Question

If $ \alpha $ and $ \beta $ are the zeros of the polynomial, such that $ \alpha + \beta = 6 $ and $ \alpha \beta = 4 $, then the quadratic polynomial is:

• A. $ x^2 - x - 6 $
• B. $ x^2 + x - 6 $
• C. $ x^2 - 6x + 4 $
• D. $ x^2 - 6x + 4 $

Hint:

To construct a quadratic polynomial given the sum and product of its roots, use Vieta's formulas: $ P(x) = x^2 - (\text{sum of roots})x + (\text{product of roots}) $ . This ensures the polynomial accurately reflects the given conditions.

Answer 1

The Correct Option is C

Step 1: Use Vieta's Formulas.
For a quadratic polynomial: \[ P(x) = x^2 - (\text{sum of roots})x + (\text{product of roots}), \] where the sum of the roots is \( \alpha + \beta = 6 \) and the product is \( \alpha \beta = 4 \). Step 2: Substitute the values.
\[ P(x) = x^2 - 6x + 4. \] Step 3: Analyze the options.
Option (1): \( x^2 - x - 6 \) — Incorrect.
Option (2): \( x^2 + x - 6 \) — Incorrect.
Option (3): \( x^2 - 6x + 4 \) — Correct.
Option (4): \( x^2 - 6x + 4 \) — Correct. Step 4: Final Answer.
\[ \boxed{x^2 - 6x + 4} \]