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

4 min read Jul 03, 2024

The Surprising Beauty of Algebra: The (a-b)^3 + (b-c)^3 + (c-a)^3 Solution

In the realm of algebra, there exist certain equations that have fascinated mathematicians for centuries. One such equation is the cube of the differences of three variables, often represented as:

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

At first glance, this equation may seem daunting, but fear not, dear reader, for we are about to embark on a journey to uncover the surprising beauty of algebra and reveal the secrets hidden within this equation.

The Problem Statement

Prove that:

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

The Solution

To tackle this problem, we will employ the power of algebraic manipulation and clever rearrangement of terms. Let us begin by expanding each cube using the binomial theorem:

$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

$(b-c)^3 = b^3 - 3b^2c + 3bc^2 - c^3$

$(c-a)^3 = c^3 - 3c^2a + 3ca^2 - a^3$

Now, add the three equations together:

$a^3 - 3a^2b + 3ab^2 - b^3 + b^3 - 3b^2c + 3bc^2 - c^3 + c^3 - 3c^2a + 3ca^2 - a^3$

Simplifying and Rearranging

Notice that the terms $a^3$, $b^3$, and $c^3$ cancel each other out. We are left with:

$-3a^2b + 3ab^2 - 3b^2c + 3bc^2 - 3c^2a + 3ca^2$

Next, we rearrange the terms to group them by variables:

$-3a^2b + 3ab^2 + 3ca^2 - 3c^2a - 3b^2c + 3bc^2$

Factor out the common terms:

$-3ab(a-b) + 3ac(c-a) - 3bc(b-c)$

And finally, collect like terms:

$3(a-b)(b-c)(c-a)$

The Conclusion

And there you have it! The seemingly complex equation has been reduced to a beautifully simple expression. The algebraic manipulations have led us to the surprising conclusion that:

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

This equation is a testament to the power and elegance of algebra, showcasing how clever rearrangement of terms can lead to unexpected and beautiful results.

Final Thoughts

The solution to this equation not only highlights the importance of algebraic manipulation but also underscores the significance of pattern recognition and creative problem-solving. As we delve deeper into the world of mathematics, we continue to uncover new and intriguing relationships between variables, waiting to be discovered and appreciated.

So the next time you encounter a daunting equation, remember the (a-b)^3 + (b-c)^3 + (c-a)^3 solution and the surprising beauty of algebra that lies within.