10 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } - 2 x ^ { 2 } - 4 x - 2\).
- Show that \(x = - 1\) is a root of \(\mathrm { f } ( x ) = 0\).
- Show that another root of \(\mathrm { f } ( x ) = 0\) lies between \(x = 1\) and \(x = 2\).
- Show that \(\mathrm { f } ( x ) = ( x + 1 ) \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } + a x + b\) and \(a\) and \(b\) are integers to be determined.
- Without further calculation, explain why \(\mathrm { g } ( x ) = 0\) has a root between \(x = 1\) and \(x = 2\).
- Use the Newton-Raphson formula to show that an iteration formula for finding roots of \(\mathrm { g } ( x ) = 0\) may be written
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 2 } { 3 x _ { n } ^ { 2 } - 2 }$$
Determine the root of \(\mathrm { g } ( x ) = 0\) which lies between \(x = 1\) and \(x = 2\) correct to 4 significant figures.