Find constant then factorise

2 The cubic polynomial \(3 x ^ { 3 } + a x ^ { 2 } - 2 x - 8\) is denoted by \(\mathrm { f } ( x )\).

1. Given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { f } ( x )\) completely.

2 The polynomial \(2 x ^ { 3 } - x ^ { 2 } + a x - 6\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

3 The polynomial \(4 x ^ { 3 } - 8 x ^ { 2 } + a x - 3\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { p } ( x )\) completely.

2 In this question you must show detailed reasoning.
The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 5 x ^ { 3 } - 4 x ^ { 2 } + a x - 2\), where \(a\) is a constant. You are given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).

1. Find the value of \(a\).
2. Find all the factors of \(\mathrm { f } ( x )\).

The polynomial \(\text{p}(x)\) is defined by $$\text{p}(x) = ax^3 - ax^2 - 15x + 18,$$ where \(a\) is a constant. It is given that \((x + 2)\) is a factor of \(\text{p}(x)\).

1. Find the value of \(a\). [2]
2. Hence factorise \(\text{p}(x)\) completely. [3]

The polynomial \(ax^3 - 20x^2 + x + 3\), where \(a\) is a constant, is denoted by \(\text{p}(x)\). It is given that \((3x + 1)\) is a factor of \(\text{p}(x)\).

1. Find the value of \(a\). [3]
2. When \(a\) has this value, factorise \(\text{p}(x)\) completely. [3]