So you want to know what happens with Newton's method?
In particular, what guesses will lead to which roots?
Here are a number of useful things to engage with that over the real numbers.

First, evaluate the following cell of preliminaries. Then scroll ahead to some interactive stuff.


This first cell allows you to experiment with different functions and what zeros they converge to.
Click to the left again to hide and once more to show the dynamic interactive window 
Try to figure out what starting values lead to which zeros. I start with $x^3x$ because it's a fun one to explore, but you should try for others as well.
In particular, try to see what intervals of starting guesses will all lead to the same zeros. Write down some of your explorations!



Our next interact is different in style.
Click to the left again to hide and once more to show the dynamic interactive window 
Here, we are using color to identify which initial guesses will end up near which roots. The parameters are
Notice that we don't want to ask Sage to guess what the roots are; instead, we create the polynomial by telling Sage what its roots are ahead of time.
Questions you may enjoy asking:
Finally, I'd ask what happens if you add in a couple of complex roots. Try the list "1,i,i" for kicks. What function does that give? Is the result surprising?
