Find factor then solve

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 11 x + 10\).

1. Use the factor theorem to find a factor of \(\mathrm { f } ( x )\).
2. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving each root in an exact form.

4 In this question you must show detailed reasoning.
Find the coordinates of the points where the curve \(y = x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6\) crosses the \(x\)-axis.

3 In this question you must show detailed reasoning.
The function \(\mathrm { f } ( \mathrm { z } )\) is given by \(\mathrm { f } ( \mathrm { z } ) = 2 \mathrm { z } ^ { 3 } - 7 \mathrm { z } ^ { 2 } + 16 \mathrm { z } - 15\).
By first evaluating \(\mathrm { f } \left( \frac { 3 } { 2 } \right)\), find the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\).

Solve the equation $$\frac { 6 x ^ { 5 } - 17 x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } } { ( 3 x + 2 ) } = 0$$

Use an algebraic method to solve the equation \(12x^3 - 29x^2 + 7x + 6 = 0\). Show all your working. [6]