Simplifying the Algebraic Expression $(x - y)^3 + (y - z)^3 + (z - x)^3$

The problem asks us to simplify the algebraic expression given by the sum of three cubed terms: $(x - y)^3$, $(y - z)^3$, and $(z - x)^3$. This specific structure suggests the use of a known algebraic identity.

Key Algebraic Identity for Sum of Cubes

There is a very useful algebraic identity related to the sum of three cubes. It states that if the sum of three terms is zero, then the sum of their cubes is equal to three times the product of the terms. Mathematically, this is written as:

If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$.

Applying the Identity to the Given Expression

Let's identify the terms in our given expression that can correspond to $a$, $b$, and $c$ in the identity:

• Let $a = (x - y)$
• Let $b = (y - z)$
• Let $c = (z - x)$

Now, let's check if the condition $a + b + c = 0$ is satisfied for these terms:

$a + b + c = (x - y) + (y - z) + (z - x)$

Let's group the terms:

$a + b + c = x - y + y - z + z - x$

$a + b + c = (x - x) + (-y + y) + (-z + z)$

$a + b + c = 0 + 0 + 0$

$a + b + c = 0$

Since the sum of the three terms $(x - y)$, $(y - z)$, and $(z - x)$ is indeed 0, we can apply the identity $a^3 + b^3 + c^3 = 3abc$.

Using the identity with $a = (x - y)$, $b = (y - z)$, and $c = (z - x)$, we get:

$(x - y)^3 + (y - z)^3 + (z - x)^3 = 3 \times (x - y) \times (y - z) \times (z - x)$

So, the simplified form of the expression is $3(x - y)(y - z)(z - x)$.

Comparing with Options

Let's look at the given options:

1. $3xyz$
2. $3(x - y)(y - z)(z - x)$
3. $(x + y + z)(x^2 + y^2 + z^2)$
4. $(x - y)(y - z)(z - x)$

Our result, $3(x - y)(y - z)(z - x)$, matches option 2.

Step-by-Step Solution

Here is a summary of the steps:

1. Recognize the form of the expression as a sum of three cubes.
2. Identify the base terms of the cubes: $(x - y)$, $(y - z)$, and $(z - x)$.
3. Check if the sum of these base terms is zero: $(x - y) + (y - z) + (z - x) = 0$.
4. Since the sum is zero, apply the identity: If $a+b+c=0$, then $a^3+b^3+c^3 = 3abc$.
5. Substitute the base terms back into the identity: $(x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(z - x)$.
6. Compare the result with the given options.

Revision Table: Key Algebraic Identities

Additional Information on Algebraic Identities and Factorization

Algebraic identities are equations that are true for all values of the variables involved. They are fundamental tools in algebra for simplifying expressions, factoring polynomials, and solving equations.

The identity $a+b+c=0 \implies a^3+b^3+c^3=3abc$ is a special case derived from the general factorization formula for the sum of three cubes, which is:

$a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)$

If $a+b+c = 0$, then the right side of the equation becomes $0 \times (a^2 + b^2 + c^2 - ab - bc - ca) = 0$.

This leads to $a^3 + b^3 + c^3 - 3abc = 0$, which simplifies to $a^3 + b^3 + c^3 = 3abc$.

This identity is particularly useful in problems where the sum of the terms being cubed is easily found to be zero, as demonstrated in this problem.