If p, q, r are all real numbers. Then `( p - q) ^(3) + (q - r) ^(3) + ( r - p) ^(3)` is equal to

Options

A. (p - q) (q - r) (r - p)
B. 3 (p - q) ( q - r) ( r - p)
C. 0
D. 1

Answer

To solve the expression \( (p - q)^3 + (q - r)^3 + (r - p)^3 \), we can use a known algebraic identity.

### Step-by-Step Solution:

1. **Identify the Variables**:
Let:
- \( a = p - q \)
- \( b = q - r \)
- \( c = r - p \)

We notice that \( a + b + c = (p - q) + (q - r) + (r - p) = 0 \).

2. **Use the Identity**:
There is a useful identity in algebra that states:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
Since \( a + b + c = 0 \), we can simplify the identity:
\[ a^3 + b^3 + c^3 = 3abc \]

3. **Calculate \( abc \)**:
Now we need to find \( abc \):
\[ abc = (p - q)(q - r)(r - p) \]

4. **Substitute Back**:
Therefore, we substitute back into our expression:
\[ (p - q)^3 + (q - r)^3 + (r - p)^3 = 3(p - q)(q - r)(r - p) \]

### Final Result:
Thus, the expression \( (p - q)^3 + (q - r)^3 + (r - p)^3 \) is equal to:
\[ 3(p - q)(q - r)(r - p) \]