Challenging +1.2 This question requires systematic application of root transformation and recurrence relations for power sums. The transformation part is a standard technique (substitution y = 2x + 1), while finding Sā and Sāā requires using Newton's identities or recurrence relations with the transformed equation. It's more involved than typical A-level questions due to the negative power sum and multi-step reasoning, but follows well-established methods without requiring novel insight.
The equation
$$8x^3 + 12x^2 + 4x - 1 = 0$$
has roots \(\alpha, \beta, \gamma\). Show that the equation with roots \(2\alpha + 1, 2\beta + 1, 2\gamma + 1\) is
$$y^3 - y - 1 = 0.$$ [3]
The sum \((2\alpha + 1)^n + (2\beta + 1)^n + (2\gamma + 1)^n\) is denoted by \(S_n\). Find the values of \(S_3\) and \(S_{-2}\). [5]
The equation
$$8x^3 + 12x^2 + 4x - 1 = 0$$
has roots $\alpha, \beta, \gamma$. Show that the equation with roots $2\alpha + 1, 2\beta + 1, 2\gamma + 1$ is
$$y^3 - y - 1 = 0.$$ [3]
The sum $(2\alpha + 1)^n + (2\beta + 1)^n + (2\gamma + 1)^n$ is denoted by $S_n$. Find the values of $S_3$ and $S_{-2}$. [5]
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q7 [8]}}