5 You are given that the equation \(x ^ { 3 } + 4 x ^ { 2 } + x - 1 = 0\) has a root, \(\alpha\), where \(- 1 < \alpha < 0\).
- Show that the Newton-Raphson iterative formula for this equation can be written in the form
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 4 x _ { n } ^ { 2 } + 1 } { 3 x _ { n } ^ { 2 } + 8 x _ { n } + 1 } .$$
- Using the initial value \(x _ { 1 } = - 0.7\), find \(x _ { 2 }\) and \(x _ { 3 }\) and find \(\alpha\) correct to 5 decimal places.
- The diagram shows a sketch of the curve \(y = x ^ { 3 } + 4 x ^ { 2 } + x - 1\) for \(- 1.5 \leqslant x \leqslant 1\).
\includegraphics[max width=\textwidth, alt={}, center]{a80eb21f-c273-4b65-8617-16cdee783305-3_602_926_749_566}
Using the copy of the diagram in your answer book, explain why the initial value \(x _ { 1 } = 0\) will fail to find \(\alpha\).