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:

To find \( (\alpha + 1)(\beta + 1) \), use the relation: \[ (\alpha + 1)(\beta + 1) = \alpha \beta + (\alpha + \beta) + 1 \] This formula simplifies the computation by leveraging known properties of the polynomial's roots.

Answer 1

The Correct Option is B

Step 1: Expand the given polynomial expression: \[ x^2 - 3(x + 1) - 5 = x^2 - 3x - 3 - 5 = x^2 - 3x - 8 \] Step 2: The sum of the zeroes \( \alpha + \beta \) is given by the coefficient of \( x \) with opposite sign, which is 3. The product of the zeroes \( \alpha \beta \) is the constant term, which is \(-8\). Step 3: Calculate \( (\alpha + 1)(\beta + 1) \): \[ (\alpha + 1)(\beta + 1) = \alpha \beta + \alpha + \beta + 1 = -8 + 3 + 1 = -4 \] Thus, the correct answer is \( -4 \), which corresponds to option (C).