The polynomials $f(x)$ and $g(x)$ are given by $f(x) = 4x^3 + ax^2 + 8x + 15$ and $g(x) = x^2 + bx + 18$, with $a$ and $b$ as constants.
(a)[2]
If $(x + 3)$ is a factor of $f(x)$, determine the value of $a$.
(b)[2]
If the remainder is $40$ when $g(x)$ is divided by $(x - 2)$, determine the value of $b$.
(c)[3]
With these values of $a$ and $b$, factorise $f(x) - g(x)$ fully.
(d)[3]
Therefore, solve the equation $f(\cosec\, \theta) - g(\cosec\, \theta) = 0$ for $0 < \theta < 2\pi$.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Replace $x$ with $-3$, set the expression equal to zero, and try to solve for $a$” …