证明:x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2能被(x+y+z)整除证明:x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2是(x+y+z)的倍数
问题描述：
证明:x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2能被(x+y+z)整除
证明:x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2是(x+y+z)的倍数
答
x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2 =x^4+y^4+z^4+2x^2y^2-2x^2z^2-2y^2z^2-4x^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) 所以x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2是(x+y+z)的倍数
答
x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2
=x^4+y^4+z^4+2x^2y^2-2x^2z^2-2y^2z^2-4x^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)
所以x^4+y^4+z^4-2x^2y^2-2x^2z^2-2y^2z^2是(x+y+z)的倍数