Hensel Lifting with a non-simple root

Question

I was doing a problem and it involves lifting a root x= 55, from mod $2^{10}$ to a solution mod $2^{19}$ but the root is non simple, i.e. $$f'(x) \equiv 0 (mod 2)$$

Here, $f(x) = x^{3} - 9x + 8 \equiv 0 $ $(mod 2^{10})$

Where x = 55 is a solution of f(x).

I am wondering how I go about trying to lift to this power modulus.

Answer 1

A general method is: If $x$ is a singular root $\mod p^j$, then either it lifts to p roots $\mod p^{j+1}$, or it lifts to no root, depending on if $f(x)$ is congruent to $0 \mod p^{j+1}$.