若xy+yz+zx=0,3xyz+x^2(y+z)+y^2(z+x)+z^2(x+y)等于？

来源：百度知道 编辑：UC知道 时间：2024/05/15 17:26:45

将3xyz+x^2(y+z)+y^2(z+x)+z^2(x+y)拆开,得:
3XYZ+X^2Y+X^2Z+Y^2Z+Y^2Z+Y^2X+Z^2X+Z^2Y

将xy+yz+zx=0乘X 乘Y 乘Z分别得:
X^2Y+XYZ+X^2Z=0 ①
Y^2X+Y^2Z+XYZ=0 ②
XYZ+Z^2Y+Z^2X=0 ③

①+②+③恰好得3XYZ+X^2Y+X^2Z+Y^2Z+Y^2Z+Y^2X+Z^2X+Z^2Y=3xyz+x^2(y+z)+y^2(z+x)+z^2(x+y)=0
所以3xyz+x^2(y+z)+y^2(z+x)+z^2(x+y)=0

解:3xyz+x^2*(y+z)+y^2*(x+z)+z^2*(x+y)=3xyz+x^2*y+x^2z+y^2*x+y^2*z+z^2*x+z^2*y=3xyz+xy(x+y)+yz(y+z)+xz(x+z)=xyz+xy(x+y)+xyz+yz(y+z)+xyz+xz(x+z)=xy(x+y+z)+yz(x+y+z)+yz(x+y+z)=(x+y+z)(xy+yz+xz)=(x+y+z)*0=0.