Let v = 2 + i and P(z) = z3 - 7z2 + 17z - 15
a) Show by substitution that P(2 + i) = 0
b) Find the other two roots of the equation P(z) = 0
c) Let i be a unit vector in the positive Re(z) direction and let j be a unit vector in the positive Im(z) direction.
Let A be the point on the Argand diagram corresponding to v = 2 + i.
Let B be the point on the Argand diagram corresponding to 1 - 2i.
Show that OA is perpendicular to OB.
d) Find a polynomial with real coefficients and with roots 3, 1 - 2i and 2 + i.
Parts a, b and c were no trouble, but I've never done anything similar (that I know of) to part d before. Could you please point me in the right direction?