11.
$$f ( x ) = 2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 48$$
- Prove that \(( x - 4 )\) is a factor of \(\mathrm { f } ( x )\).
- Hence, using algebra, show that the equation \(\mathrm { f } ( x ) = 0\) has only two distinct roots.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{deba6a2b-1821-4110-bde8-bde18a5f9be9-24_727_1059_566_504}
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\caption{Figure 2}
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Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\). - Deduce, giving reasons for your answer, the number of real roots of the equation
$$2 x ^ { 3 } - 13 x ^ { 2 } + 8 x + 46 = 0$$
Given that \(k\) is a constant and the curve with equation \(y = \mathrm { f } ( x + k )\) passes through the origin, (d) find the two possible values of \(k\).