# Source code for sympy.polys.polyroots

"""Algorithms for computing symbolic roots of polynomials. """

import math

from sympy.core.symbol import Dummy, Symbol, symbols
from sympy.core import S, I, pi
from sympy.core.mul import expand_2arg
from sympy.core.sympify import sympify
from sympy.core.numbers import Rational, igcd

from sympy.ntheory import divisors, isprime, nextprime
from sympy.functions import exp, sqrt, re, im, Abs, cos, sin

from sympy.polys.polytools import Poly, cancel, factor, gcd_list, discriminant
from sympy.polys.specialpolys import cyclotomic_poly
from sympy.polys.polyerrors import PolynomialError, GeneratorsNeeded, DomainError
from sympy.polys.polyquinticconst import PolyQuintic
from sympy.polys.rationaltools import together

from sympy.simplify import simplify, powsimp
from sympy.utilities import default_sort_key

from sympy.core.compatibility import reduce

def roots_linear(f):
"""Returns a list of roots of a linear polynomial."""
r = -f.nth(0)/f.nth(1)
dom = f.get_domain()

if not dom.is_Numerical:
if dom.is_Composite:
r = factor(r)
else:
r = simplify(r)

return [r]

def roots_quadratic(f):
"""Returns a list of roots of a quadratic polynomial."""
a, b, c = f.all_coeffs()
dom = f.get_domain()

def _simplify(expr):
if dom.is_Composite:
return factor(expr)
else:
return simplify(expr)

if c is S.Zero:
r0, r1 = S.Zero, -b/a

if not dom.is_Numerical:
r1 = _simplify(r1)
elif b is S.Zero:
r = -c/a

if not dom.is_Numerical:
R = sqrt(_simplify(r))
else:
R = sqrt(r)

r0 = R
r1 = -R
else:
d = b**2 - 4*a*c

if dom.is_Numerical:
D = sqrt(d)

r0 = (-b + D) / (2*a)
r1 = (-b - D) / (2*a)
else:
D = sqrt(_simplify(d))
A = 2*a

E = _simplify(-b/A)
F = D/A

r0 = E + F
r1 = E - F

return sorted([expand_2arg(i) for i in (r0, r1)], key=default_sort_key)

def roots_cubic(f):
"""Returns a list of roots of a cubic polynomial."""
_, a, b, c = f.monic().all_coeffs()

if c is S.Zero:
x1, x2 = roots([1, a, b], multiple=True)
return [x1, S.Zero, x2]

p = b - a**2/3
q = c - a*b/3 + 2*a**3/27

pon3 = p/3
aon3 = a/3

if p is S.Zero:
if q is S.Zero:
return [-aon3]*3
else:
if q.is_real:
if q > 0:
u1 = -q**Rational(1, 3)
else:
u1 = (-q)**Rational(1, 3)
else:
u1 = (-q)**Rational(1, 3)
elif q is S.Zero:
y1, y2 = roots([1, 0, p], multiple=True)
return [tmp - aon3 for tmp in [y1, S.Zero, y2]]
else:
u1 = (q/2 + sqrt(q**2/4 + pon3**3))**Rational(1, 3)

coeff = S.ImaginaryUnit*sqrt(3)/2

u2 = u1*(-S.Half + coeff)
u3 = u1*(-S.Half - coeff)

if p is S.Zero:
return [u1 - aon3, u2 - aon3, u3 - aon3]

soln = [
-u1 + pon3/u1 - aon3,
-u2 + pon3/u2 - aon3,
-u3 + pon3/u3 - aon3
]

return soln

def _roots_quartic_euler(p, q, r, a):
"""
Descartes-Euler solution of the quartic equation

Parameters
==========

p, q, r: coefficients of x**4 + p*x**2 + q*x + r
a: shift of the roots

Notes
=====

This is a helper function for roots_quartic.

Look for solutions of the form ::

x1 = sqrt(R) - sqrt(A + B*sqrt(R))
x2 = -sqrt(R) - sqrt(A - B*sqrt(R))
x3 = -sqrt(R) + sqrt(A - B*sqrt(R))
x4 = sqrt(R) + sqrt(A + B*sqrt(R))

To satisfy the quartic equation one must have
p = -2*(R + A); q = -4*B*R; r = (R - A)**2 - B**2*R
so that R must satisfy the Descartes-Euler resolvent equation
64*R**3 + 32*p*R**2 + (4*p**2 - 16*r)*R - q**2 = 0

If the resolvent does not have a rational solution, return None;
in that case it is likely that the Ferrari method gives a simpler
solution.

