AQA Further Paper 1 2022 June — Question 5

Exam BoardAQA
ModuleFurther Paper 1 (Further Paper 1)
Year2022
SessionJune
TopicComplex Numbers Arithmetic
TypeGiven one complex root, find all roots

5 It is given that \(z = - \frac { 3 } { 2 } + \mathrm { i } \frac { \sqrt { 11 } } { 2 }\) is a root of the equation $$z ^ { 4 } - 3 z ^ { 3 } - 5 z ^ { 2 } + k z + 40 = 0$$ where \(k\) is a real number.
5
  1. Find the other three roots.
    5
  2. Given that \(x \in \mathbb { R }\), solve $$x ^ { 4 } - 3 x ^ { 3 } - 5 x ^ { 2 } + k x + 40 < 0$$