CAIE FP1 2018 June — Question 6

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2018
SessionJune
TopicRoots of polynomials

6 The equation $$9 x ^ { 3 } - 9 x ^ { 2 } + x - 2 = 0$$ has roots \(\alpha , \beta , \gamma\).
  1. Use the substitution \(y = 3 x - 1\) to show that \(3 \alpha - 1,3 \beta - 1,3 \gamma - 1\) are the roots of the equation $$y ^ { 3 } - 2 y - 7 = 0$$ The sum \(( 3 \alpha - 1 ) ^ { n } + ( 3 \beta - 1 ) ^ { n } + ( 3 \gamma - 1 ) ^ { n }\) is denoted by \(S _ { n }\).
  2. Find the value of \(S _ { 3 }\).
  3. Find the value of \(S _ { - 2 }\).