Factoring the Sum of Two Cubes (a3 + b3)
How to solve factoring the sum of two cubes problems: formula, examples, and their solutions.
a3 + b3 = (a + b)(a2 - ab + b2)
See the blue colored signs.
If the sign of b3 term in (a3 + b3) is (+),
then the sign of b term in (a + b) is (+)
and the sign of ab term in (a2 - ab + b2) is (-).
Solve (a - b)(a2 + ab + b2).
Then a2b terms and ab2 terms are cancelled.
So (a - b)(a2 + ab + b2) = a3 - b3.
Switch both sides.
Then a3 - b3 = (a - b)(a2 + ab + b2).
x3 + 8 = x3 + 23
= (x + 2)(x2 - x⋅2 + 22)
27y3 + 1 = (3y)3 + 13
= (3y + 1)((3y)2 - 3y⋅1 + 12)