Verify factor then factorise/solve - Factor & Remainder Theorem

10 marks Moderate -0.8

5. $$f ( x ) = - 4 x ^ { 3 } + 16 x ^ { 2 } - 13 x + 3$$

Use the remainder theorem to find the remainder when $\mathrm { f } ( x )$ is divided by ( $x - 1$ ).
Use the factor theorem to show that ( $x - 3$ ) is a factor of $\mathrm { f } ( x )$.
Hence fully factorise $\mathrm { f } ( x )$. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-12_581_636_731_657} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$.
Use your answer to part (c) and the sketch to deduce the set of values of $x$ for which $\mathrm { f } ( x ) \leqslant 0$

8 marks Moderate -0.8

4. $$f ( x ) = 6 x ^ { 3 } + 13 x ^ { 2 } - 4$$

Use the remainder theorem to find the remainder when $\mathrm { f } ( x )$ is divided by ( $2 x + 3$ ).
Use the factor theorem to show that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$.
Factorise $\mathrm { f } ( x )$ completely.

7 marks Easy -1.2

The polynomial $f(x)$ is defined by $f(x) = 8x^3 + 6x^2 - 14x - 1$. Find the remainder when $f(x)$ is divided by $(4x - 1)$. [2 marks]
The polynomial $g(x)$ is defined by $g(x) = 8x^3 + 6x^2 - 14x + d$.
1. Given that $(4x - 1)$ is a factor of $g(x)$, find the value of the constant $d$. [2 marks]
2. Given that $g(x) = (4x - 1)(ax^2 + bx + c)$, find the values of the integers $a$, $b$ and $c$. [3 marks]