Final answer:
To factor the expression 27x³ - 125, we identify it as a difference of two cubes and apply the formula a³ - b³ = (a - b)(a² + ab + b²), resulting in the factored form (3x - 5)(9x² + 15x + 25).
Step-by-step explanation:
The question is asking to factor the difference of two cubes, specifically the expression 27x³ - 125. In mathematics, factoring is the process of breaking down an expression into a product of simpler expressions. The difference of two cubes can be factored using the formula a³ - b³ = (a - b)(a² + ab + b²).
Applying this formula to the given expression, we identify a as (3x) and b as 5, since 27x³ is the cube of 3x (since 3³ equals 27) and 125 is the cube of 5. Then factoring 27x³ - 125:
So the factored form is:
(3x - 5)((3x)² + (3x)(5) + 5²)
Which simplifies to:
(3x - 5)(9x² + 15x + 25)