Complex roots of cubic and quartic equationsEdexcel International A Level Further Maths: Mind map

Exam questions on Complex roots of cubic and quartic equations

1. The cubic equation $z^3-5z^2+17z-13=0$.
   Given that $z=1$ is a root, find the other two roots.2 marks
2. The cubic equation $z^3+z^2-z+15=0$, which has a root $1-2\mathrm{i}$.
   Find the quadratic factor of the cubic with real coefficients.2 marks
3. The quartic function $\text{g}(x)=x^4-6x^3+18x^2-30x+25$, which has a root $x=1+2\mathrm{i}$.
   Write down a second root of $\text{g}(x)=0$ and hence find a quadratic factor of $\text{g}(x)$ with real coefficients.3 marks