## Understanding the Difference of Cubes: (a - b)(a² + ab + b²) = a³ - b³

This equation reveals a powerful factorization pattern known as the **difference of cubes**. It helps us simplify expressions and solve equations efficiently. Let's break down the equation and explore its significance.

### The Left-Hand Side: (a - b)(a² + ab + b²)

The left side of the equation represents the product of two expressions:

**(a - b):**This is a simple binomial representing the difference between two variables, 'a' and 'b'.**(a² + ab + b²):**This is a trinomial, where each term contains a combination of 'a' and 'b' raised to different powers.

### The Right-Hand Side: a³ - b³

The right side of the equation represents the difference of two cubes, 'a³' and 'b³'.

### Expanding the Left-Hand Side

Let's expand the left side of the equation using the distributive property (also known as FOIL):

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

Now, distribute the 'a' and '-b' terms:

= a³ + a²b + ab² - a²b - ab² - b³

Notice that the middle terms (a²b and -a²b) and (ab² and -ab²) cancel out:

= a³ - b³

### The Result: A Powerful Identity

We have successfully shown that the product of (a - b) and (a² + ab + b²) simplifies to a³ - b³. This demonstrates the following identity:

**(a - b)(a² + ab + b²) = a³ - b³**

This identity is crucial for simplifying expressions and factoring polynomials. It allows us to factor the difference of two cubes into a product of two simpler expressions.

### Applications in Algebra and Beyond

The difference of cubes identity has various applications in algebra and beyond:

**Factoring polynomials:**This identity helps factor polynomials that involve the difference of two cubes.**Solving equations:**It aids in solving equations where one side involves the difference of two cubes.**Calculus:**This identity can be used to simplify derivatives and integrals involving expressions with cubes.

Understanding and applying the difference of cubes identity significantly enhances our algebraic skills, providing us with a valuable tool for simplifying and solving problems.