The function $$f(x_1,x_2,y_1,y_2,z_1,z_2)=\sqrt{x_{1}^2-2x_{1}x_{2}+x_{2}^2+y_{1}^2-2y_{1}y_{2}+y_{2}^2+z_{1}^2-2z_{1}z_{2}+z_{2}^2}$$ is not a polynomial or even a rational function (over the real numbers). Suppose it was a rational function. Then $$g(x)=f(x,0,0,0,0,0)=\sqrt{x^2}=|x|$$ would also have to be a rational function - of a single variable. So let's write it as $g(x)=\frac{P(x)}{Q(x)}$. Observe that $Q$ has no (real) zeros, so it is either always positive or always negative.
