2024 CGSS P2 Q7

It is given that \(\mathrm{f}(x) = x^3 + ax^2 - 5x + b\), where \(a\) and \(b\) are constants, has a factor of \(x + 1\) and leaves a remainder of \(-24\) when divided by \((x + 3)\).

1. Show that \(a = -1\) and \(b = -3\).[4]
2. Hence factorise \(\mathrm{f}(x)\) completely and write down the solutions for \(\mathrm{f}(x) = 0\).[3]

From part (a), it is known that \(\mathrm{f}(x) = x^3 - x^2 - 5x - 3\).
The diagram below shows part of the curve \(y = \mathrm{f}(x)\) and a line that cuts the curve at \(B\) and \(C\). \(B\) has coordinates \((1, -6)\).
\(A\) and \(C\) are points on the curve and the \(x\)-axis.

3. Find the area of the shaded region.[5]
4. Show that the equation of the tangent at the minimum point of the curve is \(y = -\frac{256}{27}\).[3]