Question

I am trying to understand how to derive the following two formulae. Would you mind giving me a hint as to how I should go about with doing so?

a^3 + b^3 = (a + b)(a^2 – ab + b^2)
a^3 – b^3 = (a – b)(a^2 + ab + b^2)

I am not sure whether or not it requires you to do long division with the formula ax^3 + bx^2 + cx + d. What I am thinking is that you would take the linear factor (a-b) or (a+b) and divide it by the cubic polynomial equation, but I am not sure as to how I will go about with that.

Answer 1

it not so much as a deriving that gets you there; its just a matter of after doing so many of them you notice a pattern that emerges which makes the process easier

Answer 2

a^2 -ab +b^2
----------------
a+b| a^3 + b^3
-a^3 -a^2b
------------
-a^2b +b^3
+a^2b +ab^2
--------------
+ab^2 +b^3
-ab^2 -b^3
-----------
0

that would work if need be

Answer 3

(a+b)^3 = a^3 +3a^2b +3ab^2 + b^3 (Pascal) so

a^3 +b^3 = (a+b)^3 -3ab(a+b)

= (a+b) ((a+b)^2 -3ab)

= (a+b)(a^2 +b^2 -ab)

Answer 4

what is wrong with just multiplying?

Answer 5

it tells me the next title is legend

Answer 6

"what is wrong with just multiplying"

Nothing at all, all these things (proofs) depend on where u are allowed to begin.

The word "derive" is the clue, tutor code for show something that looks like a proof.