CAIE P3 2022 June — Question 5

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2022
SessionJune
TopicFactor & Remainder Theorem
TypeMultiple unknowns with derivative condition

5 The polynomial \(a x ^ { 3 } - 10 x ^ { 2 } + b x + 8\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of both \(\mathrm { p } ( x )\) and \(\mathrm { p } ^ { \prime } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.