Question

The sum and product of the roots of the polynomial \(x^2 - 2x - 8\) are respectively

• A. 2, 8
• B. -2, 8
• C. -2, -8
• D. 2, -8

Hint:

Using Vieta's formulas is much faster than factoring or using the quadratic formula.
Always look at the sign: the sum of the roots is the negative of the coefficient of \( x \) (when \( a = 1 \)), and the product is the constant term.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires finding the sum and product of the roots of a quadratic polynomial using Vieta's formulas.
Key Formula or Approach:
For any quadratic polynomial of the form:
\[ ax^2 + bx + c \] with roots \( \alpha \) and \( \beta \), Vieta's formulas state that:
- Sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} \] - Product of the roots:
\[ \alpha \cdot \beta = \frac{c}{a} \]

Step 2: Detailed Explanation:
Let us identify the coefficients of the given quadratic polynomial:
\[ x^2 - 2x - 8 \] Comparing this with the standard form \( ax^2 + bx + c \):
- Coefficient of \( x^2 \), \( a = 1 \)
- Coefficient of \( x \), \( b = -2 \)
- Constant term, \( c = -8 \)
Now, calculate the sum and product of the roots using Vieta's formulas:
- Sum of the roots:
\[ \text{Sum} = -\frac{b}{a} = -\frac{-2}{1} = 2 \] - Product of the roots:
\[ \text{Product} = \frac{c}{a} = \frac{-8}{1} = -8 \] Therefore, the sum and product of the roots are \( 2 \) and \( -8 \), respectively.
We can also verify this by explicitly finding the roots of the quadratic equation:
\[ x^2 - 2x - 8 = 0 \] Factor the quadratic term:
\[ (x - 4)(x + 2) = 0 \implies x = 4 \quad \text{or} \quad x = -2 \] The two roots are \( \alpha = 4 \) and \( \beta = -2 \).
- Sum of roots: \( 4 + (-2) = 2 \).
- Product of roots: \( 4 \times (-2) = -8 \).
This confirms our calculations.

Step 3: Final Answer:
The sum and product of the roots are \( 2 \) and \( -8 \), respectively.
Therefore, the correct choice is Option (D).