## Pages

### Sum of two cubes: a^3 + b^3

The formula
a3 + b3 = (a + b)(a2 - ab + b2)
Working forwards - Factorization
For example, the expression x3 + 8y3 can be expanded as follows:
= (x)3 + (2y)3
= (x + 2y)(x2 - (x)(2y) + (2y)2)
= (x + 2y)(x2 - 2xy + 4y2)
.. which is the factored form of x3 + 8y3

Working backwards - Expansion
Since the opposite working of this formula is one step and requires almost no simplification, it is better to see an example of how this formula is applied in a problem like
 x2 - 4x + 16 x - 4
In the above example, if we multiply the fraction by
 x + 4 x + 4

 x2 - 4x + 16 x - 4
×
 x  + 4 x + 4
=
 (x2 - 4x + 16) (x + 4) (x - 4) (x + 4)

On comparing the numerator with the RHS of the above formula, it is a sum of two cubes, x3 and 43. So, factoring it, you get:
 x3 + 43 (x + 4) (x - 4)
.. which is a simplified form of the original expression.