(x+y)(y+z)(z+x)=0

Solution for (x+y)(y+z)(z+x)=0 equation:

Simplifying
(x + y)(y + z)(z + x) = 0

Reorder the terms:
(x + y)(y + z)(x + z) = 0

Multiply (x + y) * (y + z)
(x(y + z) + y(y + z))(x + z) = 0
((y * x + z * x) + y(y + z))(x + z) = 0
((xy + xz) + y(y + z))(x + z) = 0
(xy + xz + (y * y + z * y))(x + z) = 0

Reorder the terms:
(xy + xz + (yz + y2))(x + z) = 0
(xy + xz + (yz + y2))(x + z) = 0
(xy + xz + yz + y2)(x + z) = 0

Multiply (xy + xz + yz + y2) * (x + z)
(xy(x + z) + xz(x + z) + yz(x + z) + y2(x + z)) = 0
((x * xy + z * xy) + xz(x + z) + yz(x + z) + y2(x + z)) = 0

Reorder the terms:
((xyz + x2y) + xz(x + z) + yz(x + z) + y2(x + z)) = 0
((xyz + x2y) + xz(x + z) + yz(x + z) + y2(x + z)) = 0
(xyz + x2y + (x * xz + z * xz) + yz(x + z) + y2(x + z)) = 0

Reorder the terms:
(xyz + x2y + (xz2 + x2z) + yz(x + z) + y2(x + z)) = 0
(xyz + x2y + (xz2 + x2z) + yz(x + z) + y2(x + z)) = 0
(xyz + x2y + xz2 + x2z + (x * yz + z * yz) + y2(x + z)) = 0
(xyz + x2y + xz2 + x2z + (xyz + yz2) + y2(x + z)) = 0
(xyz + x2y + xz2 + x2z + xyz + yz2 + (x * y2 + z * y2)) = 0
(xyz + x2y + xz2 + x2z + xyz + yz2 + (xy2 + y2z)) = 0

Reorder the terms:
(xyz + xyz + xy2 + xz2 + x2y + x2z + yz2 + y2z) = 0

Combine like terms: xyz + xyz = 2xyz
(2xyz + xy2 + xz2 + x2y + x2z + yz2 + y2z) = 0

Solving
2xyz + xy2 + xz2 + x2y + x2z + yz2 + y2z = 0

Solving for variable 'x'.

The solution to this equation could not be determined.