Examples
========

>>> from sympy import S
>>> from sympy.polys.polyroots import _roots_quartic_euler
>>> p, q, r = -S(64)/5, -S(512)/125, -S(1024)/3125
>>> _roots_quartic_euler(p, q, r, S(0))[0]
-sqrt(32*sqrt(5)/125 + 16/5) + 4*sqrt(5)/5
"""
from sympy.solvers import solve
# solve the resolvent equation
x = Symbol('x')
eq = 64*x**3 + 32*p*x**2 + (4*p**2 - 16*r)*x - q**2
xsols = roots(Poly(eq, x), cubics=False).keys()
xsols = [sol for sol in xsols if sol.is_rational]
if not xsols:
return None
R = max(xsols)
c1 = sqrt(R)
B = -q*c1/(4*R)
A = -R - p/2
c2 = sqrt(A + B)
c3 = sqrt(A - B)
return [c1 - c2 - a, -c1 - c3 - a, -c1 + c3 - a, c1 + c2 - a]

def roots_quartic(f):
r"""
Returns a list of roots of a quartic polynomial.

There are many references for solving quartic expressions available [1-5].
This reviewer has found that many of them require one to select from among
2 or more possible sets of solutions and that some solutions work when one
is searching for real roots but don't work when searching for complex roots
(though this is not always stated clearly). The following routine has been
tested and found to be correct for 0, 2 or 4 complex roots.

The quasisymmetric case solution [6] looks for quartics that have the form
x**4 + A*x**3 + B*x**2 + C*x + D = 0 where (C/A)**2 = D.

Although there is a general solution, simpler results can be obtained for
certain values of the coefficients. In all cases, 4 roots are returned:

1) f = c + a*(a**2/8 - b/2) == 0
2) g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0
3) if f != 0 and g != 0 and p = -d + a*c/4 - b**2/12 then
a) p == 0
b) p != 0

Examples
========

>>> from sympy import Poly, symbols, I
>>> from sympy.polys.polyroots import roots_quartic

>>> r = roots_quartic(Poly('x**4-6*x**3+17*x**2-26*x+20'))

>>> # 4 complex roots: 1+-I*sqrt(3), 2+-I
>>> sorted(str(tmp.evalf(n=2)) for tmp in r)
['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + 1.0*I', '2.0 - 1.0*I']

References
==========

1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html
2. http://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method
3. http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html
4. http://staff.bath.ac.uk/masjhd/JHD-CA.pdf
5. http://www.albmath.org/files/Math_5713.pdf
6. http://www.statemaster.com/encyclopedia/Quartic-equation
7. eqworld.ipmnet.ru/en/solutions/ae/ae0108.pdf
"""
_, a, b, c, d = f.monic().all_coeffs()

if not d:
return [S.Zero] + roots([1, a, b, c], multiple=True)
elif (c/a)**2 == d:
x, m = f.gen, c/a

g = Poly(x**2 + a*x + b - 2*m, x)

z1, z2 = roots_quadratic(g)

h1 = Poly(x**2 - z1*x + m, x)
h2 = Poly(x**2 - z2*x + m, x)

r1 = roots_quadratic(h1)
r2 = roots_quadratic(h2)

return r1 + r2
else:
a2 = a**2
e = b - 3*a2/8
f = c + a*(a2/8 - b/2)
g = d - a*(a*(3*a2/256 - b/16) + c/4)
aon4 = a/4

if f is S.Zero:
y1, y2 = [sqrt(tmp) for tmp in
roots([1, e, g], multiple=True)]
return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]]
if g is S.Zero:
y = [S.Zero] + roots([1, 0, e, f], multiple=True)
return [tmp - aon4 for tmp in y]
else:
# Descartes-Euler method, see [7]
sols = _roots_quartic_euler(e, f, g, aon4)
if sols:
return sols
# Ferrari method, see [1, 2]
a2 = a**2
e = b - 3*a2/8
f = c + a*(a2/8 - b/2)
g = d - a*(a*(3*a2/256 - b/16) + c/4)
p = -e**2/12 - g
q = -e**3/108 + e*g/3 - f**2/8
TH = Rational(1, 3)
if p.is_zero:
y = -5*e/6 - q**TH
elif p.is_number and p.is_comparable:
# with p != 0 then u below is not 0
root = sqrt(q**2/4 + p**3/27)
r = -q/2 + root  # or -q/2 - root
u = r**TH  # primary root of solve(x**3-r, x)
y = -5*e/6 + u - p/u/3
else:
raise PolynomialError('cannot return general quartic solution')
w = sqrt(e + 2*y)
arg1 = 3*e + 2*y
arg2 = 2*f/w
ans = []
for s in [-1, 1]:
root = sqrt(-(arg1 + s*arg2))
for t in [-1, 1]:
ans.append((s*w - t*root)/2 - aon4)

