The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + d\), where \(d\) is a constant. When \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ), the remainder is - 2 . Use the Remainder Theorem to find the value of \(d\).
The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + 3\).
Given that \(x = - \frac { 1 } { 2 }\) is a solution of the equation \(\mathrm { g } ( x ) = 0\), write \(\mathrm { g } ( x )\) as a product of three linear factors.
The function h is defined by \(\mathrm { h } ( x ) = \frac { 4 x ^ { 2 } - 1 } { \mathrm {~g} ( x ) }\) for \(x > 2\).
Simplify \(\mathrm { h } ( x )\), and hence show that h is a decreasing function. [0pt]
[4 marks]