3 The equation
$$x + ( 1 + 3 x ) ^ { \frac { 1 } { 4 } } = 0$$
has a single root, \(\alpha\).
- Show that \(\alpha\) lies between - 0.33 and - 0.32 .
- Show that the equation \(x + ( 1 + 3 x ) ^ { \frac { 1 } { 4 } } = 0\) can be rearranged into the form
$$x = \frac { 1 } { 3 } \left( x ^ { 4 } - 1 \right)$$
- Use the iteration \(x _ { n + 1 } = \frac { \left( x _ { n } ^ { 4 } - 1 \right) } { 3 }\) with \(x _ { 1 } = - 0.3\) to find \(x _ { 4 }\), giving your answer to three significant figures.