return ans

def roots_binomial(f):
"""Returns a list of roots of a binomial polynomial."""
n = f.degree()

a, b = f.nth(n), f.nth(0)
alpha = (-cancel(b/a))**Rational(1, n)

if alpha.is_number:
alpha = alpha.expand(complex=True)

roots, I = [], S.ImaginaryUnit

for k in xrange(n):
zeta = exp(2*k*S.Pi*I/n).expand(complex=True)
roots.append((alpha*zeta).expand(power_base=False))

return sorted(roots, key=default_sort_key)

def _inv_totient_estimate(m):
"""
Find (L, U) such that L <= phi^-1(m) <= U.

Examples
========

>>> from sympy.polys.polyroots import _inv_totient_estimate

>>> _inv_totient_estimate(192)
(192, 840)
>>> _inv_totient_estimate(400)
(400, 1750)

"""
primes = [ d + 1 for d in divisors(m) if isprime(d + 1) ]

a, b = 1, 1

for p in primes:
a *= p
b *= p - 1

L = m
U = int(math.ceil(m*(float(a)/b)))

P = p = 2
primes = []

while P <= U:
p = nextprime(p)
primes.append(p)
P *= p

P //= p
b = 1

for p in primes[:-1]:
b *= p - 1

U = int(math.ceil(m*(float(P)/b)))

return L, U

def roots_cyclotomic(f, factor=False):
"""Compute roots of cyclotomic polynomials. """
L, U = _inv_totient_estimate(f.degree())

for n in xrange(L, U + 1):
g = cyclotomic_poly(n, f.gen, polys=True)

if f == g:
break
else:  # pragma: no cover
raise RuntimeError("failed to find index of a cyclotomic polynomial")

roots = []

if not factor:
for k in xrange(1, n + 1):
if igcd(k, n) == 1:
roots.append(exp(2*k*S.Pi*I/n).expand(complex=True))
else:
g = Poly(f, extension=(-1)**Rational(1, n))

for h, _ in g.factor_list()[1]:
roots.append(-h.TC())

return sorted(roots, key=default_sort_key)

def roots_quintic(f):
"""
Calulate exact roots of a solvable quintic
"""
result = []
coeff_5, coeff_4, p, q, r, s = f.all_coeffs()

# Eqn must be of the form x^5 + px^3 + qx^2 + rx + s
if coeff_4:
return result

if coeff_5 != 1:
l = [p/coeff_5, q/coeff_5, r/coeff_5, s/coeff_5]
if not all(coeff.is_Rational for coeff in l):
return result
f = Poly(f/coeff_5)
quintic = PolyQuintic(f)

# Eqn standardised. Algo for solving starts here
if not f.is_irreducible:
return result

f20 = quintic.f20
# Check if f20 has linear factors over domain Z
if f20.is_irreducible:
return result

# Now, we know that f is solvable
for _factor in f20.factor_list()[1]:
if _factor[0].is_linear:
theta = _factor[0].root(0)
break
d = discriminant(f)
delta = sqrt(d)
# zeta = a fifth root of unity
zeta1, zeta2, zeta3, zeta4 = quintic.zeta
T = quintic.T(theta, d)
tol = S(1e-10)
alpha = T[1] + T[2]*delta
alpha_bar = T[1] - T[2]*delta
beta = T[3] + T[4]*delta
beta_bar = T[3] - T[4]*delta

disc = alpha**2 - 4*beta
disc_bar = alpha_bar**2 - 4*beta_bar

l0 = quintic.l0(theta)

l1 = _quintic_simplify((-alpha + sqrt(disc)) / S(2))
l4 = _quintic_simplify((-alpha - sqrt(disc)) / S(2))

l2 = _quintic_simplify((-alpha_bar + sqrt(disc_bar)) / S(2))
l3 = _quintic_simplify((-alpha_bar - sqrt(disc_bar)) / S(2))

order = quintic.order(theta, d)
test = (order*delta.n()) - ( (l1.n() - l4.n())*(l2.n() - l3.n()) )
# Comparing floats
# Problems importing on top
from sympy.utilities.randtest import comp
if not comp(test, 0, tol):
l2, l3 = l3, l2

