Factoring a Difference of Cubes

Consider (a - b)(a² + ab + b²) = a³ - b³.

The binomial given above is called a Difference of Cubes since it is of the form a cube minus a cube.

To factor a difference of cubes:

Binomial Factor:

• Take a cube root of the first term, a³, of the binomial to get the first term ,a, of the binomial factor.

• Take a cube root of -b³ to get the last term of the binomial factor,-b.

Trinomial Factor:

• Square the first term of the binomial factor to get first term of trinomial factor

• Take the opposite of the product of first and last term of binomial factor to obtain middle term of trinomial factor.

• Square the last term of the binomial factor to obtain the last term of the trinomial factor.

                                Cube rt     Cube rt       Sq a       Opposite  Product        Sq b



a³ - b³ = (a  -  b)  (a²   + ab +   b²)

                                               Cube rt 5x    Cube rt -3y      Sq 5x              opposite product           Sq -3y



125x³ - 27y³ = (5x - 3y) ( 25x²  +  15xy  +  9y²)

                                             Cube rt 3x     Cube rt -2y    Sq 3x        opposite product        Sq -2y


27x³ - 8y³ = (3x  -  2y) ( 9x²  +  6xy  +  4y²)

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