CAIE P3 2010 November — Question 10

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2010
SessionNovember
TopicRoots of polynomials

10 The polynomial \(\mathrm { p } ( z )\) is defined by $$\mathrm { p } ( z ) = z ^ { 3 } + m z ^ { 2 } + 24 z + 32$$ where \(m\) is a constant. It is given that \(( z + 2 )\) is a factor of \(\mathrm { p } ( z )\).
  1. Find the value of \(m\).
  2. Hence, showing all your working, find
    (a) the three roots of the equation \(\mathrm { p } ( z ) = 0\),
    (b) the six roots of the equation \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).