1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 7\).
- Use the Remainder Theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by \(( 2 x + 1 )\).
(2 marks) - The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + d\), where \(d\) is a constant.
- Given that \(( 2 x + 1 )\) is a factor of \(\mathrm { g } ( x )\), show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 8 x - 4\).
(1 mark) - Given that \(\mathrm { g } ( x )\) can be written as \(\mathrm { g } ( x ) = ( 2 x + 1 ) \left( x ^ { 2 } + a \right)\), where \(a\) is an integer, express \(\mathrm { g } ( x )\) as a product of three linear factors.
- Hence, or otherwise, show that \(\frac { \mathrm { g } ( x ) } { 2 x ^ { 3 } - 3 x ^ { 2 } - 2 x } = p + \frac { q } { x }\), where \(p\) and \(q\) are integers.
(3 marks)