Extended Euclidean algorithm for polynomials
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gcdex(p
,q
, <x
>) gcdex(f
,g
,x
)
gcdex(p, q, x)
regards p
and q
as
univariate polynomials in x
and returns their greatest
common divisor as a linear combination of p
and q
.
gcdex(p, q, x)
returns a sequence g, s, t with
three elements, where the polynomial g is
the greatest common divisor of p and q.
The polynomials s and t satisfy g = s p + t q and deg(s)
< deg(q), deg(t)
< deg(p).
These data are computed by the extended Euclidean algorithm.
gcdex
only processes univariate polynomials:
If the indeterminate x
is specified,
the input polynomials are regarded as univariate polynomials in x
.
If no indeterminate is specified, the indeterminate of the polynomials is searched for internally. An error occurs if more than one indeterminate is found.
Note that x
must be specified if polynomial
expressions are used on input.
Polynomial expressions are converted to polynomials. See poly
for details. FAIL
is
returned if an argument cannot be converted to a polynomial.
The returned polynomials are polynomial expressions if the input
consists of polynomial expressions. Otherwise, polynomials of type DOM_POLY
are
returned.
The coefficient ring of the polynomials must provide the method "_divide"
.
This method must return FAIL
if domain elements cannot be divided.
Note: If the coefficient domain of the polynomial is not a field, then it may not be possible to represent a greatest common divisor as a linear combination of the input polynomials. In such a case, an error is raised. 
The greatest common divisor of two univariate polynomials in extended form can be computed as follows:
gcdex(poly(x^3 + 1), poly(x^2 + 2*x + 1))
For multivariate polynomials, an indeterminate must be specified:
gcdex(poly(x^2*y), poly(x + y), x)
gcdex(poly(x^2*y), poly(x + y), y)
gcdex(x^3 + a, x^2 + 1, x)

polynomials of
type 
 

An indeterminate: an identifier or an indexed identifier 
Sequence of three polynomials, or a sequence of three polynomial
expressions, or FAIL
.
p
, q