9231 P12 - Nov 2023 - Q04
4188
The cubic equation \(27x^3 + 18x^2 + 6x - 1 = 0\) has roots \(\alpha, \beta, \gamma\).
(a) Show that a cubic equation with roots \(3\alpha + 1, 3\beta + 1, 3\gamma + 1\) is \(y^3 - y^2 + y - 2 = 0\).
The sum \((3\alpha + 1)^n + (3\beta + 1)^n + (3\gamma + 1)^n\) is denoted by \(S_n\).
(b) Find the values of \(S_2\) and \(S_3\).
(c) Find the values of \(S_{-1}\) and \(S_{-2}\).
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