Understanding the Expansion of (a + b)³
The equation (a + b)³ = a³ + b³ + 3ab(a + b) is a fundamental algebraic identity that reveals the expansion of the cube of a binomial. This identity is widely used in algebra, calculus, and other branches of mathematics.
Derivation of the Identity
The identity can be derived using the distributive property of multiplication:

Expand the cube: (a + b)³ = (a + b)(a + b)(a + b)

Apply the distributive property twice:
 (a + b)(a + b) = a² + 2ab + b²
 (a² + 2ab + b²)(a + b) = a³ + 2a²b + ab² + a²b + 2ab² + b³

Simplify by combining like terms:
 a³ + 2a²b + ab² + a²b + 2ab² + b³ = a³ + b³ + 3a²b + 3ab²

Factor out 3ab:
 a³ + b³ + 3a²b + 3ab² = a³ + b³ + 3ab(a + b)
Applications of the Identity
The identity (a + b)³ = a³ + b³ + 3ab(a + b) finds various applications in mathematics, including:
 Simplifying algebraic expressions: The identity can be used to simplify complex algebraic expressions involving the cube of a binomial.
 Solving equations: It can be employed in solving equations where one side involves the cube of a binomial.
 Calculus: The identity is crucial in calculating derivatives and integrals involving functions of the form (a + b)³.
Example
Let's illustrate how the identity can be used to simplify an algebraic expression:
Simplify: (x + 2y)³
Solution: Using the identity, we get:
(x + 2y)³ = x³ + (2y)³ + 3(x)(2y)(x + 2y)
Simplifying further:
= x³ + 8y³ + 6xy(x + 2y)
Therefore, the simplified form of (x + 2y)³ is x³ + 8y³ + 6xy(x + 2y).
Conclusion
The identity (a + b)³ = a³ + b³ + 3ab(a + b) is a fundamental tool in algebra and beyond. Understanding its derivation and applications will significantly enhance your ability to manipulate and simplify expressions involving the cube of a binomial.