(f, g, K)
return a, f
Half extended Euclidean algorithm in `F[x]`.

    Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x = ring("x", QQ)

    >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
    >>> g = x**3 + x**2 - 4*x - 4

    >>> R.dup_half_gcdex(f, g)
    (-1/5*x + 3/5, x + 1)
Half extended Euclidean algorithm in `F[x]`.
Half extended Euclidean algorithm in F [ x ] .
def dup_half_gcdex(f, g, K):
    """
    Half extended Euclidean algorithm in `F[x]`.

    Returns ``(s, h)`` such that ``h = gcd(f, g)`` and ``s*f = h (mod g)``.

    Examples
    ========

    >>> from sympy.polys import ring, QQ
    >>> R, x = ring("x", QQ)

    >>> f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
    >>> g = x**3 + x**2 - 4*x - 4

    >>> R.dup_half_gcdex(f, g)
    (-1/5*x + 3/5, x + 1)

    """
    if not K.has_Field:
        raise DomainError("can't compute half extended GCD over %s" % K)

    a, b = [K.one], []

    while g:
        q, r = dup_div(f, g, K)
        f, g = g, r
        a, b = b, dup_sub_mul(a, q, b, K)

    a = dup_quo_ground(a, dup_LC(f, K), K)
    f = dup_monic(f, K)

    return a, f
def dup_half_gcdex ( f , g , K ) : if not K . has_Field : raise DomainError ( "can't compute half extended GCD over %s" % K ) a , b = [ K . one ] , [ ] while g : q , r = dup_div ( f , g , K ) f , g = g , r a , b = b , dup_sub_mul ( a , q , b , K ) a = dup_quo_ground ( a , dup_LC ( f , K ) , K ) f = dup_monic ( f , K ) return a , f