Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 12 / c / Solution

Divide the complex polynomial by its known factor $z-(2+i)$:The quadratic factors further as $(z-1)(z-1-i)$, because its two proposed roots sum to $2+i$ and multiply to $1+i$. Therefore the remaining solutions areAll three roots are distinct. Complex coefficients do not require nonreal roots to occur in conjugate pairs.