4 In this question you must show detailed reasoning.
The cubic polynomial \(6 x ^ { 3 } + k x ^ { 2 } + 57 x - 20\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
- Use the factor theorem to show that \(k = - 37\).
- Using this value of \(k\), factorise \(\mathrm { f } ( x )\) completely.
- Hence find the three values of \(t\) satisfying the equation \(6 \mathrm { e } ^ { - 3 t } - 37 \mathrm { e } ^ { - 2 t } + 57 \mathrm { e } ^ { - t } - 20 = 0\).
- Express the sum of the three values found in part (c)(i) as a single logarithm.