def test_roots_binomial():
    assert roots_binomial(Poly(5*x, x)) == [0]
    assert roots_binomial(Poly(5*x**4, x)) == [0, 0, 0, 0]
    assert roots_binomial(Poly(5*x + 2, x)) == [-Rational(2, 5)]

    A = 10**Rational(3, 4)/10

    assert roots_binomial(Poly(5*x**4 + 2, x)) == \
        [-A - A*I, -A + A*I, A - A*I, A + A*I]
    _check(roots_binomial(Poly(x**8 - 2)))

    a1 = Symbol('a1', nonnegative=True)
    b1 = Symbol('b1', nonnegative=True)

    r0 = roots_quadratic(Poly(a1*x**2 + b1, x))
    r1 = roots_binomial(Poly(a1*x**2 + b1, x))

    assert powsimp(r0[0]) == powsimp(r1[0])
    assert powsimp(r0[1]) == powsimp(r1[1])
    for a, b, s, n in cartes((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
        if a == b and a != 1:  # a == b == 1 is sufficient
            continue
        p = Poly(a*x**n + s*b)
        ans = roots_binomial(p)
        assert ans == _nsort(ans)

    # issue 8813
    assert roots(Poly(2*x**3 - 16*y**3, x)) == {
        2*y*(-S(1)/2 - sqrt(3)*I/2): 1,
        2*y: 1,
        2*y*(-S(1)/2 + sqrt(3)*I/2): 1}

    def _roots_trivial(cls, poly, radicals):
        """Compute roots in linear, quadratic and binomial cases. """
        if poly.degree() == 1:
            return roots_linear(poly)

        if not radicals:
            return None

        if poly.degree() == 2:
            return roots_quadratic(poly)
        elif poly.length() == 2 and poly.TC():
            return roots_binomial(poly)
        else:
            return None

def roots_trivial(poly, radicals=True):
    """Compute roots in linear, quadratic and binomial cases. """
    if poly.degree() == 1:
        return roots_linear(poly)
    else:
        if not radicals:
            return None

        if poly in _rootof_trivial_cache:
            roots = _rootof_trivial_cache[poly]
        else:
            if radicals and poly.degree() == 2:
                roots = roots_quadratic(poly)
            elif radicals and poly.length() == 2 and poly.TC():
                roots = roots_binomial(poly)
            else:
                return None

            _rootof_trivial_cache[poly] = roots

        return roots

    def _roots_trivial(cls, poly, radicals):
        """Compute roots in linear, quadratic and binomial cases. """
        if poly.degree() == 1:
            return roots_linear(poly)

        if not radicals:
            return None
        free = len(poly.free_symbols)
        sort = isinstance(poly, PurePoly) and free or free > 1
        if poly.degree() == 2:
            roots = roots_quadratic(poly, sort=sort)
        elif poly.length() == 2 and poly.TC():
            roots = roots_binomial(poly, sort=sort)
        else:
            return None
        # put roots in same order as RootOf instances
        if not sort:
            key = [r.n(2) for r in roots]
            key = [(1 if not r.is_real else 0, C.re(r), C.im(r))
                for r in key]
            _, roots = zip(*sorted(zip(key, roots)))
        return roots

def test_roots_binomial():
    assert roots_binomial(Poly(5 * x, x)) == [0]
    assert roots_binomial(Poly(5 * x ** 4, x)) == [0, 0, 0, 0]
    assert roots_binomial(Poly(5 * x + 2, x)) == [-Rational(2, 5)]

    A = 10 ** Rational(3, 4) / 10

    assert roots_binomial(Poly(5 * x ** 4 + 2, x)) == [-A - A * I, -A + A * I, A - A * I, A + A * I]

    a1 = Symbol("a1", nonnegative=True)
    b1 = Symbol("b1", nonnegative=True)

    r0 = roots_quadratic(Poly(a1 * x ** 2 + b1, x))
    r1 = roots_binomial(Poly(a1 * x ** 2 + b1, x))

    assert powsimp(r0[0]) == powsimp(r1[0])
    assert powsimp(r0[1]) == powsimp(r1[1])
    for a, b, s, n in cartes((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
        if a == b and a != 1:  # a == b == 1 is sufficient
            continue
        p = Poly(a * x ** n + s * b)
        roots = roots_binomial(p)
        assert roots == _nsort(roots)