of implicit solutions to equations:
Use RootReduce to canonicalize algebraic numbers:
Simplify combinations of Root objects:
Reduce an equation for a parameter in a Root object:
Use RootApproximant to generate Root objects from numbers:
Roots are numeric expressions:
Series at branch points may not be valid in all directions:
Canonicalization is only possible for parameter-free roots:
Parameterized roots can have complicated branch cuts in the complex parameter plane:
A non-polynomial Root object may represent a cluster of distinct roots:
Numerical computation with a higher precision yields an approximation of one of the roots:
The choice of root stays the same for subsequent computations:
A high power of a Pisot number that is nearly an integer:
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