# Factoring the Sum of Two Cubes (*a*^{3} + *b*^{3})

How to solve factoring the sum of two cubes problems: formula, examples, and their solutions.

## Formula

*a*^{3} + *b*^{3} = (*a* + *b*)(*a*^{2} - *ab* + *b*^{2})

See the blue colored signs.

If the sign of *b*^{3} term in (*a*^{3} + *b*^{3}) is (+),

then the sign of *b* term in (*a* + *b*) is (+)

and the sign of *ab* term in (*a*^{2} - *ab* + *b*^{2}) is (-).

## Proof

Solve (*a* - *b*)(*a*^{2} + *ab* + *b*^{2}).

Multiplying polynomials

Then *a*^{2}*b* terms and *ab*^{2} terms are cancelled.

So (*a* - *b*)(*a*^{2} + *ab* + *b*^{2}) = *a*^{3} - *b*^{3}.

Switch both sides.

Then *a*^{3} - *b*^{3} = (*a* - *b*)(*a*^{2} + *ab* + *b*^{2}).

## Example 1

*x*^{3} + 8 = *x*^{3} + 2^{3}

= (*x* + 2)(*x*^{2} - *x*⋅2 + 2^{2})

## Example 2

27*y*^{3} + 1 = (3*y*)^{3} + 1^{3}

= (3*y* + 1)((3*y*)^{2} - 3*y*⋅1 + 1^{2})