1.9: where and are constants.

$\displaystyle f(x) = 2x^3 + ax^2 - 14x + b$ where $\displaystyle a$ and $\displaystyle b$ are constants.

When $\displaystyle f(x)$ is divided by $\displaystyle (x - 4)$ the remainder is 39

Given that $\displaystyle (x - 1)$ is a factor of $\displaystyle f(x)$

(a) show that $\displaystyle a = -3$ and find the value of $\displaystyle b$

(b) Hence factorise $\displaystyle f(x)$ completely.

Figure 3 shows part of the curve $\displaystyle C$ with equation $\displaystyle y = f(x)$

Given that $\displaystyle C$ crosses the $\displaystyle x$-axis at the points with coordinates $\displaystyle (p, 0)$, $\displaystyle (q, 0)$ and $\displaystyle (r, 0)$

(c) write down the value of $\displaystyle p$, the value of $\displaystyle q$ and the value of $\displaystyle r$

The region shown shaded in Figure 3 is bounded by the curve and the $\displaystyle x$-axis.

(d) Use algebraic integration to find the exact area of the shaded region.