(a+b+c)^3 Expanded

3 min read Jun 16, 2024

Expanding the Cube of a Trinomial: (a + b + c)³

The expansion of (a + b + c)³ is a fundamental concept in algebra, often encountered in various mathematical contexts. Understanding the expansion is crucial for simplifying expressions, solving equations, and working with polynomials.

The Expansion

The expansion of (a + b + c)³ results in a sum of ten terms:

(a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3ac² + 3b²c + 3bc² + 6abc

Methods for Expansion

There are two primary methods to arrive at this expansion:

1. Direct Multiplication:

Begin by multiplying (a + b + c) by itself: (a + b + c) * (a + b + c)
Then multiply the result by (a + b + c) again: (a² + ab + ac + ba + b² + bc + ca + cb + c²) * (a + b + c)
Perform the multiplication and combine like terms to get the final result.

2. Binomial Theorem:

While the binomial theorem directly applies to expansions of the form (x + y)ⁿ, it can be extended to trinomials.
First, expand (a + b + c) as ((a + b) + c)³.
Apply the binomial theorem to expand ((a + b) + c)³, treating (a + b) as one term: ((a + b)³ + 3(a + b)²c + 3(a + b)c² + c³)
Expand each term individually using the binomial theorem again.

Key Points

The expansion of (a + b + c)³ is a sum of ten terms, each with a specific combination of variables.
The coefficients of each term are determined by the binomial coefficients.
The expansion can be obtained through direct multiplication or by using the binomial theorem.

Applications

The expansion of (a + b + c)³ has various applications in different fields:

Algebra: Simplifying expressions, solving equations, and working with polynomials.
Calculus: Finding derivatives and integrals of functions involving trinomials.
Physics: Solving problems related to mechanics, electricity, and magnetism.
Chemistry: Calculating chemical reactions and determining the concentration of solutions.

Understanding the expansion of (a + b + c)³ provides a strong foundation for various mathematical concepts and applications.