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若x+y+z=0,证明:(x^2+y^2+z^2)/2*(x^3+y^3+z^3)/3=(x^5+y^5+z^5)/5
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若x+y+z=0,证明:(x^2+y^2+z^2)/2*(x^3+y^3+z^3)/3=(x^5+y^5+z^5)/5
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(x+y+z)²=x²+y²+z²+2xy+2xz+2yz=0 可知 x²+y²+z²=-(2xy+2xz+2yz)
x³+y³+z³-3xyz=(x+y+z)(x²+y²+z²-xy-yz-xz)=0 可知 x³+y³+z³=3xyz
x+y=-z 可知 x^5+y^5+z^5=x^5+y^5-(x+y)^5=-5(x^4)y-10(x^3)(y^2)-10(x^2)(y^3)-5x(y^4)
右边=-(x^4)y-2(x^3)(y^2)-2(x^2)(y^3)-x(y^4)
左边=(-2xy-2yz-2xz)/2*(3xyz)/3=-(xy+yz+xz)xyz=-(xy-x²-2xy-y²)xy(-x-y)
把z=-x-y 代入左边得两边相等
x³+y³+z³-3xyz=(x+y+z)(x²+y²+z²-xy-yz-xz)=0 可知 x³+y³+z³=3xyz
x+y=-z 可知 x^5+y^5+z^5=x^5+y^5-(x+y)^5=-5(x^4)y-10(x^3)(y^2)-10(x^2)(y^3)-5x(y^4)
右边=-(x^4)y-2(x^3)(y^2)-2(x^2)(y^3)-x(y^4)
左边=(-2xy-2yz-2xz)/2*(3xyz)/3=-(xy+yz+xz)xyz=-(xy-x²-2xy-y²)xy(-x-y)
把z=-x-y 代入左边得两边相等
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