Factors of `(2x - 3y) ^(3) + ( 3y - 5z) ^(3) + ( 5z - 2x) ^(3)` are:

Options

A. `3 ( 2x -3y) ( 3y - 5z) ( 5z - 2x)`
B. `3 ( 3x - 2y) (3y - 5z) ( 5z - 2x)`
C. `3 ( 2x - 3y) ( 5y - 3z) ( 5z - 2x)`
D. `3 ( 2x - 3y) (3y - 5z) (2z - 5x)`

Answer

To factor the expression \( (2x - 3y)^3 + (3y - 5z)^3 + (5z - 2x)^3 \), we can use the identity for the sum of cubes:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

### Step 1: Identify \(a\), \(b\), and \(c\)

Let:
- \( a = 2x - 3y \)
- \( b = 3y - 5z \)
- \( c = 5z - 2x \)

### Step 2: Check if \( a + b + c = 0 \)

Now, we need to check if \( a + b + c = 0 \):

\[ a + b + c = (2x - 3y) + (3y - 5z) + (5z - 2x) \]

Combine like terms:

\[ = 2x - 2x - 3y + 3y - 5z + 5z = 0 \]

Since \( a + b + c = 0 \), we can use the identity.

### Step 3: Apply the identity

From the identity, we have:

\[ a^3 + b^3 + c^3 = 3abc \]

### Step 4: Calculate \(abc\)

Now we need to calculate \(abc\):

\[ abc = (2x - 3y)(3y - 5z)(5z - 2x) \]

### Step 5: Write the final factorization

Thus, we can write:

\[ (2x - 3y)^3 + (3y - 5z)^3 + (5z - 2x)^3 = 3(2x - 3y)(3y - 5z)(5z - 2x) \]

### Final Answer

The factors of \( (2x - 3y)^3 + (3y - 5z)^3 + (5z - 2x)^3 \) are:

\[ 3(2x - 3y)(3y - 5z)(5z - 2x) \]