1.1: Polynomial factorisation and equal areas

Given that $\displaystyle f(x)$ can be written in the form

(a) find the value of $\displaystyle a$, the value of $\displaystyle b$ and the value of $\displaystyle c$.

Figure 3 shows a sketch of part of the curve $\displaystyle C$ with equation

The curve $\displaystyle C$ crosses the $\displaystyle x$-axis at the point $\displaystyle M$, the origin and the point $\displaystyle B$. The point $\displaystyle N$ lies on the $\displaystyle x$-axis between $\displaystyle M$ and $\displaystyle O$. The point $\displaystyle A$ lies on $\displaystyle C$ such that $\displaystyle AN$ is parallel to the $\displaystyle y$-axis.

The area of the shaded region bounded by the curve and $\displaystyle OB$ is numerically equal to the area of the shaded region bounded by the curve, $\displaystyle ON$ and $\displaystyle NA$.

Given that the coordinates of $\displaystyle N$ are $\displaystyle (n,0)$,

(b) use algebraic integration to show that $\displaystyle n$ satisfies the equation

(c) Hence find the exact coordinates of $\displaystyle A$.