## The Difference of Cubes Formula: (a - b)(a² + ab + b²)

The formula **(a - b)(a² + ab + b²) = a³ - b³** is known as the **difference of cubes** formula. It provides a quick and efficient way to factorize expressions of the form a³ - b³.

### Understanding the Formula

The formula is derived from the expansion of the product:

(a - b)(a² + ab + b²) = a(a² + ab + b²) - b(a² + ab + b²)

Expanding this gives:

a³ + a²b + ab² - a²b - ab² - b³

Simplifying, we get:

**a³ - b³**

### Using the Formula

The difference of cubes formula can be applied to factorize expressions where both terms are perfect cubes.

**Here's how to use it:**

**Identify the terms:**Determine the cube root of each term in the expression.**Apply the formula:**Substitute the cube roots (a and b) into the formula.**Simplify:**The result will be the factored form of the expression.

**Example:**

Factorize the expression: x³ - 8

**Identify the terms:**The cube root of x³ is x, and the cube root of 8 is 2.**Apply the formula:**Substitute a = x and b = 2 into the formula: (x - 2)(x² + 2x + 2²)**Simplify:**The factored form is**(x - 2)(x² + 2x + 4)**

### Applications

The difference of cubes formula has various applications in algebra and other mathematical fields, including:

**Simplifying expressions:**It helps simplify complex expressions involving cubes.**Solving equations:**The formula can be used to solve equations with cubic terms.**Calculus:**The formula aids in differentiating and integrating expressions with cubic terms.

### Conclusion

The difference of cubes formula is a valuable tool for factoring expressions and simplifying calculations. Mastering this formula allows you to efficiently manipulate algebraic expressions and solve problems involving cubes.