This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)
 Documentation / Mathematica / Built-in Functions / New in Version 3.0 / Algebraic Computation  /
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 1.5.7Section 3.4.2.
  • See also: Solve, RootReduce, ToRadicals, RootSum, Extension.
  • Related package: Algebra`RootIsolation`.

    Further Examples

    The polynomial that appears in Root is automatically reduced to a minimal polynomial in the form of a pure function.

    In[1]:=

    Out[1]=

    In[2]:=

    Out[2]=

    It can be cast into the form of a polynomial in one variable.

    In[3]:=

    Out[3]=

    In[4]:=

    In the quadratic case the Root object is automatically expressed in terms of radicals.

    In[5]:=

    Out[5]=

    For degree three or higher, Root objects are not automatically expressed in terms of radicals.

    In[6]:=

    Out[6]=

    ToRadicals attempts to express all Root objects in terms of radicals.

    In[7]:=

    Out[7]=

    To get a Root object from performing algebraic operations on algebraic numbers, use RootReduce.

    In[8]:=

    Out[8]=

    In[9]:=

    Out[9]=

    Here is a more complicated expression involving algebraic numbers.

    In[10]:=

    Out[10]=

    Sometimes RootReduce gives a simple result. This is the product of the roots of the polynomial.

    In[11]:=

    Out[11]=