Adequacy: a 3+b 3 = (a+b) (a 2-ab+b 2), and the original formula is (a+b- 1) (a 2-ab+b 2) = 0, because (a 2-ab+b).

Necessity: because a+b= 1 means A 3+B 3 = (A+B) (A 2-AB+B 2) = A 2-AB+B 2 because (A 2-AB+B 2) = (A-B/B).

So ab ≠ 0 a 3+b 3+ab-a 2-b 2 = 0.