CAIE FP1 2013 November — Question 2

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2013
SessionNovember
TopicRoots of polynomials

2 The cubic equation \(x ^ { 3 } - p x - q = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha , \beta , \gamma\). Show that
  1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 p\),
  2. \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 q\),
  3. \(6 \left( \alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } \right) = 5 \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } \right)\).