The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x - 10\).
Use the Factor Theorem to show that \(x - 2\) is a factor of \(\mathrm { p } ( x )\).
Express \(\mathrm { p } ( x )\) in the form \(( x - 2 ) \left( x ^ { 2 } + a x + b \right)\), where \(a\) and \(b\) are constants.
The curve \(C\) with equation \(y = x ^ { 3 } + x - 10\), sketched below, crosses the \(x\)-axis at the point \(Q ( 2,0 )\).
\includegraphics[max width=\textwidth, alt={}, center]{22c93dd5-d96a-4e31-8507-9c802e386231-3_444_547_1781_756}
Find the gradient of the curve \(C\) at the point \(Q\).
Hence find an equation of the tangent to the curve \(C\) at the point \(Q\).
Find \(\int \left( x ^ { 3 } + x - 10 \right) \mathrm { d } x\).
Hence find the area of the shaded region bounded by the curve \(C\) and the coordinate axes.