The polynomial \(\mathrm{p}\) is such that \(\mathrm{p}(x)=5x^3+ax^2+39x+b\), where \(a\) and \(b\) are constants.
Given that \(x+3\) is a factor of both \(\mathrm{p}(x)\) and \(\mathrm{p}'(x)\), find the values of \(a\) and \(b\).[5]
Hence solve the equation \(\mathrm{p}(x)=0\).
You must show your working.[3]
Hence, using your values for \(a\) and \(b\), solve the equation
\(5\mathrm{cosec}^3 2\theta+a\mathrm{cosec}^2 2\theta+39\mathrm{cosec}\,2\theta+b=0\quad\text{for}\quad0^\circ\leq\theta\leq360^\circ\).[5]