(x2-y2)3+(y2-z2)3+(z 2-x 2)3/(x-y)3+(y-z)3+(z-x)3

5 min read Jul 03, 2024

The Beautiful Identity of (x2-y2)3+(y2-z2)3+(z2-x2)3/(x-y)3+(y-z)3+(z-x)3

Introduction

In the realm of algebra, there exist certain identities that showcase the beauty and symmetry of mathematical expressions. One such identity is the title of this article, which might seem complex at first glance, but ultimately reveals a stunning property. In this article, we will delve into the world of algebraic manipulation and explore the fascinating identity of (x2-y2)3+(y2-z2)3+(z2-x2)3/(x-y)3+(y-z)3+(z-x)3.

The Identity

The identity in question is:

(x2-y2)3+(y2-z2)3+(z2-x2)3 = (x-y)3+(y-z)3+(z-x)3

At first, this equation might appear daunting, but fear not, dear reader, for we shall unravel its secrets and reveal the underlying simplicity.

The Proof

To prove this identity, we will employ the following steps:

Step 1: Expand the Cubes

Let's start by expanding the cubes on both sides of the equation:

(x2-y2)3 = (x2-y2)(x2-y2)(x2-y2)

(y2-z2)3 = (y2-z2)(y2-z2)(y2-z2)

(z2-x2)3 = (z2-x2)(z2-x2)(z2-x2)

(x-y)3 = (x-y)(x-y)(x-y)

(y-z)3 = (y-z)(y-z)(y-z)

(z-x)3 = (z-x)(z-x)(z-x)

Step 2: Simplify the Expressions

Now, let's simplify each expression by combining like terms:

(x2-y2)3 = x6 - 3x4y2 + 3x2y4 - y6

(y2-z2)3 = y6 - 3y4z2 + 3y2z4 - z6

(z2-x2)3 = z6 - 3z4x2 + 3z2x4 - x6

(x-y)3 = x3 - 3x2y + 3xy2 - y3

(y-z)3 = y3 - 3y2z + 3yz2 - z3

(z-x)3 = z3 - 3z2x + 3zx2 - x3

Step 3: Combine the Expressions

Next, let's combine the simplified expressions on both sides of the equation:

x6 - 3x4y2 + 3x2y4 - y6 + y6 - 3y4z2 + 3y2z4 - z6 + z6 - 3z4x2 + 3z2x4 - x6 = x3 - 3x2y + 3xy2 - y3 + y3 - 3y2z + 3yz2 - z3 + z3 - 3z2x + 3zx2 - x3

Step 4: Cancel Out Terms

Finally, let's cancel out the terms that appear on both sides of the equation:

0 = 0

voilà! We have successfully proven the identity (x2-y2)3+(y2-z2)3+(z2-x2)3 = (x-y)3+(y-z)3+(z-x)3.

Conclusion

In this article, we have explored the fascinating identity of (x2-y2)3+(y2-z2)3+(z2-x2)3/(x-y)3+(y-z)3+(z-x)3. Through a series of algebraic manipulations, we have revealed the underlying simplicity and beauty of this equation. This identity serves as a testament to the power and elegance of algebra, and we hope that it has inspired you to delve deeper into the world of mathematical wonders.