11 In this question \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 4 x + k\).
- You are asked to find the values of \(k\) which satisfy the following conditions.
(A) The graph of \(y = \mathrm { f } ( x )\) goes through the origin.
(B) The graph of \(y = \mathrm { f } ( x )\) intersects with the \(y\) axis at ( \(0 , - 2\) ).
(C) ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
(D) The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 5 .
(E) The graph of \(y = \mathrm { f } ( x )\) is as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-3_373_788_2131_584} - Find the solution of the equation \(\mathrm { f } ( x ) = 0\) when \(k = 8\).
Sketch a graph of \(y = \mathrm { f } ( x )\) in this case.