Complex conjugates and polynomial rootsAQA A-Level Further Maths: Mind map

Exam questions on Complex conjugates and polynomial roots

1. The cubic equation $z^3-5z^2+17z-13=0$ has a real root $z=1$.
   Explain why the cubic has exactly one real root.2 marks
2. The cubic equation $z^3-4z^2+6z-4=0$ has real coefficients and a root $z=1+i$.
   Show by substitution that $1-i$ is a root of the equation.2 marks
3. The quartic $f(z)=z^4-2z^3+9z^2-8z+20$ has $z^2-2z+5$ as a factor.
   Find the other quadratic factor of $f(z)$.3 marks