2 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13\).
- Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x - 3 )\).
- The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 13 + d\), where \(d\) is a constant.
Given that ( \(2 x - 3\) ) is a factor of \(\mathrm { g } ( x )\), show that \(d = - 4\).
- Express \(\mathrm { g } ( x )\) in the form \(( 2 x - 3 ) \left( x ^ { 2 } + a x + b \right)\).