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\includegraphics[max width=\textwidth, alt={}, center]{adbef77f-e2ac-40ce-a56b-cf6776534ec1-3_561_732_255_705}
The diagram shows the curve \(y = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } - 4 x - 16\), which crosses the \(x\)-axis at the points \(( \alpha , 0 )\) and \(( \beta , 0 )\) where \(\alpha < \beta\). It is given that \(\alpha\) is an integer.
- Find the value of \(\alpha\).
- Show that \(\beta\) satisfies the equation \(x = \sqrt [ 3 ] { } ( 8 - 2 x )\).
- Use an iteration process based on the equation in part (ii) to find the value of \(\beta\) correct to 2 decimal places. Show the result of each iteration to 4 decimal places.