This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
 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 1.5.7, Section 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]=