Question

If \(\alpha,\beta\) are the integral roots of \[ 6x^4+5x^3-38x^2+5x+6=0, \] then \(\alpha^4+\beta^4=\)

• A. 113
• B. 35
• C. 82
• D. 97 \bigskip

Hint:

Always verify roots by substitution instead of relying only on factor theorem guesses.

Answer 1

The Correct Option is D

Concept: For integral roots, we use Rational Root Theorem and factorization. After finding roots, we compute required expression.

Step 1: Check possible integer roots.
Possible roots: \(\pm1,\pm2,\pm3,\pm6\). Testing \(x=1\): \[ 6+5-38+5+6=-16 \neq 0. \] Testing \(x=2\): \[ 96+40-152+10+6=0. \] So \(x=2\) is a root.

Step 2: Factor the polynomial.
Dividing by \((x-2)\), we get: \[ 6x^4+5x^3-38x^2+5x+6 = (x-2)(6x^3+17x^2-4x-3). \] Now test \(x=-\frac{1}{2}\) is not integer, so try \(x=3\): \[ 162+135-342+15+6= -24 \neq 0. \] Try \(x=-1\): \[ 6-5-38-5+6=-36 \neq 0. \] Try \(x=-3\): \[ 486-135-342-15+6=0. \] So \(x=-3\) is root.

Step 3: Find required values.
Thus integral roots: \[ \alpha=2,\quad \beta=-3. \]

Step 4: Compute expression.
\[ \alpha^4+\beta^4 = 2^4+(-3)^4 = 16+81 = 97. \] \[ \boxed{97} \] Hence correct option: \[ \boxed{(D)}. \]