Factoring a Sum of Cubes

 

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

The binomial given above is called a Sum of Cubes since it is of the form a cube plus a cube.

To factor a sum of cubes:

To obtain 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.

To obtain the 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 2x    Cube rt 3y      Sq 2x         opposite product         Sq 3y

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

                                               Cube rt 5x     Cube rt 2y      Sq 5x        opposite product        Sq 2y

125x³ + 8y³ = (5x + 2y) ( 25x² - 10xy + 4y²)

 

 

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