- In this question you must show detailed reasoning.
The equation \(x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
- Using the identity \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma\) find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
- Given that \(\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112\) find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 }\) and \(\gamma ^ { 3 }\).
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