Question

If \(\alpha\) and \(\beta\) (\(\alpha>\beta\)) are the multiple roots of the equation \[ 4x^4 + 4x^3 - 23x^2 - 12x + 36 = 0, \] then find \(2\alpha - \beta\).

• A. \(-1\)
• B. \(3\)
• C. \(5\)
• D. \(-7\)

Hint:

Multiple roots satisfy \(P(x) = 0\) and \(P'(x) = 0\) simultaneously.

Answer 1

The Correct Option is C

Step 1: Identify multiple roots condition
If \(\alpha\) and \(\beta\) are multiple roots, then the polynomial and its derivative both vanish at these roots. Step 2: Derivative of polynomial
\[ P(x) = 4x^4 + 4x^3 - 23x^2 - 12x + 36 \] \[ P'(x) = 16x^3 + 12x^2 - 46x - 12 \] Step 3: Solve \(P(x) = 0\) and \(P'(x) = 0\) simultaneously to find multiple roots. Step 4: Calculate \(2\alpha - \beta\)
By solving system or factorization, find roots \(\alpha\), \(\beta\) and compute \(2\alpha - \beta\). The value is 5.