Cyclic summation of roots of a cubic

Algebra Level 4

\[\large 2x^3 + x^2 - 7 = 0 \]

If \(\alpha, \beta, \gamma\) are the roots of the above equation, evaluate

\[\large \left | \displaystyle \sum_{\text{cyc}} \left( \dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha} \right )\right| \]

Note

  • \(|z|\) is the absolute value of \(z\)

  • \(\displaystyle \sum_{\text{cyc}}\) means cyclic summation.

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