If ` alpha , beta , gamma ` are the roots of the equation `x^3 +px^2 + qx +r=0` then the coefficient of x in cubic equation whose roots are
` alpha ( beta + gamma ) , beta ( gamma + alpha) and gamma ( alpha + beta)` is

Options

A. 2q
B. `q^2 +pr`
C. `p^2 -qr`
D. `r(pq-r)`

Answer

To solve the problem, we need to find the coefficient of \( x \) in the cubic equation whose roots are \( \alpha (\beta + \gamma) \), \( \beta (\gamma + \alpha) \), and \( \gamma (\alpha + \beta) \), given that \( \alpha, \beta, \gamma \) are the roots of the equation \( x^3 + px^2 + qx + r = 0 \).

### Step 1: Identify the roots
The roots of the new cubic equation are:
- \( L = \alpha (\beta + \gamma) \)
- \( M = \beta (\gamma + \alpha) \)
- \( N = \gamma (\alpha + \beta) \)

### Step 2: Use Vieta's Formulas
From the original cubic equation \( x^3 + px^2 + qx + r = 0 \), we can use Vieta's formulas:
1. \( \alpha + \beta + \gamma = -p \)
2. \( \alpha\beta + \beta\gamma + \gamma\alpha = q \)
3. \( \alpha\beta\gamma = -r \)

### Step 3: Calculate \( L + M + N \)
We calculate:
\[ L + M + N = \alpha(\beta + \gamma) + \beta(\gamma + \alpha) + \gamma(\alpha + \beta) \]
Using \( \beta + \gamma = -p - \alpha \) (from Vieta's):
\[ L + M + N = \alpha(-p - \alpha) + \beta(-p - \beta) + \gamma(-p - \gamma) \]
This simplifies to:
\[ = -p(\alpha + \beta + \gamma) - (\alpha^2 + \beta^2 + \gamma^2) \]
Using \( \alpha + \beta + \gamma = -p \):
\[ = -p(-p) - (\alpha^2 + \beta^2 + \gamma^2) = p^2 - (\alpha^2 + \beta^2 + \gamma^2) \]

### Step 4: Calculate \( \alpha^2 + \beta^2 + \gamma^2 \)
Using the identity:
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]
Substituting the values:
\[ = (-p)^2 - 2q = p^2 - 2q \]
Thus,
\[ L + M + N = p^2 - (p^2 - 2q) = 2q \]

### Step 5: Calculate \( LM + MN + NL \)
Next, we find \( LM + MN + NL \):
\[ LM = \alpha(\beta + \gamma) \cdot \beta(\gamma + \alpha) = \alpha\beta(\beta\gamma + \alpha\beta + \gamma\alpha) \]
Using \( \beta\gamma + \alpha\beta + \gamma\alpha = q \):
\[ LM = \alpha\beta q \]
Similarly, we can derive:
\[ MN = \beta\gamma q \quad \text{and} \quad NL = \gamma\alpha q \]
Thus,
\[ LM + MN + NL = q(\alpha\beta + \beta\gamma + \gamma\alpha) = q \cdot q = q^2 \]

### Step 6: Calculate \( LMN \)
Finally, we calculate \( LMN \):
\[ LMN = \alpha(\beta + \gamma) \cdot \beta(\gamma + \alpha) \cdot \gamma(\alpha + \beta) \]
This can be expressed as:
\[ = \alpha\beta\gamma \cdot (\beta + \gamma)(\gamma + \alpha)(\alpha + \beta) \]
Using \( \alpha\beta\gamma = -r \) and expanding the product of sums, we can derive:
\[ LMN = -r \cdot \text{(some expression)} \]

### Step 7: Form the cubic equation
The cubic equation with roots \( L, M, N \) is:
\[ x^3 - (L + M + N)x^2 + (LM + MN + NL)x - LMN = 0 \]
The coefficient of \( x \) is \( LM + MN + NL \), which we found to be \( q^2 \).

### Final Answer
The coefficient of \( x \) in the cubic equation is:
\[ \boxed{q^2} \]