5.
$$f ( x ) = x ^ { 3 } + ( p + 3 ) x ^ { 2 } - x + q$$
where \(p\) and \(q\) are constants and \(p > 0\)
Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\)
- show that
$$9 p + q = - 51$$
Given also that when \(\mathrm { f } ( x )\) is divided by ( \(x + p\) ) the remainder is 9
- show that
$$3 p ^ { 2 } + p + q - 9 = 0$$
- Hence find the value of \(p\) and the value of \(q\).
- Hence find a quadratic expression \(\mathrm { g } ( x )\) such that
$$f ( x ) = ( x - 3 ) g ( x )$$