What is the value of the expression ?
`((a-b)^(3)+(b-c)^(3)+(c-a)^(3))/(3(a-b)(b-c)(c-a))=?`

Options

A. 4
B. 1
C. 2
D. 0

Answer

To solve the expression

\[ \frac{(a-b)^3 + (b-c)^3 + (c-a)^3}{3(a-b)(b-c)(c-a)}, \]

we can use a well-known algebraic identity.

### Step 1: Recognize the identity
We know that for any three numbers \( x, y, z \):

\[ x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - xy - xz - yz). \]

In our case, we can set \( x = a-b \), \( y = b-c \), and \( z = c-a \).

### Step 2: Calculate \( x + y + z \)
Let's compute \( x + y + z \):

\[ x + y + z = (a-b) + (b-c) + (c-a) = a - b + b - c + c - a = 0. \]

### Step 3: Apply the identity
Since \( x + y + z = 0 \), we can simplify our expression using the identity:

\[ x^3 + y^3 + z^3 = 3xyz. \]

Thus, we have:

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

### Step 4: Substitute back into the original expression
Now, substituting this back into the original expression gives:

\[ \frac{3(a-b)(b-c)(c-a)}{3(a-b)(b-c)(c-a)}. \]

### Step 5: Simplify the expression
This simplifies to:

\[ 1. \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{1}. \]