The Formula (a + b)(a²  ab + b²) and its Significance
In mathematics, the expression (a + b)(a²  ab + b²) is a fundamental formula used to simplify and expand algebraic expressions. It is commonly known as the sum of cubes formula or the difference of cubes formula, depending on the context.
Understanding the Formula
The formula states that the product of the sum of two terms (a + b) and the expression (a²  ab + b²) equals the cube of the first term (a) plus the cube of the second term (b). This can be represented as:
(a + b)(a²  ab + b²) = a³ + b³
How to Derive the Formula
The formula can be derived using the distributive property of multiplication:

Expand the brackets: (a + b)(a²  ab + b²) = a(a²  ab + b²) + b(a²  ab + b²)

Apply the distributive property: = a³  a²b + ab² + ba²  ab² + b³

Simplify by combining like terms: = a³ + b³
Applications of the Formula
The sum of cubes formula has wide applications in various areas of mathematics, including:
 Factoring algebraic expressions: The formula allows us to factor expressions of the form a³ + b³.
 Solving equations: The formula can be used to solve cubic equations by factoring them into the form a³ + b³ = 0.
 Simplifying expressions: The formula can be used to simplify complex algebraic expressions.
 Calculus: The formula is used in calculus to find the derivative of functions.
Example of Application
Let's consider the expression x³ + 8. We can factor this expression using the sum of cubes formula:

Identify the values of a and b: a = x and b = 2 (since 8 = 2³)

Apply the formula: x³ + 8 = (x + 2)(x²  2x + 4)
Therefore, the factored form of x³ + 8 is (x + 2)(x²  2x + 4).
Conclusion
The formula (a + b)(a²  ab + b²) is a powerful tool in mathematics that simplifies calculations, allows for factoring of expressions, and helps in solving equations. Understanding its derivation and applications is crucial for mastering algebraic manipulations and problemsolving.