def eval(cls, x, k):
        x = sympify(x)
        k = sympify(k)

        if x is S.NaN or k is S.NaN:
            return S.NaN
        elif k.is_integer and x == k:
            return factorial(x)
        elif k.is_Integer:
            if k is S.Zero:
                return S.One
            else:
                if k.is_positive:
                    if x is S.Infinity:
                        return S.Infinity
                    elif x is S.NegativeInfinity:
                        if k.is_odd:
                            return S.NegativeInfinity
                        else:
                            return S.Infinity
                    else:
                        try:
                            F, opt = poly_from_expr(x)
                        except PolificationFailed:
                            return reduce(lambda r, i: r*(x - i),
                                          range(0, int(k)), 1)
                        if len(opt.gens) > 1 or F.degree() <= 1:
                            return reduce(lambda r, i: r*(x - i),
                                          range(0, int(k)), 1)
                        else:
                            v = opt.gens[0]
                            return reduce(lambda r, i:
                                          r*(F.subs(v, v - i).expand()),
                                          range(0, int(k)), 1)
                else:
                    if x is S.Infinity:
                        return S.Infinity
                    elif x is S.NegativeInfinity:
                        return S.Infinity
                    else:
                        try:
                            F, opt = poly_from_expr(x)
                        except PolificationFailed:
                            return 1/reduce(lambda r, i: r*(x + i),
                                            range(1, abs(int(k)) + 1), 1)
                        if len(opt.gens) > 1 or F.degree() <= 1:
                            return 1/reduce(lambda r, i: r*(x + i),
                                            range(1, abs(int(k)) + 1), 1)
                        else:
                            v = opt.gens[0]
                            return 1/reduce(lambda r, i:
                                            r*(F.subs(v, v + i).expand()),
                                            range(1, abs(int(k)) + 1), 1)

def horner(f, *gens, **args):
    """
    Rewrite a polynomial in Horner form.

    Examples
    ========

    >>> from sympy.polys.polyfuncs import horner
    >>> from sympy.abc import x, y, a, b, c, d, e

    >>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
    x*(x*(x*(9*x + 8) + 7) + 6) + 5

    >>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
    e + x*(d + x*(c + x*(a*x + b)))

    >>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y

    >>> horner(f, wrt=x)
    x*(x*y*(4*y + 2) + y*(2*y + 1))

    >>> horner(f, wrt=y)
    y*(x*y*(4*x + 2) + x*(2*x + 1))

    """
    allowed_flags(args, [])

    try:
        F, opt = poly_from_expr(f, *gens, **args)
    except PolificationFailed, exc:
        return exc.expr

def horner(f, *gens, **args):
    """
    Rewrite a polynomial in Horner form.

    **Examples**

    >>> from sympy.polys.polyfuncs import horner
    >>> from sympy.abc import x, y, a, b, c, d, e

    >>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
    5 + x*(6 + x*(7 + x*(8 + 9*x)))

    >>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
    e + x*(d + x*(c + x*(b + a*x)))

    >>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y

    >>> horner(f, wrt=x)
    x*(y*(1 + 2*y) + x*y*(2 + 4*y))

    >>> horner(f, wrt=y)
    y*(x*(1 + 2*x) + x*y*(2 + 4*x))

    """
    allowed_flags(args, [])

    try:
        F, opt = poly_from_expr(f, *gens, **args)
    except PolificationFailed, exc:
        return exc.expr

    def get_dixon_polynomial(self):
        r"""
        Returns
        =======

        dixon_polynomial: polynomial
            Dixon's polynomial is calculated as:

            delta = Delta(A) / ((x_1 - a_1) ... (x_n - a_n)) where,

            A =  |p_1(x_1,... x_n), ..., p_n(x_1,... x_n)|
                 |p_1(a_1,... x_n), ..., p_n(a_1,... x_n)|
                 |...             , ...,              ...|
                 |p_1(a_1,... a_n), ..., p_n(a_1,... a_n)|
        """
        if self.m != (self.n + 1):
            raise ValueError('Method invalid for given combination.')

