# Revision history [back]

Expanding on Snark's comment: use algebraic numbers to get an exact answer.

sage: R.<x> = QQbar['x']
sage: ff = x^3 - 2
sage: factor(ff)
(x - 1.259921049894873?) * (x + 0.6299605249474365? - 1.091123635971722?*I) * (x + 0.6299605249474365? + 1.091123635971722?*I)
sage: (1/ff).partial_fraction_decomposition()
(0,
[(0.2099868416491456? + 0.?e-19*I)/(x - 1.259921049894873?),
(-0.10499342082457277? + 0.1818539393286203?*I)/(x + 0.6299605249474365? - 1.091123635971722?*I),
(-0.10499342082457277? - 0.1818539393286203?*I)/(x + 0.6299605249474365? + 1.091123635971722?*I)])


Expanding on Snark's comment: use algebraic numbers to get an exact answer.

sage: R.<x> = QQbar['x']
sage: ff = x^3 - 2
sage: factor(ff)
(x - 1.259921049894873?) * (x + 0.6299605249474365? - 1.091123635971722?*I) * (x + 0.6299605249474365? + 1.091123635971722?*I)
sage: (1/ff).partial_fraction_decomposition()
(0,
[(0.2099868416491456? + 0.?e-19*I)/(x - 1.259921049894873?),
(-0.10499342082457277? + 0.1818539393286203?*I)/(x + 0.6299605249474365? - 1.091123635971722?*I),
(-0.10499342082457277? - 0.1818539393286203?*I)/(x + 0.6299605249474365? + 1.091123635971722?*I)])


The entries are exact; they live in QQbar. To access the entries:

sage: for q in factor(ff):
...       q = q[0]
...       print 'x - a =', q
...       a = QQbar(x - q)
...       print '  a =', a
...       print '  minpoly:', a.minpoly()
x - a = x - 1.259921049894873?
a = 1.259921049894873?
minpoly: x^3 - 2
x - a = x + 0.6299605249474365? - 1.091123635971722?*I
a = -0.6299605249474365? + 1.091123635971722?*I
minpoly: x^3 - 2
x - a = x + 0.6299605249474365? + 1.091123635971722?*I
a = -0.6299605249474365? - 1.091123635971722?*I
minpoly: x^3 - 2


and

sage: for q in (1/ff).partial_fraction_decomposition()[1]:
...       print 'a / (x - b) =', q
...       a = QQbar(q.numerator())
...       print '  a =', a
...       print '  minpoly:', a.minpoly()
...       b = QQbar(x - q.denominator())
...       print '  b =', b
...       print '  minpoly:', b.minpoly()
a / (x - b) = (0.2099868416491456? + 0.?e-19*I)/(x - 1.259921049894873?)
a = 0.2099868416491456? + 0.?e-19*I
minpoly: x^3 - 1/108
b = 1.259921049894873?
minpoly: x^3 - 2
a / (x - b) = (-0.10499342082457277? + 0.1818539393286203?*I)/(x + 0.6299605249474365? - 1.091123635971722?*I)
a = -0.10499342082457277? + 0.1818539393286203?*I
minpoly: x^3 - 1/108
b = -0.6299605249474365? + 1.091123635971722?*I
minpoly: x^3 - 2
a / (x - b) = (-0.10499342082457277? - 0.1818539393286203?*I)/(x + 0.6299605249474365? + 1.091123635971722?*I)
a = -0.10499342082457277? - 0.1818539393286203?*I
minpoly: x^3 - 1/108
b = -0.6299605249474365? - 1.091123635971722?*I
minpoly: x^3 - 2


Expanding on Snark's comment: use algebraic numbers to get an exact answer.

Algebraic numbers can be explored, you can ask their minimal polynomial or a radical expression.

sage: R.<x> = QQbar['x']
sage: ff = x^3 - 2
sage: factor(ff)
(x - 1.259921049894873?) * (x + 0.6299605249474365? - 1.091123635971722?*I) * (x + 0.6299605249474365? + 1.091123635971722?*I)
sage: (1/ff).partial_fraction_decomposition()
(0,
[(0.2099868416491456? + 0.?e-19*I)/(x - 1.259921049894873?),
(-0.10499342082457277? + 0.1818539393286203?*I)/(x + 0.6299605249474365? - 1.091123635971722?*I),
(-0.10499342082457277? - 0.1818539393286203?*I)/(x + 0.6299605249474365? + 1.091123635971722?*I)])


The entries are exact; they live in QQbar. To access investigate the entries:

sage: for q in factor(ff):
...   ....:     q = q[0]
...   ....:     print 'x - a =', q
...   ....:     a = QQbar(x - q)
...   ....:     print '  a =', a
...   ....:     print '  minpoly:', a.minpoly()
....:
x - a = x - 1.259921049894873?
a = 1.259921049894873?
minpoly: x^3 - 2
x - a = x + 0.6299605249474365? - 1.091123635971722?*I
a = -0.6299605249474365? + 1.091123635971722?*I
minpoly: x^3 - 2
x - a = x + 0.6299605249474365? + 1.091123635971722?*I
a = -0.6299605249474365? - 1.091123635971722?*I
minpoly: x^3 - 2


and

sage: for q in (1/ff).partial_fraction_decomposition()[1]:
...   ....:     print 'a / (x - b) =', q
...   ....:     a = QQbar(q.numerator())
...   ....:     print '  a =', a
...   ....:     print '  minpoly:', a.minpoly()
....:     b = QQbar(x - q.denominator())
...   ....:     print '  b =', b
...   ....:     print '  minpoly:', b.minpoly()
....:
a / (x - b) = (0.2099868416491456? + 0.?e-19*I)/(x 0.2099868416491456?/(x - 1.259921049894873?)
a = 0.2099868416491456? + 0.?e-19*I
0.2099868416491456?
minpoly: x^3 - 1/108
b = 1.259921049894873?
minpoly: x^3 - 2
a / (x - b) = (-0.10499342082457277? + 0.1818539393286203?*I)/(x + 0.6299605249474365? - 1.091123635971722?*I)
a = -0.10499342082457277? + 0.1818539393286203?*I
minpoly: x^3 - 1/108