3. The quadratic equation
$$x ^ { 2 } - 2 x + 3 = 0$$
has roots \(\alpha\) and \(\beta\).
Without solving the equation,
- write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
- show that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 2\)
- find the value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
- show that \(\alpha ^ { 4 } + \beta ^ { 4 } = \left( \alpha ^ { 2 } + \beta ^ { 2 } \right) ^ { 2 } - 2 ( \alpha \beta ) ^ { 2 }\)
- find a quadratic equation which has roots
$$\text { ( } \alpha ^ { 3 } - \beta \text { ) and ( } \beta ^ { 3 } - \alpha \text { ) }$$
giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.