Express as product with specific form

9. $$g ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - 15 x + 50$$

1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\).
2. Hence show that \(\mathrm { g } ( x )\) can be written in the form \(\mathrm { g } ( x ) = ( x + 2 ) ( a x + b ) ^ { 2 }\), where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-22_517_807_607_621} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { g } ( x )\)
3. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
   1. \(\mathrm { g } ( x ) \leqslant 0\)
   2. \(\mathrm { g } ( 2 x ) = 0\)

6. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 1\) a. (i) Show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
(ii) Express \(\mathrm { f } ( x )\) in the form \(( 2 x - 1 ) ( x + a ) ^ { 2 }\) where \(a\) is an integer. Using the answer to part a) (ii)
b. show that the equation \(2 p ^ { 6 } + 3 p ^ { 4 } - 1\) has exactly two real solutions and state the values of these roots.
c. deduce the number of real solutions, for \(5 \pi \leq \theta \leq 8 \pi\), to the equation $$2 \cos ^ { 3 } \theta + 3 \cos ^ { 2 } \theta - 1 = 0$$

9. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16 .$$

1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$ where \(p\) and \(q\) are integers to be found.
3. Sketch the curve \(y = \mathrm { f } ( x )\).
4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.

Express \(3x^3 + 5x^2 - 27x + 10\) in the form \((x - 2)(ax^2 + bx + c)\), where \(a\), \(b\) and \(c\) are integers. [3 marks]

\(f(x) = -2x^3 - x^2 + 4x + 3\)

1. Use the factor theorem to show that \((3 - 2x)\) is a factor of \(f(x)\). [2]
2. Hence show that \(f(x)\) can be written in the form \(f(x) = (3 - 2x)(x + a)^2\) where \(a\) is an integer to be found. [4]

\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve with equation \(y = f(x)\).

1. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
   1. \(f(x) \leq 0\)
   2. \(f'(\frac{x}{2}) = 0\)
   [3]