Understanding the (a + b)³ Formula
The formula (a + b)³ = a³ + 3a²b + 3ab² + b³ is a fundamental concept in algebra, particularly when working with binomial expansions. This formula helps us to expand the cube of a binomial expression (a + b) without needing to perform the multiplication directly.
How to Derive the Formula
The formula can be derived by expanding the expression (a + b)³ using the distributive property:
(a + b)³ = (a + b)(a + b)(a + b)

Expand the first two brackets: (a + b)(a + b) = a² + 2ab + b²

Multiply the result by (a + b): (a² + 2ab + b²)(a + b) = a³ + 2a²b + ab² + a²b + 2ab² + b³

Combine like terms: a³ + 3a²b + 3ab² + b³
Examples of Using the (a + b)³ Formula
Let's look at some examples of how to use the formula:
Example 1:
(x + 2)³
Using the formula, we get:
x³ + 3(x²)(2) + 3(x)(2²) + 2³ = x³ + 6x² + 12x + 8
Example 2:
(2y  3)³
We can apply the formula by treating (2y) as 'a' and 3 as 'b':
(2y)³ + 3(2y)²(3) + 3(2y)(3)² + (3)³ = 8y³  36y² + 54y  27
Example 3:
(a + 2b)³
Following the same pattern:
a³ + 3(a²)(2b) + 3(a)(2b)² + (2b)³ = a³ + 6a²b + 12ab² + 8b³
Key Points to Remember
 The formula applies to binomials (expressions with two terms) only.
 The coefficients in the expanded expression follow Pascal's Triangle.
 The formula can be used to simplify complex expressions and solve problems involving cubic equations.
By understanding and applying the (a + b)³ formula, you can easily expand cubic expressions, simplify algebraic expressions, and develop a deeper understanding of algebraic concepts.