        # First row
        rows = [self.polynomials]

        temp = list(self.variables)

        for idx in range(self.n):
            temp[idx] = self.dummy_variables[idx]
            substitution = {var: t for var, t in zip(self.variables, temp)}
            rows.append([f.subs(substitution) for f in self.polynomials])

        A = Matrix(rows)

        terms = zip(self.variables, self.dummy_variables)
        product_of_differences = Mul(*[a - b for a, b in terms])
        dixon_polynomial = (A.det() / product_of_differences).factor()

        return poly_from_expr(dixon_polynomial, self.dummy_variables)[0]

def _muly(p, x, y):
    """
    Returns ``_mexpand(y**deg*p.subs({x:x / y}))``
    """
    p1 = poly_from_expr(p, x)[0]

    n = degree(p1)
    a = [c * x**i * y**(n - i) for (i,), c in p1.terms()]
    return Add(*a)

def _invertx(p, x):
    """
    Returns ``expand_mul(x**degree(p, x)*p.subs(x, 1/x))``
    """
    p1 = poly_from_expr(p, x)[0]

    n = degree(p1)
    a = [c * x**(n - i) for (i,), c in p1.terms()]
    return Add(*a)

def _invertx(p, x):
    """
    Returns ``expand_mul(x**degree(p, x)*p.subs(x, 1/x))``
    """
    p1 = poly_from_expr(p, x)[0]

    terms = p1.terms()
    n = terms[0][0][0]
    a = [c*x**(n - i[0]) for i, c in terms]
    return Add(*a)

def viete(f, roots=None, *gens, **args):
    """
    Generate Viete's formulas for ``f``.

    Examples
    ========

    >>> from sympy.polys.polyfuncs import viete
    >>> from sympy import symbols

    >>> x, a, b, c, r1, r2 = symbols('x,a:c,r1:3')

    >>> viete(a*x**2 + b*x + c, [r1, r2], x)
    [(r1 + r2, -b/a), (r1*r2, c/a)]

    """
    allowed_flags(args, [])

    if isinstance(roots, Basic):
        gens, roots = (roots,) + gens, None

    try:
        f, opt = poly_from_expr(f, *gens, **args)
    except PolificationFailed as exc:
        raise ComputationFailed('viete', 1, exc)

    if f.is_multivariate:
        raise MultivariatePolynomialError(
            "multivariate polynomials are not allowed")

    n = f.degree()

    if n < 1:
        raise ValueError(
            "can't derive Viete's formulas for a constant polynomial")

    if roots is None:
        roots = numbered_symbols('r', start=1)

    roots = take(roots, n)

    if n != len(roots):
        raise ValueError("required %s roots, got %s" % (n, len(roots)))

    lc, coeffs = f.LC(), f.all_coeffs()
    result, sign = [], -1

    for i, coeff in enumerate(coeffs[1:]):
        poly = symmetric_poly(i + 1, roots)
        coeff = sign*(coeff/lc)
        result.append((poly, coeff))
        sign = -sign

    return result

def horner(f, *gens, **args):
    """
    Rewrite a polynomial in Horner form.

    Among other applications, evaluation of a polynomial at a point is optimal
    when it is applied using the Horner scheme ([1]).

    Examples
    ========

    >>> from sympy.polys.polyfuncs import horner
    >>> from sympy.abc import x, y, a, b, c, d, e

    >>> horner(9*x**4 + 8*x**3 + 7*x**2 + 6*x + 5)
    x*(x*(x*(9*x + 8) + 7) + 6) + 5

    >>> horner(a*x**4 + b*x**3 + c*x**2 + d*x + e)
    e + x*(d + x*(c + x*(a*x + b)))

    >>> f = 4*x**2*y**2 + 2*x**2*y + 2*x*y**2 + x*y

    >>> horner(f, wrt=x)
    x*(x*y*(4*y + 2) + y*(2*y + 1))

    >>> horner(f, wrt=y)
    y*(x*y*(4*x + 2) + x*(2*x + 1))

    References
    ==========
    [1] - http://en.wikipedia.org/wiki/Horner_scheme

    """
    allowed_flags(args, [])

    try:
        F, opt = poly_from_expr(f, *gens, **args)
    except PolificationFailed as exc:
        return exc.expr

    form, gen = S.Zero, F.gen

    if F.is_univariate:
        for coeff in F.all_coeffs():
            form = form*gen + coeff
    else:
        F, gens = Poly(F, gen), gens[1:]

        for coeff in F.all_coeffs():
            form = form*gen + horner(coeff, *gens, **args)

    return form