8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-26_718_1076_260_434}
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\caption{Figure 4}
\end{figure}
Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where
$$f ( x ) = ( 3 x - 2 ) ^ { 2 } ( x - 4 )$$
- Deduce the values of \(x\) for which \(\mathrm { f } ( x ) > 0\)
- Expand f(x) to the form
$$a x ^ { 3 } + b x ^ { 2 } + c x + d$$
where \(a\), \(b\), \(c\) and \(d\) are integers to be found.
The line \(l\), also shown in Figure 4, passes through the \(y\) intercept of \(C\) and is parallel to the \(x\)-axis.
The line \(l\) cuts \(C\) again at points \(P\) and \(Q\), also shown in Figure 4 .
- Using algebra and showing your working, find the length of line \(P Q\). Write your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be found.
(Solutions relying entirely on calculator technology are not acceptable.)