Evaluation Using Zeros of a Cubic Polynomial
Video Explanation
Question
If \( \alpha, \beta, \gamma \) are the zeroes of the polynomial
\[ f(x) = ax^3 + bx^2 + cx + d, \]
find
\[ \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}. \]
Solution
Step 1: Write Relations Between Zeroes and Coefficients
For the cubic polynomial \(ax^3 + bx^2 + cx + d\),
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \]
\[ \alpha\beta\gamma = -\frac{d}{a} \]
Step 2: Evaluate the Required Expression
\[ \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{\alpha\beta + \beta\gamma + \gamma\alpha}{\alpha\beta\gamma} \]
Substitute the values:
\[ = \frac{\frac{c}{a}}{-\frac{d}{a}} \]
\[ = -\frac{c}{d} \]
Conclusion
\[ \boxed{\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = -\frac{c}{d}} \]