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


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.


represents the exact root of the general equation f[x]0 near x=x0.


represents n roots of the equation f[x]0 near x=x0.

Details and Options

  • 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[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 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.
  • For other polynomials, ToRadicals can be used to convert to explicit radicals.
  • In Root[{f1,f2,},{k1,k2,}], fi must be a Function object with i formal parameters, and fi[a1,,ai-1,x] should be a polynomial in x of degree at least ki.
  • If for all i, fi[x1,,xi] is a polynomial in x1,,xi 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 f[x]0, which can be transcendental.
  • In Root[{f,x0}], x0 must be an approximate real or complex number such that exactly one root of f[x] 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 x0.
  • 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.


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Basic Examples  (3)

Solution to a quintic:

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Numerical values:

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Real solutions to an exp-log equation:

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Real solution to a system of equations:

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Scope  (12)

Generalizations & Extensions  (2)

Options  (1)

Applications  (17)

Properties & Relations  (7)

Possible Issues  (5)

Neat Examples  (1)

Introduced in 1996
Updated in 2012