The value of the expression `((a - b)^(2))/((b - c) ( c - a))+ ((b - c)^(2))/(( a - b) ( c - a)) + ((c - a)^(2))/( (a - b) ( b - c))` is

Options

A. 0
B. 3
C. `(1)/(3)`
D. 2

Answer

To find the value of the expression

\[ \frac{(a - b)^{2}}{(b - c)(c - a)} + \frac{(b - c)^{2}}{(a - b)(c - a)} + \frac{(c - a)^{2}}{(a - b)(b - c)}, \]

we will follow these steps:

### Step 1: Identify the common denominator
The common denominator for the three fractions is \((a - b)(b - c)(c - a)\).

### Step 2: Rewrite each fraction with the common denominator
To combine the fractions, we rewrite each term with the common denominator:

1. For the first term:
\[ \frac{(a - b)^{2}}{(b - c)(c - a)} = \frac{(a - b)^{2} \cdot (a - b)}{(b - c)(c - a)(a - b)} = \frac{(a - b)^{3}}{(b - c)(c - a)(a - b)} \]

2. For the second term:
\[ \frac{(b - c)^{2}}{(a - b)(c - a)} = \frac{(b - c)^{2} \cdot (b - c)}{(a - b)(c - a)(b - c)} = \frac{(b - c)^{3}}{(a - b)(c - a)(b - c)} \]

3. For the third term:
\[ \frac{(c - a)^{2}}{(a - b)(b - c)} = \frac{(c - a)^{2} \cdot (c - a)}{(a - b)(b - c)(c - a)} = \frac{(c - a)^{3}}{(a - b)(b - c)(c - a)} \]

### Step 3: Combine the fractions
Now, we can combine the three fractions:

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

### Step 4: Use the identity for the sum of cubes
We know that if \(x + y + z = 0\), then \(x^3 + y^3 + z^3 = 3xyz\). Here, let:

- \(x = a - b\)
- \(y = b - c\)
- \(z = c - a\)

We can see that:

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

Thus, we can apply the identity:

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

### Step 5: Substitute back into the expression
Substituting this result back into our expression gives:

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

### Step 6: Simplify the expression
Now, we can cancel out the common terms in the numerator and denominator:

\[ = 3. \]

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

\[ \boxed{3}. \]