9. The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + k x ^ { 2 } - 7 x - 15$$ where \(k\) is a constant.
When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) the remainder is \(r\).
When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) the remainder is \(3 r\).

1. Find the value of \(k\).
2. Find the value of \(r\).
3. Show that \(( x - 5 )\) is a factor of \(\mathrm { f } ( x )\).
4. Show that there is only one real solution to the equation \(\mathrm { f } ( x ) = 0\).