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.

For polynomials the starting point does not really matter, since Newton's method is contractive for large arguments, but with slow convergence. What that really means that the iteration remains bounded, and converges with high probability to one of the roots. So take any random point inside the computed root radius.
You have to check for cycles in the iteration, some polynomials have stable cycles for certain intervals of initial values. In general, after 5 iterations the method should be in the quadratically convergent phase, so check if the step size ratio falls below 0.5 and if that fails, pick another initial value and repeat.
