Question

If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( x^2 + 5x + 8 \), then the value of \( \alpha^2 + \beta^2 + 2\alpha \beta \) is:

• A. 25
• B. 5
• C. 8
• D. 64

Hint:

Use the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \) to find expressions involving the zeroes of a polynomial.

Answer 1

The Correct Option is C

We are given the polynomial \( x^2 + 5x + 8 \), which has the following relationships for the sum and product of the zeroes: - \( \alpha + \beta = -\frac{b}{a} = -\frac{5}{1} = -5 \), - \( \alpha \beta = \frac{c}{a} = \frac{8}{1} = 8 \). Now, calculate \( \alpha^2 + \beta^2 + 2\alpha \beta \). We use the identity: \[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta. \] Substituting the values of \( \alpha + \beta \) and \( \alpha \beta \): \[ \alpha^2 + \beta^2 = (-5)^2 - 2 \times 8 = 25 - 16 = 9. \] Thus, the value of \( \alpha^2 + \beta^2 + 2\alpha \beta \) is: \[ 9 + 2 \times 8 = 9 + 16 = 25. \] Thus, the correct answer is \( \boxed{25} \).