Question

3a) factor $1 - 125n^3$

Explanation:

Step1: Recall the difference of cubes formula

The formula for the difference of two cubes is \(a^3 - b^3=(a - b)(a^2+ab + b^2)\). Here, we can rewrite \(1-125n^3\) as \(1^3-(5n)^3\), so \(a = 1\) and \(b=5n\).

Step2: Apply the difference of cubes formula

Substitute \(a = 1\) and \(b = 5n\) into the formula \(a^3 - b^3=(a - b)(a^2+ab + b^2)\).
We get \((1-(5n))(1^2+(1\times(5n))+(5n)^2)\).
Simplify each part: \(1 - 5n\) for the first factor, and for the second factor, \(1^2 = 1\), \(1\times5n=5n\), \((5n)^2 = 25n^2\), so the second factor is \(1 + 5n+25n^2\).

Answer:

\((1 - 5n)(1 + 5n + 25n^2)\)