Sums or Differences of Cubes

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) (A² – AB + B²) = A(A² – AB + B²) + B(A² – AB +B²)

= A³ – A²B + AB² + A²B – AB² + B³

= A³ + B³

and (A – B) (A² + AB + B²) = A(A² + AB +B²) – B(A² + AB + B²)

= A³ + A²B +AB² – A²B – AB² – B³

= A³ – B³.

 

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

 

IMPORTANT FACT SUM OR DIFFERENCE OF CUBES

A³ + B³ = (A + B)(A² – AB + B²);

A³ – B³ = (A – B)(A² + AB + B²)

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 A³ + B³. The table of cubes in the margin is helpful.