# Thread: Perfect cube decomposition question

1. ## Perfect cube decomposition question

can someone explain to me how they got $\displaystyle (4x - y)$ in the numerator?

problem: $\displaystyle ( 64x^3 - y^3 ) / ( 4x^2 - 33xy + 8y^2 )$

first step: $\displaystyle [ ( 4x - y ) ( 16x^2 + 4xy + y^2 ) ] / [ ( 4x - y ) ( x - 8y ) ]$

I know they used the perfect cube formula, but why use 16 and 4. and where did the $\displaystyle (4x - y)$ come from?

2. Originally Posted by vd853
can someone explain to me how they got $\displaystyle (4x - y)$ in the numerator?

problem: $\displaystyle ( 64x^3 - y^3 ) / ( 4x^2 - 33xy + 8y^2 )$

first step: $\displaystyle [ ( 4x - y ) ( 16x^2 + 4xy + y^2 ) ] / [ ( 4x - y ) ( x - 8y ) ]$

I know they used the perfect cube formula, but why use 16 and 4. and where did the $\displaystyle (4x - y)$ come from?
Hi vd853,

The general factorization model for the differernce of two cubes is:

$\displaystyle (a^3-b^3)=(a-b)(a^2+ab+b^2)$

Your numerator could be re-written as $\displaystyle ((4x)^3-y^3)$

Using this and the model, we come up with $\displaystyle (4x-y)(16x^2+4xy+y^2)$

3. Thanks! did not know about the difference formula.