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\includegraphics[max width=\textwidth, alt={}, center]{c473f577-1e96-4d11-a0d5-cdfa4873c295-12_650_720_260_708} A curve has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } + 5 x - 15\). As shown in the diagram, the curve crosses the \(x\)-axis at the points \(A\) and \(B\) with coordinates \(( a , 0 )\) and \(( b , 0 )\) respectively.

1. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. By first finding the quotient when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\), show that $$a = - \sqrt { \frac { 5 } { 2 - a } } .$$
3. Use an iterative formula, based on the equation in part (b), to find the value of \(a\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
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