7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).
- Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\).
- Hence find the three roots of the equation \(\mathrm { f } ( x ) = 0\).
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The diagram shows the curve \(C\) with equation \(y = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + 12 x + 9\). - Show that the \(x\)-coordinates of the stationary points on \(C\) are given by \(x ^ { 3 } - 3 x ^ { 2 } - x + 3 = 0\).
- Use integration to find the exact area of the region enclosed by \(C\) and the \(x\)-axis.