CAIE P3 2012 June — Question 3

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2012
SessionJune
TopicFactor & Remainder Theorem
TypeSolve p(algebraic transform) = 0

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } - 3 a x + 4 a$$ where \(a\) is a constant.
  1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
  2. When \(a\) has this value,
    (a) factorise \(\mathrm { p } ( x )\) completely,
    (b) find all the roots of the equation \(\mathrm { p } \left( x ^ { 2 } \right) = 0\).