Question

factor (216t^{3}-125) completely.

Explanation:

Step1: Identify the formula

The expression \(216t^3 - 125\) is a difference of cubes. The formula for factoring the difference of cubes is \(a^3 - b^3=(a - b)(a^2+ab + b^2)\).

Step2: Find \(a\) and \(b\)

We can rewrite \(216t^3\) as \((6t)^3\) and \(125\) as \(5^3\). So, \(a = 6t\) and \(b = 5\).

Step3: Apply the formula

Substitute \(a = 6t\) and \(b = 5\) into the difference of cubes formula:
\[

$$\begin{align*} (6t)^3-5^3&=(6t - 5)((6t)^2+(6t)(5)+5^2)\\ &=(6t - 5)(36t^2 + 30t+25) \end{align*}$$

\]

Answer:

\((6t - 5)(36t^2 + 30t + 25)\)