Let $p(x)$ denote the polynomial $p(x) = ax^3 + bx^2 - ax + 8$, with $a$ and $b$ as constants. It is stated that $(x + 2)$ divides $p(x)$ exactly, and that the remainder is $24$ when $p(x)$ is divided by $(x - 2)$.
(a)[4]
Determine the values of $a$ and $b$.
(b)[3]
Factorise $p(x)$ and thus demonstrate that the equation $p(x) = 0$ has exactly one real root.
(c)[3]
Solve the equation $p\!\left(\frac{1}{2}\cosec\theta\right)=0$ within the range $-90^\circ < \theta < 90^\circ$.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Put $x=-2$ into the expression and set it equal to zero” …