Writing $x^2+y^2+z^2$ as a polynomial combination of $xyz$, $x+y+z$, and $\frac1x+\frac1y+\frac1z$ Can we write $x^2+y^2+z^2$ as a polynomial combination of $xyz$, $x+y+z$, and $\dfrac1x+\dfrac1y+\dfrac1z$?
What about $x^3+y^3+z^3$?

Let our three expressions be $a$, $b$, and $c$. Note that $xy+yz+zx=ac$. 
For $x^2+y^2+z^2$, use the identity 
$$(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2xz.$$
For $x^3+y^3+z^3$, use the not difficult to verify identity
$$x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-xz).$$ 
