Use polynomial Identities to factor 8-27n^3

To factor the expression $8 - 27n^3$, we can use the sum of cubes formula. The sum of cubes formula is given by:<br /><br />\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]<br /><br />In this case, we need to rewrite the expression in the form of $a^3 - b^3$, which is the difference of cubes. The difference of cubes formula is:<br /><br />\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]<br /><br />Let's rewrite $8 - 27n^3$ as the difference of cubes:<br /><br />\[8 - 27n^3 = 2^3 - (3n)^3\]<br /><br />Now, we can apply the difference of cubes formula:<br /><br />\[a = 2 \quad \text{and} \quad b =n\]<br /><br />So,<br /><br />\[2^3 - (3n)^3 = (2 - 3n)(2^2 + 2 \cdot 3n + (3n)^2)\]<br /><br />Simplify the terms inside the parentheses:<br /><br />\[= (2 - 3n)(4 + 6n + 9n^2)\]<br /><br />Therefore, the factored form of $8 -n^3$ is:<br /><br />\[(2 - 3n)(4 + 6n + 9n^2)\]