4.05b Transform equations: substitution for new roots

The cubic equation \(4x^3 - 12x^2 + 9x - 16 = 0\) has roots \(r_1\), \(r_2\) and \(r_3\). A second cubic equation, with integer coefficients, has roots \(R_1 = \frac{r_2 + r_3}{r_1}\), \(R_2 = \frac{r_3 + r_1}{r_2}\) and \(R_3 = \frac{r_1 + r_2}{r_3}\).

1. Show that \(1 + R_1 = \frac{3}{r_1}\) and write down the corresponding results for the other roots. [2]
2. Using a substitution based on this result, or otherwise, find this second cubic equation. [6]

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]