Show that 2+i is a root of z^3 - 5z^2 + 9z - 5=0. Find the other 2 roots

one can put z = 2 + i and expand it but it shall be long and prone to error

you can see that it has rational coefficients so if z = 2 + i is a root then z = 2 -i must be a root

and so (z-(2+i))(z-(2 -i) ) or (z-2)^2 +1 or z^2-4z + 5 must devide z^3 - 5z^2 + 9z - 5

by division you see z^3 - 5z^2 + 9z - 5 = (z^2-4z + 5)(z-1)

so z^2-4z + 5 devides z^3 - 5z^2 + 9z - 5

so z = 2 + i is a root
other roots are 2 -i  and 1