The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\).
Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
Express \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\) as the product of three linear factors.
The curve with equation \(y = x ^ { 3 } - 7 x - 6\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{de4f827d-f237-488a-9177-3d85d0cb1771-4_403_762_651_641}
The curve cuts the \(x\)-axis at the point \(A\) and the points \(B ( - 1,0 )\) and \(C ( 3,0 )\).
State the coordinates of the point \(A\).
Find \(\int _ { - 1 } ^ { 3 } \left( x ^ { 3 } - 7 x - 6 \right) \mathrm { d } x\).
Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 7 x - 6\) and the \(x\)-axis between \(B\) and \(C\).
Find the gradient of the curve \(y = x ^ { 3 } - 7 x - 6\) at the point \(B\).
Hence find an equation of the normal to the curve at the point \(B\).