Calculating xyz from Given Equations

[aznluster]

Given:

x+y+z=1 x^{2}+y^{2}+z^{2}=2x^{4}+y^{4}+z^{4}=4

Find:x^{3}+y^{3}+z^{3} Attempt at Solving:Note: x^{3}+y^{3}+z^{3}-3xyz=(x+y+z)(x^{2}+y^{2}+z^{2}-xy-xz-yz)(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2xy+2xz+2zy=1xy+xz+zy=-1/2 Plugging in the unknown, we get:x^{3}+y^{3}+z^{3}-3xyz=(1)(2-(-1/2)) Now I need to find xyz.

 

[phyzguy]

Using:
(x^2+y^2+z^2)^2=x^4+y^4+z^4+ 2 (x^2 y^2+x^2 z^2+y^2<br /> z^2)
(x+y+z)^4=x^4+y^4+z^4+4 (x y+x z+y z) (x^2+y^2+z^2)+6<br /> (x^2 y^2+x^2 z^2+y^2 z^2)+8 x y z (x+y+z)

You can solve the first for x^2 y^2+x^2 z^2+y^2 z^2, and then the second for xyz.

 

[aznluster]

Thank you. :)