Package sage :: Package rings :: Module polynomial :: Class Polynomial_rational_dense
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Type Polynomial_rational_dense

object --+        
         |        
   Element --+    
             |    
    Polynomial --+
                 |
                Polynomial_rational_dense


Method Summary
  __init__(self, parent, x, check, is_gen)
  __add__(self, right)
  __getitem__(self, n)
  __getslice__(self, i, j)
  __mul__(self, right)
EXAMPLES:
  __reduce__(self)
  __setitem__(self, n, value)
  __sub__(self, right)
The difference self - right.
  copy(self)
Return a copy of self.
  degree(self)
Return the degree of this polynomial.
  discriminant(self)
  list(self)
  resultant(self, other)
    Inherited from Polynomial
  __call__(self, x)
Compute value of this polynomial at x.
  __cmp__(self, other)
  __div__(self, other)
Division with remainder.
  __float__(self)
  __floordiv__(self, right)
Quotient of division of self by other.
  __hash__(self)
  __int__(self)
  __long__(self)
  __mod__(self, other)
Remainder of division of self by other.
  __neg__(self)
  __pos__(self)
  __pow__(self, right)
  __radd__(self, left)
  __rdiv__(self, left)
  __repr__(self)
  __rmul__(self, left)
  __rsub__(self, left)
  base_ring(self)
Return the base ring of the parent of self.
  complex_roots(self, flag)
Returns the complex roots of this polynomial.
  denominator(self)
Return the least common multiple of the denominators of the entries of self, when this makes sense, i.e., when the coefficients have a denominator function.
  derivative(self)
  dict(self)
  factor(self)
Return polynomials f1, ..., fn and exponents e1, ..., en such that the gcd fo the coefficients of the fi is 1, and prod fi**ei is equal to a scalar multiple of self.
  gcd(self, other)
Greatest common divisor of self and polynomial other.
  is_gen(self)
  is_irreducible(self)
  is_monic(self)
Returns True if this polynomial is monic.
  is_nonzero(self)
  is_zero(self)
  leading(self)
  pari(self)
  polynomial(self, *args, **kwds)
  reverse(self)
  valuation(self)
If f = a_r x^r + a_{r+1}x^{r+1} + higher terms ..., with a_r nonzero then the valuation of f is r.
  variable(self)
  xgcd(self, other)
Extended gcd of self and polynomial other.
    Inherited from object
  __delattr__(...)
x.__delattr__('name') <==> del x.name
  __getattribute__(...)
x.__getattribute__('name') <==> x.name
  __reduce_ex__(...)
helper for pickle
  __setattr__(...)
x.__setattr__('name', value) <==> x.name = value
  __str__(x)
x.__str__() <==> str(x)

Method Details

__mul__(self, right)

EXAMPLES:
>>> import rings
>>> Q = rings.RationalField()
>>> x = rings.PolynomialRing(Q).gen()
>>> (x - '2/3')*(x**2 - 8*x + 16)
x^3 - 26/3*x^2 + 64/3*x - 32/3
Overrides:
sage.rings.polynomial.Polynomial.__mul__

__sub__(self, right)
(Subtraction operator)

The difference self - right.

EXAMPLES:
>>> import rings
>>> Q = rings.RationalField()
>>> x = rings.PolynomialRing(Q).gen()
>>> x - (x+1)
-1
Overrides:
sage.rings.polynomial.Polynomial.__sub__ (inherited documentation)

copy(self)

Return a copy of self.

EXAMPLES: We create the polynomial f=x+3, then set g=f, and change the coefficient of x in g, which also changes the coefficient of x in f. If we instead copy f, then changing the coefficient of x of g does not change f.
>>> x = PolynomialRing(IntegerRing()).gen()
>>> f = x+3
>>> g = f
>>> g[1]=3
>>> f
3*x + 3

>>> g = f.copy()
>>> g[1]=5
>>> f
3*x + 3

>>> g
5*x + 3
Overrides:
sage.rings.polynomial.Polynomial.copy (inherited documentation)

degree(self)

Return the degree of this polynomial. The zero polynomial has degree -1. EXAMPLES:
>>> x = PolynomialRing(IntegerRing()).gen()
>>> f = x**93 + 2*x + 1
>>> f.degree()
93

>>> x = PolynomialRing(RationalField(), sparse=True).gen()
>>> f = x**100000 
>>> f.degree()
100000
Overrides:
sage.rings.polynomial.Polynomial.degree (inherited documentation)

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