证明x⁴+y⁴+z⁴-2x²y²-2x²z²-2y²z²能被x+y+z整除

证明x⁴+y⁴+z⁴-2x²y²-2x²z²-2y²z²能被x+y+z整除
证明x⁴+y⁴+z⁴-2x²y²-2x²z²-2y²z²能被x+y+z整除

证明x⁴+y⁴+z⁴-2x²y²-2x²z²-2y²z²能被x+y+z整除
原式=(x^2+y^2)^2 -2z^2(x^2+y^2) + z^4 -4x^2y^2 = (x^2+y^2-z^2)^2-4x^2y^2=
(x^2+y^2-z^2+2xy)(x^2+y^2-z^2-2xy)=[(x+y)^2-z^2][(x-y)^2-z^2]=(x+y+z)(x+y-z)(x-y+z)(x-y-z)证明完毕