已知X+Y+Z=0,求证X^3 +Y^3+Z^3=3XYZ
来源:学生作业帮 编辑:拍题作业网作业帮 分类:数学作业 时间:2024/04/29 17:33:25
已知X+Y+Z=0,求证X^3 +Y^3+Z^3=3XYZ
X+Y+Z=0
(X+Y+Z)^3 = 0
(X²+Y²+Z²+2XY+2XZ+2YZ )(X+Y+Z) = 0
(X²+Y²+Z²) (X+Y+Z) = 0
即:X³+Y³+Z³+X²Y+X²Z+XY²+Y²Z+XZ²+YZ² = 0…………………………1式
将(X+Y+Z)^3=0完全展开得:
X³+Y³+Z³+3X²Y+3X²Z+3XY²+3Y²Z+3XZ²+3YZ²+6XYZ = 0……………2式
1式乘3,再减去2式,得:
2X³+2Y³+2Z³ = 6xyz
即:X³+Y³+Z³ = 3xyz,得证
(X+Y+Z)^3 = 0
(X²+Y²+Z²+2XY+2XZ+2YZ )(X+Y+Z) = 0
(X²+Y²+Z²) (X+Y+Z) = 0
即:X³+Y³+Z³+X²Y+X²Z+XY²+Y²Z+XZ²+YZ² = 0…………………………1式
将(X+Y+Z)^3=0完全展开得:
X³+Y³+Z³+3X²Y+3X²Z+3XY²+3Y²Z+3XZ²+3YZ²+6XYZ = 0……………2式
1式乘3,再减去2式,得:
2X³+2Y³+2Z³ = 6xyz
即:X³+Y³+Z³ = 3xyz,得证
已知x,y,z满足xyz=1,求证x^3/(x+y)+y^3/(y+z)+z^3/(z+x)大于等于3
已知xyz属于R+,x+y+z=1,求证x^3/(y(1-y))+y^3/(z(1-z))+z^3/(x(1-x))大于
已知x,y,z都是正数,且xyz=1,求证:x^2/(y+z)+y^2/(x+z)+z^2/(x+y)≥3/2
已知X,Y,Z为3个互不相等的实数,且X+1/Y=Y+1/Z=Z+1/Z求证(xyz)^2=1
已知2x-3y+z=o,3x-2y-6z=0,xyz≠0,求x^2+y^2+z^2/2x^2+y^2-z^2的值
已知方程组4x-y+3z=0 2x+y+6z=0且xyz不等于0,则x/y+y/z+z/x是多少
已知4x-3y+z=0,x+2y-8z=0,xyz不等于0,求x+y-z/x-y+2z的值
已知x,y,z都是正数, 且x^3+y^3+z^3=3xyz, 求证:x=y=z.
求教初二数学难题已知:(1/x+1/y+1/z)^(1/3)=1/x+1/y+1/z,且xyz>0,求证:1/x+1/y
已知x,y,z 大于0,x+y+z=2,求证 xz/y(y+z)+zy/x(x+y)+yx/z(z+x)大于等于2/3
已知xyz满足|3x-2y+z|+|2x+y+2z|=0(xyz不等于o)求x+y除以z
已知x+y+z=0,求多项式x^3+x^2z-xyz+y^3的值