If a,b,c are real and distinct nubmer , then the value of `((a -b)^(3) + (b - c)^(3) + (c -a)^(3))/((a - b)(b - c)(c - a))` is

Options

A. 1
B. a b c
C. 2
D. 3

Answer

To solve the problem, we need to evaluate the expression:

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

### Step 1: Identify the terms
Let:
- \( P = a - b \)
- \( Q = b - c \)
- \( R = c - a \)

### Step 2: Check the sum of the terms
Now, we can calculate \( P + Q + R \):

\[ P + Q + R = (a - b) + (b - c) + (c - a) \]

When we simplify this expression, we find:

\[ P + Q + R = a - b + b - c + c - a = 0 \]

### Step 3: Apply the identity
Since \( P + Q + R = 0 \), we can use the identity:

\[ P^3 + Q^3 + R^3 = 3PQR \]

### Step 4: Substitute back into the expression
Now we substitute back into our expression:

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

### Step 5: Substitute into the original expression
Thus, the original expression becomes:

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

### Step 6: Simplify the expression
Since \( (a - b)(b - c)(c - a) \) is in both the numerator and the denominator, we can simplify:

\[ = 3 \]

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

\[ \boxed{3} \]