2.
$$f ( n ) = n ^ { 3 } + p n ^ { 2 } + 11 n + 9 , \text { where } p \text { is a constant. }$$
- Given that \(\mathrm { f } ( n )\) has a remainder of 3 when it is divided by ( \(n + 2\) ), prove that \(p = 6\).
- Show that \(\mathrm { f } ( n )\) can be written in the form \(( n + 2 ) ( n + q ) ( n + r ) + 3\), where \(q\) and \(r\) are integers to be found.
- Hence show that \(\mathrm { f } ( n )\) is divisible by 3 for all positive integer values of \(n\).