# Now we have correct order of l's
R1 = l0 + l1*zeta1 + l2*zeta2 + l3*zeta3 + l4*zeta4
R2 = l0 + l3*zeta1 + l1*zeta2 + l4*zeta3 + l2*zeta4
R3 = l0 + l2*zeta1 + l4*zeta2 + l1*zeta3 + l3*zeta4
R4 = l0 + l4*zeta1 + l3*zeta2 + l2*zeta3 + l1*zeta4

Res = [None, [None]*5, [None]*5, [None]*5, [None]*5]
Res_n = [None, [None]*5, [None]*5, [None]*5, [None]*5]
sol = Symbol('sol')

# Simplifying improves performace a lot for exact expressions
R1 = _quintic_simplify(R1)
R2 = _quintic_simplify(R2)
R3 = _quintic_simplify(R3)
R4 = _quintic_simplify(R4)

# Solve imported here. Causing problems if imported as 'solve'
# and hence the changed name
from sympy.solvers.solvers import solve as _solve
a, b = symbols('a b', cls=Dummy)
_sol = _solve( sol**5 - a - I*b, sol)
for i in range(5):
_sol[i] = factor(_sol[i])
R1 = R1.as_real_imag()
R2 = R2.as_real_imag()
R3 = R3.as_real_imag()
R4 = R4.as_real_imag()

for i, root in enumerate(_sol):
Res[1][i] = _quintic_simplify(root.subs({ a: R1[0], b: R1[1] }))
Res[2][i] = _quintic_simplify(root.subs({ a: R2[0], b: R2[1] }))
Res[3][i] = _quintic_simplify(root.subs({ a: R3[0], b: R3[1] }))
Res[4][i] = _quintic_simplify(root.subs({ a: R4[0], b: R4[1] }))

for i in range(1, 5):
for j in range(5):
Res_n[i][j] = Res[i][j].n()
Res[i][j] = _quintic_simplify(Res[i][j])
r1 = Res[1][0]
r1_n = Res_n[1][0]

for i in range(5):
if comp(im(r1_n*Res_n[4][i]), 0, tol):
r4 = Res[4][i]
break

u, v = quintic.uv(theta, d)
sqrt5 = math.sqrt(5)

# Now we have various Res values. Each will be a list of five
# values. We have to pick one r value from those five for each Res
u, v = quintic.uv(theta, d)
testplus = (u + v*delta*sqrt(5)).n()
testminus = (u - v*delta*sqrt(5)).n()

# Evaluated numbers suffixed with _n
# We will use evaluated numbers for calculation. Much faster.
r4_n = r4.n()
r2 = r3 = None

for i in range(5):
r2temp_n = Res_n[2][i]
for j in range(5):
# Again storing away the exact number and using
# evaluated numbers in computations
r3temp_n = Res_n[3][j]

if( comp( r1_n*r2temp_n**2 + r4_n*r3temp_n**2 - testplus, 0, tol) and
comp( r3temp_n*r1_n**2 + r2temp_n*r4_n**2 - testminus, 0, tol ) ):
r2 = Res[2][i]
r3 = Res[3][j]
break
if r2:
break

# Now, we have r's so we can get roots
x1 = (r1 + r2 + r3 + r4)/5
x2 = (r1*zeta4 + r2*zeta3 + r3*zeta2 + r4*zeta1)/5
x3 = (r1*zeta3 + r2*zeta1 + r3*zeta4 + r4*zeta2)/5
x4 = (r1*zeta2 + r2*zeta4 + r3*zeta1 + r4*zeta3)/5
x5 = (r1*zeta1 + r2*zeta2 + r3*zeta3 + r4*zeta4)/5
result = [x1, x2, x3, x4, x5]

# Now check if solutions are distinct

result_n = []
for root in result:
result_n.append(root.n(5))
result_n = sorted(result_n)

prev_entry = None
for r in result_n:
if r == prev_entry:
# Roots are identical. Abort. Return []
# and fall back to usual solve
return []
prev_entry = r

return result

def _quintic_simplify(expr):
expr = powsimp(expr)
expr = cancel(expr)
return together(expr)

def _integer_basis(poly):
"""Compute coefficient basis for a polynomial over integers.

