If ` alpha , beta , gamma ` are the roots of the equation `x^3 +px^2 +qx +r=0` then ` sum alpha^2 ( beta + gamma)=`

Options

A. `p^2 -2q`
B. `-p^3 +3pq -3r`
C. `p^4 -4p^2 q+4pr +2q^2`
D. `3r-pq`

Answer

To find the value of \( \sum \alpha^2 (\beta + \gamma) \) where \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + px^2 + qx + r = 0 \), we can follow these steps:

### Step 1: Use Vieta's Formulas
From Vieta's formulas, we know:
- \( \alpha + \beta + \gamma = -p \)
- \( \alpha\beta + \beta\gamma + \gamma\alpha = q \)
- \( \alpha\beta\gamma = -r \)

### Step 2: Express \( \sum \alpha^2 (\beta + \gamma) \)
We can rewrite the expression \( \sum \alpha^2 (\beta + \gamma) \) as:
\[ \sum \alpha^2 (\beta + \gamma) = \alpha^2 (\beta + \gamma) + \beta^2 (\alpha + \gamma) + \gamma^2 (\alpha + \beta) \]
Notice that \( \beta + \gamma = -p - \alpha \), \( \alpha + \gamma = -p - \beta \), and \( \alpha + \beta = -p - \gamma \).

### Step 3: Substitute the Values
Substituting these into the expression gives:
\[ = \alpha^2 (-p - \alpha) + \beta^2 (-p - \beta) + \gamma^2 (-p - \gamma) \]
This simplifies to:
\[ = -p(\alpha^2 + \beta^2 + \gamma^2) - (\alpha^3 + \beta^3 + \gamma^3) \]

### Step 4: Find \( \alpha^2 + \beta^2 + \gamma^2 \)
We can find \( \alpha^2 + \beta^2 + \gamma^2 \) using the identity:
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]
Substituting the values from Vieta's:
\[ = (-p)^2 - 2q = p^2 - 2q \]

### Step 5: Find \( \alpha^3 + \beta^3 + \gamma^3 \)
Using the identity for the sum of cubes:
\[ \alpha^3 + \beta^3 + \gamma^3 = 3\alpha\beta\gamma + (\alpha + \beta + \gamma)(\alpha^2 + \beta^2 + \gamma^2) \]
Substituting the values we have:
\[ = 3(-r) + (-p)(p^2 - 2q) = -3r - p(p^2 - 2q) \]

### Step 6: Combine Everything
Now substituting back into our expression:
\[ \sum \alpha^2 (\beta + \gamma) = -p(p^2 - 2q) + 3r + 3r \]
This simplifies to:
\[ = -p(p^2 - 2q) + 3r \]

### Final Expression
Thus, the final expression for \( \sum \alpha^2 (\beta + \gamma) \) is:
\[ \sum \alpha^2 (\beta + \gamma) = 3r - pq \]