Evaluation Using Zeros of a Cubic Polynomial
Video Explanation
Question
If \( \alpha, \beta, \gamma \) are the zeroes of the polynomial
\[ f(x) = x^3 – px^2 + qx – r, \]
find
\[ \frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha}. \]
Solution
Step 1: Write Relations Between Zeroes and Coefficients
For the cubic polynomial \(x^3 – px^2 + qx – r\),
\[ \alpha + \beta + \gamma = p \]
\[ \alpha\beta + \beta\gamma + \gamma\alpha = q \]
\[ \alpha\beta\gamma = r \]
Step 2: Evaluate the Required Expression
\[ \frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} \]
\[ = \frac{\gamma + \alpha + \beta}{\alpha\beta\gamma} \]
Step 3: Substitute the Values
\[ = \frac{p}{r} \]
Conclusion
\[ \boxed{ \frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} = \frac{p}{r} } \]