Returns the integer div such that substituting x = div*y
p(x) = m*q(y) where the coefficients of q are smaller
than those of p.

For example x**5 + 512*x + 1024 = 0
with div = 4 becomes y**5 + 2*y + 1 = 0

Returns the integer div or None if there is no possible scaling.

Examples
========

>>> from sympy.polys import Poly
>>> from sympy.abc import x
>>> from sympy.polys.polyroots import _integer_basis
>>> p = Poly(x**5 + 512*x + 1024, x, domain='ZZ')
>>> _integer_basis(p)
4
"""
monoms, coeffs = zip(*poly.terms())

monoms, = zip(*monoms)
coeffs = map(abs, coeffs)

if coeffs[0] < coeffs[-1]:
coeffs = list(reversed(coeffs))
n = monoms[0]
monoms = [n - i for i in reversed(monoms)]
else:
return None

monoms = monoms[:-1]
coeffs = coeffs[:-1]

divs = reversed(divisors(gcd_list(coeffs))[1:])

try:
div = divs.next()
except StopIteration:
return None

while True:
for monom, coeff in zip(monoms, coeffs):
if coeff % div**monom != 0:
try:
div = divs.next()
except StopIteration:
return None
else:
break
else:
return div

def preprocess_roots(poly):
"""Try to get rid of symbolic coefficients from poly. """
coeff = S.One

try:
_, poly = poly.clear_denoms(convert=True)
except DomainError:
return coeff, poly

poly = poly.primitive()[1]
poly = poly.retract()

if poly.get_domain().is_Poly and all(c.is_monomial for c in poly.rep.coeffs()):
poly = poly.inject()

strips = zip(*poly.monoms())
gens = list(poly.gens[1:])

base, strips = strips[0], strips[1:]

for gen, strip in zip(list(gens), strips):
reverse = False

if strip[0] < strip[-1]:
strip = reversed(strip)
reverse = True

ratio = None

for a, b in zip(base, strip):
if not a and not b:
continue
elif not a or not b:
break
elif b % a != 0:
break
else:
_ratio = b // a

if ratio is None:
ratio = _ratio
elif ratio != _ratio:
break
else:
if reverse:
ratio = -ratio

poly = poly.eval(gen, 1)
coeff *= gen**(-ratio)
gens.remove(gen)

if gens:
poly = poly.eject(*gens)

if poly.is_univariate and poly.get_domain().is_ZZ:
basis = _integer_basis(poly)

if basis is not None:
n = poly.degree()

def func(k, coeff):
return coeff//basis**(n - k[0])

poly = poly.termwise(func)
coeff *= basis

return coeff, poly

[docs]def roots(f, *gens, **flags):
"""
Computes symbolic roots of a univariate polynomial.

Given a univariate polynomial f with symbolic coefficients (or
a list of the polynomial's coefficients), returns a dictionary
with its roots and their multiplicities.

Only roots expressible via radicals will be returned.  To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set cubics=False or quartics=False respectively.

To get roots from a specific domain set the filter flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting filter='C').

By default a dictionary is returned giving a compact result in
case of multiple roots.  However to get a tuple containing all
those roots set the multiple flag to True.

Examples
========

>>> from sympy import Poly, roots
>>> from sympy.abc import x, y

>>> roots(x**2 - 1, x)
{-1: 1, 1: 1}

>>> p = Poly(x**2-1, x)
>>> roots(p)
{-1: 1, 1: 1}

>>> p = Poly(x**2-y, x, y)

