Unknown constant, verify then factorise

Questions where a constant in the polynomial is unknown and must be found using the given factor, then the polynomial is factorised completely.

3 questions

OCR C2 2006 January Q8
8 The cubic polynomial \(2 x ^ { 3 } + k x ^ { 2 } - x + 6\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
  1. Show that \(k = - 5\), and factorise \(\mathrm { f } ( x )\) completely.
  2. Find \(\int _ { - 1 } ^ { 2 } f ( x ) \mathrm { d } x\).
  3. Explain with the aid of a sketch why the answer to part (ii) does not give the area of the region between the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 1 \leqslant x \leqslant 2\). \section*{[Question 9 is printed overleaf.]}
OCR C2 Q5
5.
\includegraphics[max width=\textwidth, alt={}]{de1a3480-0d83-43c2-a5a2-2f117b8a50fd-2_515_771_246_438}
The diagram shows the curve \(y = \mathrm { f } ( x )\) where $$f ( x ) = 4 + 5 x + k x ^ { 2 } - 2 x ^ { 3 }$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A , B\) and \(C\).
Given that \(A\) has coordinates \(( - 4,0 )\),
  1. show that \(k = - 7\),
  2. find the coordinates of \(B\) and \(C\).
Edexcel C2 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ccac020a-c378-45db-80f4-c63b5c213e1d-2_513_775_945_388} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) where $$f ( x ) = 4 + 5 x + k x ^ { 2 } - 2 x ^ { 3 }$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A , B\) and \(C\).
Given that \(A\) has coordinates ( \(- 4,0\) ),
  1. show that \(k = - 7\),
  2. find the coordinates of \(B\) and \(C\).