Deriving the Formula for (a + b + c)³
The formula for the expansion of (a + b + c)³ is a fundamental concept in algebra with various applications in solving equations, simplifying expressions, and understanding polynomial behavior. Let's explore the derivation of this formula step by step.
Understanding the Concept
The expansion of (a + b + c)³ represents the cube of the trinomial (a + b + c). In simpler terms, it means multiplying the trinomial by itself three times.
Derivation Steps

Initial Expansion: We start by expanding the first two terms: (a + b + c)³ = (a + b + c) * (a + b + c) * (a + b + c) = [(a + b) + c] * [(a + b) + c] * (a + b + c)

Applying the Distributive Property: Now, we expand the first two brackets using the distributive property (also known as FOIL): = [(a + b)² + 2c(a + b) + c²] * (a + b + c)

Expanding Further: We distribute the remaining terms: = (a² + 2ab + b² + 2ac + 2bc + c²) * (a + b + c)

Final Distribution: Finally, we expand the entire expression using the distributive property again: = a³ + 3a²b + 3ab² + b³ + 3a²c + 6abc + 3b²c + 3ac² + 3bc² + c³
The Final Formula
The final expansion of (a + b + c)³ is: (a + b + c)³ = a³ + 3a²b + 3ab² + b³ + 3a²c + 6abc + 3b²c + 3ac² + 3bc² + c³
Important Points to Note
 Symmetry: Notice the symmetry in the terms. The coefficients of the terms with the same degree are equal. For example, the coefficients of a²b, ab², and a²c are all 3.
 Binomial Theorem: This formula can be derived using the binomial theorem, which provides a general method for expanding expressions of the form (x + y)ⁿ.
 Applications: This formula is frequently used in various fields like physics, engineering, and economics, especially when dealing with equations involving multiple variables.
By understanding the derivation of the formula for (a + b + c)³, you gain a deeper understanding of how algebraic expressions are expanded and manipulated. This knowledge proves invaluable in solving complex mathematical problems and gaining a more comprehensive understanding of algebraic concepts.