Question

factor ( v^3 - 1 ) completely.

Explanation:

Step1: Identify difference of cubes

$v^3 - 1 = v^3 - 1^3$

Step2: Apply difference of cubes formula

Recall formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
Substitute $a=v$, $b=1$:
$v^3 - 1^3 = (v - 1)(v^2 + (v)(1) + 1^2)$

Step3: Simplify the quadratic term

$(v - 1)(v^2 + v + 1)$

Answer:

$(v - 1)(v^2 + v + 1)$