How to choose the starting point in Newton's method? 
How to choose the starting point in Newton's method ?
If $p(x)=x^3-11x^2+32x-22$

We only learnt that the algorithm $x_{n+1}:=x_n-\frac{f(x_n)}{f'(x_n)}$ converges only in some $\epsilon$-neighbourhood of a root and that if $z$ is a root then $|z|\le 1+\max\limits_{k=0,...,n-1}\frac{|a_k|}{|a_n|}$
but in this case, $a_n=1\Rightarrow 1+\max\limits_{k=0,...,n-1}\frac{|a_k|}{|a_n|}=1+\frac{32}{1}=33$ this is too large isn't it ?
$\underline{\textbf{My attempt}}$
I think first root can be guessed by plugging in some values $-1,0,1,2$ and at $x=1$ we've a root
then the polynomial can be reduced to $x^3-11x^2+32x-22=(x-1)(x^2-10x+22)$
now the new polynomial to be examined is $g(x)=(x^2-10x+22)$
this is a parabola and $g'(x)=0$ is attained at $x=5$
and the advantage of the parabola is that we can consider any interval $[a,5)$ with $a<5$
any point in this interval as starpoint would converge to the $2^{nd}$ root
the same is valid for $(5,b]$, Hence we obtain our $3$ roots.
BUT in this case we had a bit luck, and if you know a general approach, can you please tell me
Thanks in advance.

The general case is very complicated. See for instance:


*

*Newton fractal

*How to find all roots of complex polynomials by Newton's method  by Hubbard et al.
Invent. Math. 146 (2001), no. 1, 1–33. pdf
