Express \(x^3 - 3x - 2\) as the product of three linear factors.
Show that \(x^3 - 3x - 2\) may also be expressed in the form \((x-1)^2(x+p) + q\), where \(p\) and \(q\) are constants to be found. Hence, or otherwise, show that the curve whose equation is \(y = x^3 - 3x - 2\) has a minimum point where \(x = 1\).
Sketch the curve whose equation is \[\begin{equation*} y = \frac{1}{x^3 - 3x - 2}, \end{equation*}\] indicating clearly all its asymptotes. Hence obtain the number of real roots of the equation \[\begin{equation*} (x+1)(x^3 - 3x - 2) = 1. \end{equation*}\]
