Determine the (2,2) Pade approximation to sqrt cube to x+8 and estimate its error

Question

here is the full question its about pade approximation and I serch the internet and couldn't find the answer anyone can help it will be great

Answer 1

As a welcome gift, I shall put you on the track.

Assuming that you want it around $x=0$, a $[2,2]$ Padé pproximant write $$f(x)=\frac{a_0+a_1x+a_2x^2}{1+b_1x+b_2 x^2}$$ Rewrite $$(1+b_1x+b_2 x^2)\,f(x)=a_0+a_1x+a_2x^2$$ Now, use either Taylor expansion or, in your case, the binomial expansion and replace $f(x)$ by the result. Group terms and solve the linear equations (you need five of them since you have five parameters - but one of them is obvious : guess which one).