Question

If \( \alpha, \beta, \gamma \) are the roots of the equation \[ x^3 + px^2 + qx + r = 0, \] then \[ (\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) =\ ? \]

• A. \( p - qr \)
• B. \( q - rp \)
• C. \( r - pq \)
• D. \( r + pq \)

Hint:

Remember key identity: \((\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) = (\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha) - \alpha\beta\gamma\)

Answer 1

The Correct Option is C

Step 1: Given the roots of the cubic equation are \( \alpha, \beta, \gamma \), we apply Vieta’s formulas: \begin{align*} \alpha + \beta + \gamma = -p
\alpha\beta + \beta\gamma + \gamma\alpha = q
\alpha\beta\gamma = -r \] Step 2: Evaluate the required expression: \[ (\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) \] Step 3: Expand step-by-step: Let’s denote: \[ (\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) = S \] First, use identity: \[ (\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) = (\alpha + \beta + \gamma)(\alpha\beta + \beta\gamma + \gamma\alpha) - \alpha\beta\gamma \] Substitute the known values: \[ = (-p)(q) - (-r) = -pq + r = r - pq \] Step 4: Final Answer: \[ \boxed{r - pq} \] % Tip