Question

If a, B are the zeros of the quadratic polynomial x²+x+1, then \(\frac{1}{α}+\frac{1}{β}\) is

• A. 1
• B. -1
• C. 0
• D. None of these

Answer 1

The Correct Option is B

We are tasked with finding the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \), where \( \alpha \) and \( \beta \) are the zeros of the quadratic polynomial \( x^2 + x + 1 \).

Step 1: Recall the relationships between the roots and coefficients of a quadratic polynomial.

For a quadratic polynomial \( ax^2 + bx + c = 0 \), the sum and product of the roots are given by:

\[ \alpha + \beta = -\frac{b}{a}, \quad \alpha \beta = \frac{c}{a}. \]

Here, the polynomial is \( x^2 + x + 1 \), so \( a = 1 \), \( b = 1 \), and \( c = 1 \). Thus:

\[ \alpha + \beta = -\frac{1}{1} = -1, \quad \alpha \beta = \frac{1}{1} = 1. \]

Step 2: Simplify \( \frac{1}{\alpha} + \frac{1}{\beta} \).

Using the formula for the sum of reciprocals:

\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}. \]

Substitute the values of \( \alpha + \beta \) and \( \alpha \beta \):

\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{-1}{1} = -1. \]

Final Answer: The value of \( \frac{1}{\alpha} + \frac{1}{\beta} \) is \( \mathbf{-1} \), which corresponds to option \( \mathbf{(2)} \).