Question

factor the following binomial completely: $125x^3 - 8$.

1. $(25x - 4)(5x^2 - 2)$
2. $(5x - 2)(25x^2 + 10x + 4)$
3. $(5x - 2)(25x^2 - 10x + 4)$
4. $(5x - 2)(25x^2 + 10x - 4)$

Explanation:

Step1: Recall the difference of cubes formula

The formula for factoring the difference of cubes is \(a^3 - b^3=(a - b)(a^2+ab + b^2)\).
Here, \(125x^3=(5x)^3\) and \(8 = 2^3\), so \(a = 5x\) and \(b = 2\).

Step2: Apply the formula

Substitute \(a = 5x\) and \(b = 2\) into the formula:
\((5x)^3-2^3=(5x - 2)[(5x)^2+(5x)(2)+2^2]\)
Simplify each part:
\((5x)^2 = 25x^2\), \((5x)(2)=10x\), \(2^2 = 4\)
So we get \((5x - 2)(25x^2+10x + 4)\)

Answer:

1. \((5x - 2)(25x^2 + 10x + 4)\)