Single unknown from factor condition

3 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 14$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).

1. Find the value of \(a\).
2. Show that, when \(a\) has this value, the equation \(\mathrm { f } ( x ) = 0\) has only one real root.

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + 3 x ^ { 2 } + 4 a x - 5 ,$$ where \(a\) is a constant. It is given that ( \(2 x - 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Use the factor theorem to find the value of \(a\).
2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has only one real root.
3. Use logarithms to solve the equation \(\mathrm { p } \left( 6 ^ { y } \right) = 0\) correct to 3 significant figures.

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 \(\mathrm { f } ( x )\),

1. show that \(c = - 14\)
2. Write \(\mathrm { f } ( x )\) in the form $$\mathrm { f } ( x ) = ( x + 3 ) \mathrm { Q } ( x )$$ where \(\mathrm { Q } ( x )\) is a quadratic function.
3. 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} \captionsetup{labelformat=empty} \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
4. 1. \(y = \mathrm { f } ( 3 x )\)
   2. \(y = - \mathrm { f } ( \mathrm { x } )\) On each diagram show clearly the coordinates of the points where the curve crosses the coordinate axes.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$f ( x ) = 4 x ^ { 3 } - 8 x ^ { 2 } + 5 x + a$$ where \(a\) is a constant.
Given that ( \(2 x - 3\) ) is a factor of \(\mathrm { f } ( x )\),

1. use the factor theorem to show that \(a = - 3\)
2. Hence show that the equation \(\mathrm { f } ( x ) = 0\) has only one real root.

13. \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 7 x + c\), where \(c\) is a constant. Given that \(\mathrm { f } ( 4 ) = 0\),

1. find the value of \(c\),
2. factorise \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(\mathrm { f } ( x ) = 0\).