Approximating a complicated multi-variable function over an interval?

Question

Consider $$ F(\mathbf{r})=F(x,y,z) = \frac{2z^2 - x^2 - y^2}{(x^2+y^2+z^2)^{5/2}} $$ where $x,y,$ and $z, $ are all $n^{\text{th}}$ order polynomial functions of a parameter $t$ with arbitrary coefficients. I want to approximate $F$ to $n^{\text{th}}$ order accuracy in $t$ with a polynomial function of $t$ on the interval $t\in[0,h]$. Basically, I need to know the best way to do this, I'm thinking it's probably best to find an $n^{\text{th}}$ order approximation as a function of $x, y$ and $z$ and then just sub in whatever polynomials they are, but I don't know. Also, I want to avoid dealing with the derivatives of $F$ if that's at all possible. This might be a bit of a stretch, but I'd also really like the result to be a function of the coefficients of $x, y$ and $z$ and not just true for one specific case, but I'll take what I can get.

Answer 1

Use the trinomial expansion to get the Taylor of $(x^2+y^2+z^2)^{-5/2}$ and multiply by $2z^2-x^2-y^2$.