I still don't really know how am i to investigate the different roots.

For one root i took (x-1)[cubed] (as you said) and then i found the average root, which is 1 and then the tangent is 0!!! So I don't exactly understand what it means. I guess i'm suppossed to see if my conjecture is true here as well.

i can't find a two roots one.

For one real and two complex i took x^3 - 4x^2 + x + 8

but the root that i get from it is not a whole number and i don't really know how to find the complex roots.

apart from that i have another question, my conjecture is: in cubics the tangent at the average of two roots intersects the curve at the third root. What similar cubics can i use to prove it and they should not be exactly the same, for other proofs you usually have to try functions with different powers, e.g. fractions, negative, but here you can't really do it, so do you have any ideas.

Can you think of any algebric way to prove my conjecture?