5.
$$f ( x ) = 3 x ^ { 3 } + A x ^ { 2 } + B x - 10$$
where \(A\) and \(B\) are integers.
Given that
- when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is \(k\)
- when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is \(- 10 k\)
- \(k\) is a constant
- show that
$$11 A + 9 B = 83$$
Given also that \(( 3 x - 2 )\) is a factor of \(\mathrm { f } ( x )\),
find the value of \(A\) and the value of \(B\).Hence find the quadratic expression \(\mathrm { g } ( x )\) such that
$$f ( x ) = ( 3 x - 2 ) g ( x )$$