If `p ^(3) - q ^(3) = ((p-q) (p-q) ^(2) + xpq)` then value of x is

Options

A. 1
B. `-1`
C. 3
D. 2

Answer

To solve the equation \( p^3 - q^3 = (p - q)(p - q)^2 + xpq \) and find the value of \( x \), we can follow these steps:

### Step 1: Understand the left-hand side
The left-hand side of the equation is \( p^3 - q^3 \). We can factor this expression using the difference of cubes formula:
\[ p^3 - q^3 = (p - q)(p^2 + pq + q^2) \]

### Step 2: Rewrite the equation
Now, we rewrite the equation using the factored form:
\[ (p - q)(p^2 + pq + q^2) = (p - q)(p - q)^2 + xpq \]

### Step 3: Cancel \( (p - q) \) from both sides
Assuming \( p \neq q \), we can divide both sides by \( (p - q) \):
\[ p^2 + pq + q^2 = (p - q) + xpq \]

### Step 4: Expand the right-hand side
Now, we can expand the right-hand side:
\[ p^2 + pq + q^2 = p - q + xpq \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ p^2 + pq + q^2 - p + q - xpq = 0 \]

### Step 6: Combine like terms
Combine the terms:
\[ p^2 + (1 - x)pq + q^2 + q - p = 0 \]

### Step 7: Group terms
We can group the terms involving \( p \) and \( q \):
\[ p^2 - p + q^2 + q + (1 - x)pq = 0 \]

### Step 8: Set coefficients to zero
For this equation to hold for all values of \( p \) and \( q \), the coefficient of \( pq \) must equal zero:
\[ 1 - x = 0 \]

### Step 9: Solve for \( x \)
Solving for \( x \):
\[ x = 1 \]

### Step 10: Verify the solution
To verify, substitute \( x = 3 \) back into the equation and check if both sides are equal.

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{3} \]