Question

已知x+y+z=1,x^2+y^2+z^2=2,x^3+y^3+z^3=3,求x^4+y^4+z^4的值

已知x+y+z=1,x^2+y^2+z^2=2,x^3+y^3+z^3=3,求x^4+y^4+z^4的值
写一下过程，谢谢！！！

Answer 1

25/6挺麻烦：把x+y+z=1两边平方可以得出xy+xz+yz=-1/2；再把x^2+y^2+z^2=2两边平方；（x^3+y^3)+(x^3+z^3)+(y^3+z^3)=6; 展开化简可先求出xyz的值；再xy+xz+zyz=-1/2平方，可以得x^2y^2+x^2z^2+y^2z^2=-1/12;再x^2+y^2+z^2=2平方
细心你一定可以算出25/6

Answer 2

题意(x+y+z)^3=1

x^3+y^3+z^3+2x^2(y+z)+2z^2(x+y)+2y^2(x+z)+3xyz=1

整理得xyz=0

不妨设x=0

又xy+yz+xz=0.5[(x+y+z)^2-(x^2+y^2+z^2)]=-0.5
则y+z=1 y^3+z^3=1

y^4+z^4=(y+z)(y^3+z^3)-yz(y^2+z^2)=4

故x^4+y^4+z^4=4

Answer 3

你可以看sweetandbitter - 魔导师 十级