13. \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + c x + 12\), where \(c\) is a constant
Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),
- show that \(c = - 14\)
- Write \(\mathrm { f } ( x )\) in the form
$$\mathrm { f } ( x ) = ( x + 3 ) \mathrm { Q } ( x )$$
where \(\mathrm { Q } ( x )\) is a quadratic function.
- Use the answer to part (b) to prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real solution.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-32_595_915_1034_518}
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\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
On separate diagrams sketch the curve with equation - \(y = \mathrm { f } ( 3 x )\)
- \(y = - \mathrm { f } ( \mathrm { x } )\)
On each diagram show clearly the coordinates of the points where the curve crosses the coordinate axes.