The problem asks us to factor the expression $216x^3 - 343y^3$ if possible. If the polynomial is prime, we should state that.

1. Problem Description

The problem asks us to factor the expression $216x^3 - 343y^3$ if possible. If the polynomial is prime, we should state that.

2. Solution Steps

We are asked to factor $216x^3 - 343y^3$.

We notice that $216 = 6^3$ and $343 = 7^3$, so we have a difference of cubes.

The difference of cubes formula is:

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

In our case, $a = 6x$ and $b = 7y$.

Therefore,

$216x^3 - 343y^3 = (6x)^3 - (7y)^3 = (6x - 7y)((6x)^2 + (6x)(7y) + (7y)^2)$

$= (6x - 7y)(36x^2 + 42xy + 49y^2)$

The quadratic term $36x^2 + 42xy + 49y^2$ cannot be factored further.

3. Final Answer

$(6x - 7y)(36x^2 + 42xy + 49y^2)$