Question

If the sum of two roots \( \alpha, \beta \) of the equation \[ x^4 - x^3 - 8x^2 + 2x + 12 = 0 \] is zero and \( \gamma, \delta \) (\( \gamma\delta \)) are its other roots, then \( 3\gamma + 2\delta \) is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( 3 \)
• D. \( 5 \)

Hint:

For quartic equations, use symmetric sum properties to express unknown roots in terms of known sums.

Answer 1

The Correct Option is D

Step 1: Use Sum of Roots Property Sum of all roots: \[ \alpha + \beta + \gamma + \delta = 1 \] Given \( \alpha + \beta = 0 \), \[ \gamma + \delta = 1 \] Step 2: Compute \( 3\gamma + 2\delta \) Given that \( \gamma\delta \), we solve: \[ 3\gamma + 2\delta = 5 \] Thus, the correct answer is 5.