Problem

Figure 4 shows a sketch of part of the curve $C$ with equation $y=f(x)$, where

$$\begin{align*} f(x)=(3x-2)^2(x-4). \end{align*}$$

Figure 4

(a) Deduce the values of $x$ for which $f(x)>0$.

(b) Expand $f(x)$ to the form

$$\begin{align*} ax^3+bx^2+cx+d \end{align*}$$

where $a$, $b$, $c$ and $d$ are integers to be found.

The line $l$, also shown in Figure 4, passes through the $y$ intercept of $C$ and is parallel to the $x$-axis.

The line $l$ cuts $C$ again at points $P$ and $Q$, also shown in Figure 4.

(c) Using algebra and showing your working, find the length of line $PQ$. Write your answer in the form $k\sqrt3$, where $k$ is a constant to be found.

(Solutions relying entirely on calculator technology are not acceptable.)