Start with looking at the polynomial near zero. Get a good feeling for that.
Start with doing each division until it is completely rewritten. Show graphs of it shifted and the one with subbed in.
Then do linear, quadratic, etc.
Using these, do Newton's method demonstrations on them both for zeros and for max/min
Then discuss the patterns here and derive the derivative formulas. Note the antiderivative coming from this as well.
Lastly, explore a little bit of area related to antiderivatives.