7 The polynomial \(\mathrm { p } ( x )\) is defined by
$$\mathrm { p } ( x ) = a x ^ { 3 } - 11 x ^ { 2 } - 19 x - a$$
where \(a\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).
- Find the value of \(a\).
- When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.
- Hence find the exact values of \(y\) that satisfy the equation \(\mathrm { p } \left( \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } \right) = 0\).
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.