Explanation:
We have the equation 2x³ + 5x² - 6 = 0 with roots alpha, beta, and gamma.
Let y = x³. Then 2y - 6 = -5x².
=> 8(y - 3)³ = -125x⁶ = -125y²
=> 8(y³ - 9y² + 27y - 27) + 125y² = 0
=> 8y³ + 53y² + 216y - 216 = 0.
This new equation has roots alpha³, beta³, and gamma³.
Consequently, the equation -216y³ + 216y² + 53y + 8 = 0 has roots 1/alpha³, 1/beta³, and 1/gamma³.
By Vieta's, Sum(1/alpha⁶) = (-b/a)² - 2(c/a) = (-216/-216)² - 2(53/-216) = 1² + 53/108 = 161/108.
Sum(1/alpha⁹) = [216 * Sum(1/alpha⁶) + 53 * Sum(1/alpha³) + 8]/216 = [216 * (161/108) + 53(1) + 8]/216 = 383/216.