**Sums or Differences of Cubes**

We can factor the sum or the difference of two expressions that are cubes.

Consider the following products:

(A + B) (A^{2} – AB + B^{2}) = A(A^{2} – AB + B^{2}) + B(A^{2} – AB +B^{2})

= A^{3} – A^{2}B + AB^{2} + A^{2}B – AB^{2} + B^{3}

= A^{3} + B^{3}

and (A – B) (A^{2} + AB + B^{2}) = A(A^{2} + AB +B^{2}) – B(A^{2} + AB + B^{2})

= A^{3} + A^{2}B +AB^{2} – A^{2}B – AB^{2} – B^{3}

= A^{3} – B^{3}.

The above equations (reversed) show how we can factor a sum or a difference of two cubes.

**IMPORTANT FACT SUM OR DIFFERENCE OF CUBES**

A^{3} + B^{3} = (A + B)(A^{2} – AB + B^{2});

A^{3} – B^{3} = (A – B)(A^{2} + AB + B^{2})

Note that what we are considering here is a sum or difference of cubes. We are not cubing a binomial. For example, (A + B)^{3} is not the same as A^{3} + B^{3}. The table of cubes in the margin is helpful.