Factoring the Difference of Cubes: (x^3  27)
The expression (x^3  27) represents the difference of cubes. This is a special type of binomial that can be factored using a specific formula.
The Formula
The formula for factoring the difference of cubes is:
a³  b³ = (a  b)(a² + ab + b²)
Applying the Formula to (x³  27)

Identify a and b:
 a = x (the cube root of x³)
 b = 3 (the cube root of 27)

Substitute a and b into the formula:
 (x³  27) = (x  3)(x² + 3x + 9)
The Factored Form
Therefore, the factored form of (x³  27) is (x  3)(x² + 3x + 9).
Important Note
The quadratic factor (x² + 3x + 9) cannot be factored further using real numbers. This is because it has no real roots.
Example: Solving an Equation
Let's say we need to solve the equation x³  27 = 0.

Factor the equation: (x  3)(x² + 3x + 9) = 0

Set each factor to zero:
 x  3 = 0
 x² + 3x + 9 = 0

Solve for x:
 x = 3
 The quadratic equation has no real roots.
Therefore, the only real solution to the equation x³  27 = 0 is x = 3.