%5E3%2B(b-c)%5E3%2B(c-a)%5E3%2F9(a-b)(b-c)(c-a).jpeg "(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:
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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)
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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.