Understanding the Expansion of (a + b)^3
The expansion of (a + b)^3 is a fundamental concept in algebra, often encountered in various mathematical applications. It represents the cube of the binomial (a + b), which essentially means multiplying the binomial by itself three times.
Expanding the Expression
To understand the expansion, we can perform the multiplication stepbystep:
(a + b)^3 = (a + b) * (a + b) * (a + b)

First Multiplication: (a + b) * (a + b) = a^2 + 2ab + b^2

Second Multiplication: (a^2 + 2ab + b^2) * (a + b) = a^3 + 3a^2b + 3ab^2 + b^3
Therefore, the complete expansion of (a + b)^3 is:
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
The Binomial Theorem
The expansion of (a + b)^3 can also be derived using the binomial theorem, a powerful tool for expanding expressions of the form (a + b)^n.
The binomial theorem states that:
(a + b)^n = Σ (n choose k) * a^(nk) * b^k
Where:
 n is the power of the binomial
 k ranges from 0 to n
 (n choose k) is the binomial coefficient, calculated as n! / (k! * (nk)!)
Applying the theorem to (a + b)^3, we get:
(a + b)^3 = (3 choose 0) * a^3 * b^0 + (3 choose 1) * a^2 * b^1 + (3 choose 2) * a^1 * b^2 + (3 choose 3) * a^0 * b^3
Evaluating the binomial coefficients and simplifying:
(a + b)^3 = 1 * a^3 + 3 * a^2b + 3 * ab^2 + 1 * b^3
This yields the same result as the stepbystep multiplication method.
Key Points
 The expansion of (a + b)^3 follows a specific pattern: a^3 + 3a^2b + 3ab^2 + b^3
 The coefficients of the terms are determined by the binomial theorem, which utilizes binomial coefficients.
 Understanding the expansion of (a + b)^3 is crucial for solving various mathematical problems involving binomials and their powers.
Applications
The expansion of (a + b)^3 finds applications in various fields, including:
 Algebra: Simplifying algebraic expressions, solving equations, and working with polynomials.
 Calculus: Finding derivatives and integrals of polynomial functions.
 Probability: Calculating probabilities in situations involving multiple independent events.
 Physics: Modeling physical phenomena, such as the motion of objects under constant acceleration.
By understanding the expansion of (a + b)^3 and its various applications, you can gain a deeper understanding of algebraic concepts and their relevance to various disciplines.