9 The equation \(x ^ { 3 } - x ^ { 2 } - 5 x + 10 = 0\) has exactly one real root \(\alpha\).
- Show that the Newton-Raphson iterative formula for finding this root can be written as
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } - x _ { n } ^ { 2 } - 10 } { 3 x _ { n } ^ { 2 } - 2 x _ { n } - 5 }$$
- Apply the iterative formula in part (a) with initial value \(x _ { 1 } = - 3\) to find \(x _ { 2 } , x _ { 3 } , x _ { 4 }\) correct to 4 significant figures.
- Use a change of sign method to show that \(\alpha = - 2.533\) is correct to 4 significant figures.
- Explain why the Newton-Raphson method with initial value \(x _ { 1 } = - 1\) would not converge to \(\alpha\).