>>> roots(Poly(p, x))
{-sqrt(y): 1, sqrt(y): 1}

>>> roots(x**2 - y, x)
{-sqrt(y): 1, sqrt(y): 1}

>>> roots([1, 0, -1])
{-1: 1, 1: 1}

"""
from sympy.polys.polytools import to_rational_coeffs
flags = dict(flags)

auto = flags.pop('auto', True)
cubics = flags.pop('cubics', True)
quartics = flags.pop('quartics', True)
quintics = flags.pop('quintics', False)
multiple = flags.pop('multiple', False)
filter = flags.pop('filter', None)
predicate = flags.pop('predicate', None)

if isinstance(f, list):
if gens:
raise ValueError('redundant generators given')

x = Dummy('x')

poly, i = {}, len(f) - 1

for coeff in f:
poly[i], i = sympify(coeff), i - 1

f = Poly(poly, x, field=True)
else:
try:
f = Poly(f, *gens, **flags)
except GeneratorsNeeded:
if multiple:
return []
else:
return {}

if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')

def _update_dict(result, root, k):
if root in result:
result[root] += k
else:
result[root] = k

def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []

for root in _try_heuristics(factors[0]):
roots.append(root)

for factor in factors[1:]:
previous, roots = list(roots), []

for root in previous:
g = factor - Poly(root, f.gen)

for root in _try_heuristics(g):
roots.append(root)

return roots

def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S(0)]*f.degree()

if f.length() == 2:
if f.degree() == 1:
return map(cancel, roots_linear(f))
else:
return roots_binomial(f)

result = []

for i in [-1, 1]:
if not f.eval(i):
f = f.quo(Poly(f.gen - i, f.gen))
result.append(i)
break

n = f.degree()

if n == 1:
result += map(cancel, roots_linear(f))
elif n == 2:
result += map(cancel, roots_quadratic(f))
elif f.is_cyclotomic:
result += roots_cyclotomic(f)
elif n == 3 and cubics:
result += roots_cubic(f)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)

return result

(k,), f = f.terms_gcd()

if not k:
zeros = {}
else:
zeros = {S(0): k}

coeff, f = preprocess_roots(f)

if auto and f.get_domain().has_Ring:
f = f.to_field()

rescale_x = None
translate_x = None

result = {}

if not f.is_ground:
if not f.get_domain().is_Exact:
for r in f.nroots():
_update_dict(result, r, 1)
elif f.degree() == 1:
result[roots_linear(f)[0]] = 1
elif f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, r, 1)
elif f.length() == 2:
for r in roots_binomial(f):
_update_dict(result, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()

if len(factors) == 1 and factors[0][1] == 1:
if f.get_domain().is_EX:
res = to_rational_coeffs(f)
if res:
if res[0] is None:
translate_x, f = res[2:]
else:
rescale_x, f = res[1], res[-1]
result = roots(f)
if not result:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for root in _try_decompose(f):
_update_dict(result, root, 1)
else:
for factor, k in factors:
for r in _try_heuristics(Poly(factor, f.gen, field=True)):
_update_dict(result, r, k)

if coeff is not S.One:
_result, result, = result, {}

for root, k in _result.iteritems():
result[coeff*root] = k

result.update(zeros)

if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: r.is_real,
'I': lambda r: r.is_imaginary,
}

try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)

for zero in dict(result).iterkeys():
if not query(zero):
del result[zero]

if predicate is not None:
for zero in dict(result).iterkeys():
if not predicate(zero):
del result[zero]
if rescale_x:
result1 = {}
for k, v in result.items():
result1[k*rescale_x] = v
result = result1
if translate_x:
result1 = {}
for k, v in result.items():
result1[k + translate_x] = v
result = result1

if not multiple:
return result
else:
zeros = []

for zero, k in result.iteritems():
zeros.extend([zero]*k)

return sorted(zeros, key=default_sort_key)

def root_factors(f, *gens, **args):
"""
Returns all factors of a univariate polynomial.

Examples
========

>>> from sympy.abc import x, y
>>> from sympy.polys.polyroots import root_factors

>>> root_factors(x**2 - y, x)
[x - sqrt(y), x + sqrt(y)]

"""
args = dict(args)
filter = args.pop('filter', None)

F = Poly(f, *gens, **args)

if not F.is_Poly:
return [f]

if F.is_multivariate:
raise ValueError('multivariate polynomials not supported')

x = F.gens[0]

zeros = roots(F, filter=filter)

if not zeros:
factors = [F]
else:
factors, N = [], 0

for r, n in zeros.iteritems():
factors, N = factors + [Poly(x - r, x)]*n, N + n

if N < F.degree():
G = reduce(lambda p, q: p*q, factors)
factors.append(F.quo(G))

if not isinstance(f, Poly):
factors = [ f.as_expr() for f in factors ]

return sorted(factors, key=default_sort_key)