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已知x+y+z=0,求：X∧4+Y∧4+Z∧4-2X∧2Y∧2-2Y∧2Z∧2-2Z∧2X∧2
题目详情
已知x+y+z=0,求：X∧4+Y∧4+Z∧4-2X∧2Y∧2-2Y∧2Z∧2-2Z∧2X∧2
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答案和解析
x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2
=(x^4-2x^2y^2+y^4)+z^4-2y^2z^2+2z^2x^2-4z^2x^2
=(x^2-y^2)^2+2z^2(x^2-y^2)+z^4-4z^2x^2
=(x^2-y^2+z^2)^2-4z^2x^2
=(x^2-y^2+z^2+2zx)(x^2-y^2+z^2-2zx)
=[(x+z)^2-y^2][(x-z)^2-y^2]
=(x+y+z)(x-y+z)(x+y-z)(x-y-z)
=0
=(x^4-2x^2y^2+y^4)+z^4-2y^2z^2+2z^2x^2-4z^2x^2
=(x^2-y^2)^2+2z^2(x^2-y^2)+z^4-4z^2x^2
=(x^2-y^2+z^2)^2-4z^2x^2
=(x^2-y^2+z^2+2zx)(x^2-y^2+z^2-2zx)
=[(x+z)^2-y^2][(x-z)^2-y^2]
=(x+y+z)(x-y+z)(x+y-z)(x-y-z)
=0
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