Question

If \(\alpha, \beta, \gamma\) are the roots of the equation \(2x^3 - 3x^2 + 5x - 7 = 0\), then \(\Sigma \alpha^2 \beta^2\) is:

• A. \(-\frac{17}{4}\)
• B. \(\frac{17}{4}\)
• C. \(-\frac{13}{4}\)
• D. \(\frac{13}{4}\)

Hint:

Vieta's formulas are crucial for relating the roots of polynomial equations back to their coefficients, particularly in polynomial identities and root transformations.

Answer 1

The Correct Option is A

By Vieta's formulas, we have:

\[ \alpha + \beta + \gamma = \frac{3}{2} \] \[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{5}{2} \] \[ \alpha\beta\gamma = \frac{7}{2} \]

We want to find \( \Sigma \alpha^2 \beta^2 = \alpha^2 \beta^2 + \beta^2 \gamma^2 + \gamma^2 \alpha^2 \).

We know that:

\[ (\alpha\beta + \beta\gamma + \gamma\alpha)^2 = \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 + 2\alpha\beta\gamma(\alpha + \beta + \gamma) \]

So,

\[ \Sigma \alpha^2 \beta^2 = (\alpha\beta + \beta\gamma + \gamma\alpha)^2 - 2\alpha\beta\gamma(\alpha + \beta + \gamma) \]

Substituting the values from Vieta's formulas:

\[ \Sigma \alpha^2 \beta^2 = \left(\frac{5}{2}\right)^2 - 2\left(\frac{7}{2}\right)\left(\frac{3}{2}\right) \] \[ = \frac{25}{4} - 2\left(\frac{21}{4}\right) \] \[ = \frac{25}{4} - \frac{42}{4} \] \[ = \frac{25 - 42}{4} \] \[ = -\frac{17}{4} \]

Therefore, \( \Sigma \alpha^2 \beta^2 = -\frac{17}{4} \).

Final Answer: The final answer is \( \boxed{(1)} \).