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

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/16 08:02:40

(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

x/2=y/3=z/5 x+3y-z/x-3y+z 试证明(x+y-2z)+(y+z-2x)+(z+x-2y)=3(x+y-2z)(y+z-2x)(z+x-2y) {x+y+z=1;x+3y+7z=-1;z+5y+8z=-2 1.（x+y-z)+(x-y+z)-(x-y-z) 2.2x-(3x-2y+3)-(5y-2) x+y-z=0 2x-3y+2z=5 x+2y-z=3 若x/3=y/2=z/5,则2x-3y-z/x-y+z= 若x/3=y/4=z/5,求x+y+z/3x-2y+z的值. 已知方程组4x-5y+2z=0 x+2y=3z 则x:y:z 2x+5y+4z=0,3x+y-7z=0,则x+y-z=? 已知3x-2y-5z=0,2x-5y+4z=0,且x,y,z均不为0,求3x*x+2y*y+5z*z/5x*x+y*y- 若4x-3y-6z=0,x+2y-7z=0,求代数式5x*5x+2y*2y-z*z/2x*2x-3y*3y-10z*10 已知x+y-z=0,2x-y+2z=0 求（3x-2y+5z)/(5x-3y+2z)的值