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Root[{f,c}]

represents the exact root of the general equation f[x]0 near x=c.

Root[{{f1,…,fn},{c1,…,cn}},j]

represents the j^(th) coordinate of the exact root of the system of equations {f1[x1,…,xn]0,…,fn[x1,…,xn]0} near {x1,…,xn}={c1,…,cn}.

Root[f,k]

represents the exact k^(th) root of the polynomial equation f[x]0.

Root[{f1,f2,…},{k1,k2,…}]

represents the last coordinate of the exact vector {a1,a2,…} such that ai is the ki^(th) root of the polynomial equation fi[a1,…,ai-1,x]0.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Roots of Univariate Functions  
Roots of Multivariate Systems  
Roots of Univariate Polynomials  
Roots of Triangular Polynomial Systems  
Options  
ExactRootIsolation  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
Related Guides
Related Links
History
Cite this Page

Root

Root[{f,c}]

represents the exact root of the general equation f[x]0 near x=c.

Root[{{f1,…,fn},{c1,…,cn}},j]

represents the j^(th) coordinate of the exact root of the system of equations {f1[x1,…,xn]0,…,fn[x1,…,xn]0} near {x1,…,xn}={c1,…,cn}.

Root[f,k]

represents the exact k^(th) root of the polynomial equation f[x]0.

Root[{f1,f2,…},{k1,k2,…}]

represents the last coordinate of the exact vector {a1,a2,…} such that ai is the ki^(th) root of the polynomial equation fi[a1,…,ai-1,x]0.

Details and Options

  • Root is also known as an algebraic number when f is polynomial with integer coefficients or a transcendental number when there is no such polynomial f possible.
  • Root is typically used to represent an exact number and is automatically generated by a variety of algebra, calculus, optimization and geometry functions.
  • Root represents an exact number as a solution to an equation f[x]0 with additional information specifying which of the roots is intended.
  • Root numbers can be used like any other numbers, both in exact and approximate computations.
  • Root numbers are formatted as where approx is a numerical approximation. An approximation to precision p can be computed using N[,p].
  • For most uses, Root objects are automatically generated and can be directly used. For advanced uses, when your code is going to directly generate Root objects, a deeper understanding of the different representations is necessary.
  • There are two distinct mechanisms used to specify which root of an equation is represented, the neighborhood representation Root[{f,c}] and the indexing representation Root[f,k].
  • The root neighborhood representation Root[{f,c}] specifies the equation f[x]0 as well as the neighborhood rectangle centered at c and width and height .
  • The root neighborhood representation for systems Root[{{f1,…,fn},{c1,…,cn}},j] similarly specifies a system of equations {f1[x1,…,xn]0,…,fn[x1,…,xn]0} and a neighborhood given by the product of rectangles from ci for the different coordinates.
  • The root neighborhood representation Root[{f,c,m}] specifies that f[x]0 has a root of multiplicity m in the neighborhood given by c. However, it may be a cluster of closely spaced roots and by refining the neighborhood c, i.e. higher precision root approximation, they may separate. »
  • The root indexing representation Root[f,k] applies to polynomial functions f only. The indexing of roots takes the real roots first, in increasing order. For polynomials with rational coefficients, the complex conjugate pairs of roots have consecutive indices.
  • The root indexing representation for systems Root[{f1,f2,…,fk},{k1,k2,…,kn}] applies to triangular systems of polynomial equations only. Given equations f1[x1]0, f2[x1,x2]0, …, fn[x1,x2,…,xn]0, we recursively define r1 as the k1^(th) root of f1[x1]0, r2 as the k2^(th) root of f2[r1,x2]0 and finally rn as the kn^(th) root of fn[r1,…,rn-1,xn]0. The represented root is rn.

Examples

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Basic Examples  (4)

Solution to a quintic:

Numerical values:

Real solutions to an exp-log equation:

Real solution to a system of equations:

A solution instance to a system of transcendental equations:

Scope  (22)

Basic Uses  (5)

Some exact values are generated automatically:

Evaluate to high precision:

Exact comparisons:

Convert radicals to a Root object:

Convert a Root object to radicals:

Disable the elided formatting of Root:

Re-enable the elided formatting:

Roots of Univariate Functions  (4)

Real roots of an exp-log function:

The Root representation involves a univariate function and an approximation that isolates the root:

The Root object is an exact numeric expression:

Roots of an analytic function in a bounded region:

A triple root:

This representation is used for roots of polynomials of degrees that exceed $MaxRootDegree:

The approximation used to represent the root is equal to , but the root is not:

