- The cubic equation
$$4 x ^ { 3 } + p x ^ { 2 } - 14 x + q = 0$$
where \(p\) and \(q\) are real positive constants, has roots \(\alpha , \beta\) and \(\gamma\)
Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 16\)
- show that \(p = 12\)
Given that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } = \frac { 14 } { 3 }\)
- determine the value of \(q\)
Without solving the cubic equation,
- determine the value of \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)