If `alpha, beta and gamma` are the roots of the equation `x^(3)-px^(2)+qx-r=0`, then the value of `alpha^(2)beta+alpha^(2)gamma+beta^(2)alpha+beta^(2)gamma+gamma^(2)alpha+gamma^(2)beta` is equal to

Options

A. `pq+3r`
B. `pq+r`
C. `pq-3r`
D. `(q^(2))/(r )`

Answer

To find the value of \( \alpha^2 \beta + \alpha^2 \gamma + \beta^2 \alpha + \beta^2 \gamma + \gamma^2 \alpha + \gamma^2 \beta \) where \( \alpha, \beta, \gamma \) are the roots of the polynomial \( x^3 - px^2 + qx - r = 0 \), we can use Vieta's formulas and some algebraic manipulation.

### Step-by-Step Solution:

1. **Identify the coefficients using Vieta's formulas**:
- From the polynomial \( x^3 - px^2 + qx - r = 0 \), we have:
- \( \alpha + \beta + \gamma = p \)
- \( \alpha\beta + \beta\gamma + \gamma\alpha = q \)
- \( \alpha\beta\gamma = r \)

2. **Rewrite the expression**:
- We need to find:
\[ S = \alpha^2 \beta + \alpha^2 \gamma + \beta^2 \alpha + \beta^2 \gamma + \gamma^2 \alpha + \gamma^2 \beta \]
- This can be grouped as:
\[ S = \alpha(\alpha\beta + \alpha\gamma) + \beta(\beta\alpha + \beta\gamma) + \gamma(\gamma\alpha + \gamma\beta) \]

3. **Factor out common terms**:
- Notice that:
\[ S = \alpha(\beta + \gamma)\alpha + \beta(\alpha + \gamma)\beta + \gamma(\alpha + \beta)\gamma \]
- We can express \( \beta + \gamma \) as \( p - \alpha \), \( \alpha + \gamma \) as \( p - \beta \), and \( \alpha + \beta \) as \( p - \gamma \).

4. **Substituting back**:
- Substitute these into \( S \):
\[ S = \alpha(p - \alpha) + \beta(p - \beta) + \gamma(p - \gamma) \]
- Expanding this gives:
\[ S = \alpha p - \alpha^2 + \beta p - \beta^2 + \gamma p - \gamma^2 \]
- Combine like terms:
\[ S = p(\alpha + \beta + \gamma) - (\alpha^2 + \beta^2 + \gamma^2) \]

5. **Using Vieta's formulas**:
- We know \( \alpha + \beta + \gamma = p \), so:
\[ S = p^2 - (\alpha^2 + \beta^2 + \gamma^2) \]

6. **Finding \( \alpha^2 + \beta^2 + \gamma^2 \)**:
- We can use the identity:
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]
- Substituting the known values:
\[ \alpha^2 + \beta^2 + \gamma^2 = p^2 - 2q \]

7. **Final substitution**:
- Substitute \( \alpha^2 + \beta^2 + \gamma^2 \) back into the equation for \( S \):
\[ S = p^2 - (p^2 - 2q) = 2q \]

### Conclusion:
Thus, the value of \( \alpha^2 \beta + \alpha^2 \gamma + \beta^2 \alpha + \beta^2 \gamma + \gamma^2 \alpha + \gamma^2 \beta \) is \( 2q \).