Roots of Multivariate Systems  (1)

Find a solution instance to a system of transcendental equations:

The roots are exact numeric expressions:

The Root representation involves a multivariate system and an approximation that isolates the root:

Roots of Univariate Polynomials  (10)

Roots of a univariate polynomial with rational coefficients are algebraic numbers:

The polynomials are automatically reduced:

A minimal polynomial is always irreducible and primitive:

Use MinimalPolynomial to extract the minimal polynomial:

A Root object representing an algebraic number has three arguments:

The third argument is 0 (default value) or 1 and indicates the root isolation method to be used:

The root isolation method used may affect the ordering of non-real roots:

Roots of polynomials of degree at 1 and 2 simplify automatically:

An algebraic combination of algebraic numbers is an algebraic number:

Use RootReduce to represent the result as a single Root object:

The canonicalization is not done automatically since minimal polynomials can grow rapidly:

Complex components of algebraic numbers:

Root of a polynomial with exact numeric coefficients is an exact numeric object:

Roots of a polynomial with coefficients involving symbolic parameters:

Plot the real-valued roots:

Find the series with respect to the parameter:

Find Puiseux series of a root at a branch point:

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:

Roots of Triangular Polynomial Systems  (2)

Real root of a triangular system of equations:

The Root representation involves a triangular polynomial system and root indices:

The Root object is an exact numeric expression:

Roots of systems of polynomials with rational coefficients have algebraic number coordinates:

The degree of the minimal polynomial is generally the product of degrees of the system polynomials:

This representation is used for roots of polynomials with algebraic number coefficients:

Convert the root to the canonical algebraic number representation:

Options  (1)

ExactRootIsolation  (1)

Root[f,k] by default isolates the complex roots of a polynomial using validated numerical methods. Setting ExactRootIsolationTrue will make Root use symbolic methods that are usually much slower.

The setting of ExactRootIsolation 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:

Applications  (19)

Solve polynomial equations of any degree in closed form in terms of Root:

Solve the characteristic equation of a Hilbert matrix:

Using Eigenvalues:

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:

Find a solution of a triangular system of equations:

Represent the solution in the Root[f,k] form:

Here, the Root[f,k] representation would involve a polynomial of degree 1000000:

Compute an approximate value of the solution:

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:

Solve a convex optimization problem:

Find a solution instance of a system of transcendental equations and inequalities:

Properties & Relations  (11)

Root objects represent exact numbers:

Compute approximations to arbitrary precision:

Use MinimalPolynomial to find minimal polynomials of algebraic numbers:

Use ToRadicals to attempt to convert a Root object to an arithmetic combination of radicals:

All roots of polynomials of degree not exceeding 4 are representable in radicals:

Use RootReduce to canonicalize algebraic numbers, including from operations:

From a triangular system:

Use AlgebraicNumber for computations within a fixed simple extension of the rationals:

Rational operations on AlgebraicNumber objects produce AlgebraicNumber objects:

Use ToNumberField to express given algebraic numbers as elements of the same simple extension:

Perform computations within the common simple extension or rationals:

RootSum represents the sum of values of a function over roots of a polynomial:

Use Normal to represent the sum using explicit roots:

For rational functions, the sum can be computed without finding the roots:

Simplify combinations of Root objects:

Solve an equation for a parameter in a Root object:

Use ImplicitD to compute derivatives of implicit solutions of equations:

Compare with the result obtained by computing the derivative of directly:

Compute series expansions of implicit solutions to equations:

Use AsymptoticSolve to find series expansions of all roots:

Use RootApproximant to generate Root objects that approximate given numbers:

Possible Issues  (5)

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 larger setting for $MaxExtraPrecision can be needed for roots with noninteger coefficients:

Neat Examples  (1)

A high power of a Pisot number that is nearly an integer:

Wolfram Research (1996), Root, Wolfram Language function, https://reference.wolfram.com/language/ref/Root.html (updated 2021).

Text

Wolfram Research (1996), Root, Wolfram Language function, https://reference.wolfram.com/language/ref/Root.html (updated 2021).

CMS

Wolfram Language. 1996. "Root." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/Root.html.

APA

Wolfram Language. (1996). Root. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Root.html

BibTeX

@misc{reference.wolfram_2025_root, author="Wolfram Research", title="{Root}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Root.html}", note=[Accessed: 01-March-2026]}

BibLaTeX

@online{reference.wolfram_2025_root, organization={Wolfram Research}, title={Root}, year={2021}, url={https://reference.wolfram.com/language/ref/Root.html}, note=[Accessed: 01-March-2026]}