(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3

Jasonbradley

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Jun 16, 2024

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Factoring the Expression (a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3

This article explores the factorization of the expression:

(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3

This expression appears complex, but it can be elegantly factored using a specific algebraic identity. Let's break down the process.

The Key Identity

The key to factoring this expression lies in recognizing the following algebraic identity:

x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)

This identity holds true for any values of x, y, and z.

Applying the Identity

Let's apply this identity to our expression. First, we need to identify the corresponding values of x, y, and z:

• x = a^2 - b^2
• y = b^2 - c^2
• z = c^2 - a^2

Now, let's substitute these values into the identity:

(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 - 3(a^2 - b^2)(b^2 - c^2)(c^2 - a^2) = (a^2 - b^2 + b^2 - c^2 + c^2 - a^2)[(a^2 - b^2)^2 + (b^2 - c^2)^2 + (c^2 - a^2)^2 - (a^2 - b^2)(b^2 - c^2) - (a^2 - b^2)(c^2 - a^2) - (b^2 - c^2)(c^2 - a^2)]

Notice that the first factor on the right-hand side simplifies to 0. Therefore, the entire expression equals 0.

The Final Result

We've proven that (a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3 = 0. This factorization illustrates the power of algebraic identities in simplifying complex expressions.