Proving the Cube of a Binomial: (a + b)³ = a³ + b³ + 3ab(a + b)
This article will provide a stepbystep proof of the algebraic identity: (a + b)³ = a³ + b³ + 3ab(a + b).
Understanding the Equation
The equation represents the cube of a binomial (a + b). It states that expanding this cube results in the sum of the cubes of 'a' and 'b', plus three times the product of 'a', 'b', and the original binomial (a + b).
Proof by Expansion
We can prove this identity using the distributive property and simplification:

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

Expand the first two factors: (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 + 2a²b + ab² + a²b + 2ab² + b³ = a³ + b³ + 3a²b + 3ab²

Factor out 3ab: a³ + b³ + 3a²b + 3ab² = a³ + b³ + 3ab(a + b)
Therefore, we have proven that (a + b)³ = a³ + b³ + 3ab(a + b).
Applications of the Identity
This identity has various applications in algebra, including:
 Simplifying expressions: It allows us to quickly expand cubes of binomials.
 Solving equations: It can be used to simplify equations containing cubes of binomials.
 Calculus: It is helpful in differentiating and integrating expressions involving cubes of binomials.
Conclusion
The proof demonstrates the algebraic expansion of (a + b)³ using the distributive property and simplification. This identity is a valuable tool in algebra, providing a simplified way to represent the cube of a binomial and enabling further manipulation and simplification of expressions.