x+y+z=1,x平方加y平方加z平方=2,x立方加y立方加z立方=3,x四次方加y四次方加z四次方=?

Question

x+y+z=1,
x平方加y平方加z平方=2,
x立方加y立方加z立方=3,
x四次方加y四次方加z四次方=?

Answer 1

(x+y+z)^2=x^2+y^2+z^2+2(xy+xz+yz)
可得 xy+xz+yz=-1/2
(x+y+z)^3=x^3+y^3+z^3+6xyz+3x^2(y+z)+3y^2(x+z)+3z^2(x+y)
=x^3+y^3+z^3+6xyz+3x^2(1-x)+3y^2(1-y)+3z^2(1-z)
解得
xyz=1/6
(xy+yz+xz)^2=x^2*y^2+x^2*z^2+y^2*z^2+2xyz(x+y+z)
可得
x^2*y^2+x^2*z^2+y^2*z^2=-1/12
(x^2+y^2+z^2)^2=x^4+y^4+z^4+2(x^2*y^2+x^2*z^2+y^2*z^2)
解得 x^4+y^4+z^4=25/6
所以答案应该是25/6
欢迎各位大虾指正!