代数式化简已知z^2=x^2+y^2化简：(x+y+z)(x-y+z)(-x+y+z)(x+y-z)

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代数式化简已知z^2=x^2+y^2化简：(x+y+z)(x-y+z)(-x+y+z)(x+y-z)

代数式化简已知z^2=x^2+y^2化简：(x+y+z)(x-y+z)(-x+y+z)(x+y-z)
代数式化简
已知z^2=x^2+y^2
化简：(x+y+z)(x-y+z)(-x+y+z)(x+y-z)

代数式化简已知z^2=x^2+y^2化简：(x+y+z)(x-y+z)(-x+y+z)(x+y-z)
先分组,在用平方差公式
(x+y+z)(x-y+z)(-x+y+z)(x+y-z)
=[(x+y+z)(x+y-z)][(x-y+z)(-x+y+z)]
=[(x+y)^2-z^2][z^2-(y-x)^2]
因为z^2=x^2+y^2
=2xy*2xy=4x^2y^2

(x+y+z)(x-y+z)(-x+y+z)(x+y-z)
=[(x+y+z)(x+y-z)][(x-y+z)(-x+y+z)]
=[(x+y)^2-z^2][z^2-(y-x)^2]
因为z^2=x^2+y^2
=2xy*2xy=4x^2y^2

(x+y+z)(-x+y+z)=(y+z)^2-x^2=y^2+2yz+z^2-x^2=2y^2+2yz
(x-y+z)(x+y-z)=x^2-(y-z)^2=2yz-2y^2
(2yz-2y^2)(2y^2+2yz)=4(yz)^2-4y^4=4y^2(z^2-y^2)
=4x^2y^2

(x+y+z)(x-y+z)(-x+y+z)(x+y-z)
=[(x+y+z)*(x+y-z)]*[(x-y+z)*(-(x-y)+z)]
=[(x+y)^2-z^2]*[(x-y)^2-z^2]
z^2=x^2+y^2
(x+y+z)(x-y+z)(-x+y+z)(x+y-z)
=(2xy)*(-2xy)
=-4x^2*y^2

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