12. The cubic equation
IGNORE
has roots \(\alpha , \beta\), and \(\psi\)
Without solving the cubic equation,
$$3 x ^ { 3 } + x ^ { 2 } - 4 x + 1 = 0$$
- determine the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\)
- find a cubic equation that has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\), giving your answer in \(x ^ { 3 } + a x ^ { 2 } - b x + c = 0\), where \(a , b\) and \(c\) are integers to be determined.
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