The fourth proportional to `p^(2) - pq + q^(2), p^(3) + q^(3), p- q` is

Options

A. `(p+q)`
B. `(p-q)`
C. `(p^(2)- q^(2))`
D. `(q^(2)- p^(2))`

Answer

To find the fourth proportional to the terms \( p^2 - pq + q^2 \), \( p^3 + q^3 \), and \( p - q \), we can follow these steps:

### Step 1: Understand the concept of fourth proportional
The fourth proportional \( x \) to three numbers \( a \), \( b \), and \( c \) is defined such that:
\[ \frac{a}{b} = \frac{c}{x} \]
This means we can express the relationship as:
\[ a \cdot x = b \cdot c \]

### Step 2: Identify the values
Let:
- \( a = p^2 - pq + q^2 \)
- \( b = p^3 + q^3 \)
- \( c = p - q \)

### Step 3: Set up the equation
Using the relationship from Step 1, we can write:
\[ (p^2 - pq + q^2) \cdot x = (p^3 + q^3) \cdot (p - q) \]

### Step 4: Solve for \( x \)
To find \( x \), we rearrange the equation:
\[ x = \frac{(p^3 + q^3)(p - q)}{(p^2 - pq + q^2)} \]

### Step 5: Simplify the expression
Now, we can simplify the expression for \( x \). We know that:
\[ p^3 + q^3 = (p + q)(p^2 - pq + q^2) \]
Thus, substituting this into our expression for \( x \):
\[ x = \frac{(p + q)(p^2 - pq + q^2)(p - q)}{(p^2 - pq + q^2)} \]
The \( (p^2 - pq + q^2) \) terms cancel out:
\[ x = (p + q)(p - q) \]

### Step 6: Final expression
Now we can express the final result:
\[ x = p^2 - q^2 \]

### Conclusion
The fourth proportional to \( p^2 - pq + q^2 \), \( p^3 + q^3 \), and \( p - q \) is:
\[ \boxed{p^2 - q^2} \]