How Do You Factor 27x^3 – 8?

To factor the expression 27x³ – 8, we can recognize that both terms are perfect cubes. The first term, 27x³, can be rewritten as (3x)³, and the second term, 8, can be rewritten as 2³. This gives us the difference of cubes formula, which states:

a³ – b³ = (a – b)(a² + ab + b²)

In our case, a is 3x and b is 2. Plugging these values into the formula:

1. a – b: 3x – 2
2. a²: (3x)² = 9x²
3. ab: (3x)(2) = 6x
4. b²: 2² = 4

Now we can assemble the factored form:

27x³ – 8 = (3x – 2)(9x² + 6x + 4)

This gives us the completely factored form of the original expression.