The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 4 x - 5\).
Use the Factor Theorem to show that \(x - 1\) is a factor of \(\mathrm { f } ( x )\).
Express \(\mathrm { f } ( x )\) in the form \(( x - 1 ) \left( x ^ { 2 } + p x + q \right)\), where \(p\) and \(q\) are integers.
Hence show that the equation \(\mathrm { f } ( x ) = 0\) has exactly one real root and state its value.
The curve with equation \(y = x ^ { 3 } + 4 x - 5\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{23f34515-3373-4644-a8a1-82b45809d934-4_505_959_868_529}
The curve cuts the \(x\)-axis at the point \(A ( 1,0 )\) and the point \(B ( 2,11 )\) lies on the curve.
Find \(\int \left( x ^ { 3 } + 4 x - 5 \right) \mathrm { d } x\).
Hence find the area of the shaded region bounded by the curve and the line \(A B\).