The function f is defined for all values of \(x\) by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\).
Find the remainder when \(\mathrm { f } ( x )\) is divided by \(x + 1\).
Given that \(\mathrm { f } ( 1 ) = 0\) and \(\mathrm { f } ( - 2 ) = 0\), write down two linear factors of \(\mathrm { f } ( x )\).
Hence express \(x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) as the product of three linear factors.
The curve with equation \(y = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) is sketched below.
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The curve intersects the \(y\)-axis at the point \(A\). Find the \(y\)-coordinate of \(A\).
The curve crosses the \(x\)-axis when \(x = - 2\), when \(x = 1\) and also at the point \(B\). Use the results from part (a) to find the \(x\)-coordinate of \(B\).
Find \(\int \left( x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8 \right) d x\).
Hence find the area of the shaded region bounded by the curve and the \(x\)-axis.