Question

If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( x^2 - 3(x+1) - 5 \), then the value of \( (\alpha +1)(\beta +1) \) is:

• A. \( 3 \)
• B. \( -3 \)
• C. \( -4 \)
• D. \( 4 \)

Hint:

Use the identity: \[ (\alpha + k)(\beta + k) = \alpha\beta + k(\alpha + \beta) + k^2. \]

Answer 1

The Correct Option is D

Expanding the given polynomial:
\[ x^2 - 3x - 3 - 5 = x^2 - 3x - 8. \] Sum and product of roots:
\[ \alpha + \beta = -\left(\frac{-3}{1}\right) = 3, \quad \alpha \beta = \frac{-8}{1} = -8. \] Now, we compute:
\[ (\alpha + 1)(\beta + 1) = \alpha \beta + \alpha + \beta + 1. \] \[ = (-8) + 3 + 1 + 1 = 4. \]