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
\[ \alpha^2 + \beta^2 + \gamma^2. \]
Solution
Step 1: Write Relations Between Zeroes and Coefficients
For the cubic polynomial \( ax^3 + bx^2 + cx + d \),
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \]
Step 2: Use the Identity
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 – 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]
Step 3: Substitute the Values
\[ = \left(-\frac{b}{a}\right)^2 – 2\left(\frac{c}{a}\right) \]
\[ = \frac{b^2}{a^2} – \frac{2c}{a} \]
Conclusion
\[ \boxed{\alpha^2 + \beta^2 + \gamma^2 = \frac{b^2}{a^2} – \frac{2c}{a}} \]