code: def test_symmetrize():
      assert symmetrize(0, x, y, z) == (0, 0)
      assert symmetrize(1, x, y, z) == (1, 0)
  
      s1 = x + y + z
      s2 = x * y + x * z + y * z
  
      assert symmetrize(1) == (1, 0)
      assert symmetrize(1, formal=True) == (1, 0, [])
  
      assert symmetrize(x) == (x, 0)
      assert symmetrize(x + 1) == (x + 1, 0)
  
      assert symmetrize(x, x, y) == (x + y, -y)
      assert symmetrize(x + 1, x, y) == (x + y + 1, -y)
  
      assert symmetrize(x, x, y, z) == (s1, -y - z)
      assert symmetrize(x + 1, x, y, z) == (s1 + 1, -y - z)
  
      assert symmetrize(x**2, x, y, z) == (
          s1**2 - 2 * s2,
          -(y**2) - z**2,
      )
  
      assert symmetrize(x**2 + y**2) == (
          -2 * x * y + (x + y) ** 2,
          0,
      )
      assert symmetrize(x**2 - y**2) == (
          -2 * x * y + (x + y) ** 2,
          -2 * y**2,
      )
  
      assert symmetrize(
          x**3 + y**2 + a * x**2 + b * y**3, x, y
      ) == (
          -3 * x * y * (x + y)
          - 2 * a * x * y
          + a * (x + y) ** 2
          + (x + y) ** 3,
          y**2 * (1 - a) + y**3 * (b - 1),
      )
  
      U = [u0, u1, u2] = symbols("u:3")
  
      assert symmetrize(x + 1, x, y, z, formal=True, symbols=U) == (
          u0 + 1,
          -y - z,
          [
              (u0, x + y + z),
              (u1, x * y + x * z + y * z),
              (u2, x * y * z),
          ],
      )
  
      assert symmetrize([1, 2, 3]) == [(1, 0), (2, 0), (3, 0)]
      assert symmetrize([1, 2, 3], formal=True) == (
          [(1, 0), (2, 0), (3, 0)],
          [],
      )
  
      assert symmetrize([x + y, x - y]) == [(x + y, 0), (x + y, -2 * y)]
  
file_path: sympy/sympy/polys/tests/test_polyfuncs.py
imports: from sympy.polys.polyfuncs import symmetrize, horner, interpolate, rational_interpolate, viete
  from sympy.polys.polyerrors import MultivariatePolynomialError
  from sympy.core.singleton import S
  from sympy.core.symbol import symbols
  from sympy.testing.pytest import raises
  from sympy.abc import a, b, c, d, e, x, y, z
label_desc: Use image node_id 1 for calling a global function with example usage: test_symmetrize() without return types