There is probably a better way of asking this question. There is a pretty simple formula to figure out $x^2$ from $(x-1)^2$.
$$x^2 = (x-1)^2 + x + (x-1)$$
You can see how easy this formula is here:
$$4^2 = 3^2 + 4 + 3 = 9 + 4 + 3 = 16$$
My question is this: Is there a similar formula for figuring out cubes of numbers? This is the best that I can come up with and it seems a little forced:
$$x^3 = (x-1)^3 + 2x^2 + (x-1)^2 + (-x)$$
Again, you can see here that it works:
$$4^3 = 3^3 + 2(4^2) + 3^2 + (-4) = 27 + 32 + 9 - 4 = 64$$
Maybe this is something that everybody already knows and I'll feel like an idiot, but if somebody has a cleaner equation for what I'm looking for here, please let me know.
$$ x^3=((x-1)+1)^3=(x-1)^3+3(x-1)^2+3(x-1)+1 $$ by the binomial formula.