9.
$$f ( x ) = - 2 x ^ { 3 } - x ^ { 2 } + 4 x + 3$$
a. Use the factor theorem to show that ( \(3 - 2 x\) ) is a factor of \(\mathrm { f } ( x )\).
b. Hence show that \(\mathrm { f } ( x )\) can be written in the form \(\mathrm { f } ( x ) = ( 3 - 2 x ) ( x + a ) ^ { 2 }\) where \(a\) is an integer to be found.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{725966d1-d29d-4c9d-b850-c67d55cdd6e8-15_657_1024_278_450}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
c. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
i. \(\mathrm { f } ( x ) \leq 0\)
ii. \(\mathrm { f } \left( \frac { x } { 2 } \right) = 0\)