Evaluation Using Zeros of a Quadratic Polynomial

Video Explanation

Question

If \( \alpha \) and \( \beta \) are the zeroes of the polynomial

\[ f(x) = x^2 + x + 1, \]

find

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

Solution

Step 1: Use Relations Between Zeroes and Coefficients

For the quadratic polynomial \(x^2 + x + 1\),

\[ \alpha + \beta = -\frac{b}{a} = -1, \quad \alpha\beta = \frac{c}{a} = 1 \]

Step 2: Evaluate the Required Expression

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

\[ = \frac{-1}{1} = -1 \]

Conclusion

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