- In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
$$\mathrm { f } ( z ) = z ^ { 3 } - 13 z ^ { 2 } + 59 z + p \quad p \in \mathbb { Z }$$
Given that \(z = 3\) is a root of the equation \(f ( z ) = 0\)
- show that \(p = - 87\)
- Use algebra to determine the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\), giving your answers in simplest form.
On an Argand diagram
- the root \(z = 3\) is represented by the point \(P\)
- the other roots of \(\mathrm { f } ( \mathrm { z } ) = 0\) are represented by the points \(Q\) and \(R\)
- the number \(z = - 9\) is represented by the point \(S\)
- Show on a single Argand diagram the positions of \(P , Q , R\) and \(S\)
- Determine the perimeter of the quadrilateral \(P Q S R\), giving your answer as a simplified surd.