(a-b)^3+(b-c)^3+(c-a)^3/9(a-b)(b-c)(c-a)

3 min read Jun 16, 2024
(a-b)^3+(b-c)^3+(c-a)^3/9(a-b)(b-c)(c-a)

Exploring the Algebraic Identity: (a-b)^3 + (b-c)^3 + (c-a)^3 / 9(a-b)(b-c)(c-a)

This article delves into the interesting algebraic identity:

(a-b)^3 + (b-c)^3 + (c-a)^3 / 9(a-b)(b-c)(c-a)

We will explore its derivation, significance, and applications.

Derivation of the Identity

The identity can be derived using the following steps:

  1. Factoring the numerator: We can factor the numerator using the sum of cubes formula:

    (a-b)^3 + (b-c)^3 + (c-a)^3 = 3(a-b)(b-c)(c-a)

  2. Simplifying the expression: Substituting the factored numerator in the original expression:

    3(a-b)(b-c)(c-a) / 9(a-b)(b-c)(c-a) = 1/3

Therefore, the identity simplifies to 1/3.

Significance of the Identity

This identity is significant because it reveals a relationship between the cubes of differences and the product of those differences. It provides a concise and elegant way to express this relationship.

Applications

The identity can be applied in various mathematical contexts, including:

  • Algebraic manipulation: It can simplify complex expressions involving cubes of differences.
  • Proofs: The identity can be used to prove other mathematical theorems or identities.
  • Problem solving: It can help in solving problems related to algebraic expressions involving cubes of differences.

Example

Let's consider a simple example:

If a = 2, b = 1, and c = 0, then:

(2-1)^3 + (1-0)^3 + (0-2)^3 / 9(2-1)(1-0)(0-2) = 1^3 + 1^3 + (-2)^3 / 9(1)(-1)(-2) = 1 + 1 - 8 / 18 = -6/18 = -1/3.

This confirms that the identity holds true for these values.

Conclusion

The algebraic identity (a-b)^3 + (b-c)^3 + (c-a)^3 / 9(a-b)(b-c)(c-a) = 1/3 is a valuable tool in algebra. It simplifies expressions, provides elegant relationships, and can be applied in diverse mathematical situations.

Related Post


Featured Posts