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 \( \alpha^2 \beta + \alpha^2 \gamma + \beta^2 \alpha + \beta^2 \gamma + \gamma^2 \alpha + \gamma^2 \beta \), we can use the relationships between the roots and the coefficients of the polynomial given by Vieta's formulas.

### Step 1: Identify the coefficients
Given the polynomial \( x^3 - px^2 + qx - r = 0 \), we can identify the coefficients:
- \( A = 1 \)
- \( B = -p \)
- \( C = q \)
- \( D = -r \)

From Vieta's formulas, we know:
- \( \alpha + \beta + \gamma = p \)
- \( \alpha \beta + \beta \gamma + \gamma \alpha = q \)
- \( \alpha \beta \gamma = r \)

### Step 2: Rewrite the expression
We can rewrite the expression \( \alpha^2 \beta + \alpha^2 \gamma + \beta^2 \alpha + \beta^2 \gamma + \gamma^2 \alpha + \gamma^2 \beta \) as:
\[ \alpha^2 (\beta + \gamma) + \beta^2 (\alpha + \gamma) + \gamma^2 (\alpha + \beta) \]

### Step 3: Substitute the sums
Using the fact that \( \beta + \gamma = p - \alpha \), \( \alpha + \gamma = p - \beta \), and \( \alpha + \beta = p - \gamma \), we can substitute these into our expression:
\[ = \alpha^2 (p - \alpha) + \beta^2 (p - \beta) + \gamma^2 (p - \gamma) \]

### Step 4: Expand the expression
Expanding the expression gives:
\[ = \alpha^2 p - \alpha^3 + \beta^2 p - \beta^3 + \gamma^2 p - \gamma^3 \]
Combining like terms, we have:
\[ = p(\alpha^2 + \beta^2 + \gamma^2) - (\alpha^3 + \beta^3 + \gamma^3) \]

### Step 5: Use the identity for squares
We can express \( \alpha^2 + \beta^2 + \gamma^2 \) in terms of \( p \) and \( q \):
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha \beta + \beta \gamma + \gamma \alpha) = p^2 - 2q \]

### Step 6: Use the identity for cubes
Using the identity for the sum of cubes:
\[ \alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)(\alpha^2 + \beta^2 + \gamma^2 - \alpha \beta - \beta \gamma - \gamma \alpha) = p((p^2 - 2q) - q) = p(p^2 - 3q) \]

### Step 7: Substitute back into the expression
Now substituting back into our expression:
\[ = p(p^2 - 2q) - p(p^2 - 3q) \]
This simplifies to:
\[ = p(p^2 - 2q - p^2 + 3q) = p(q) \]

### Final Result
Thus, the value of \( \alpha^2 \beta + \alpha^2 \gamma + \beta^2 \alpha + \beta^2 \gamma + \gamma^2 \alpha + \gamma^2 \beta \) is:
\[ \boxed{pq - 3r} \]