# Thread: Complex Numbers -- Finding a Polynomial

1. ## Complex Numbers -- Finding a Polynomial

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?

2. ## Re: Complex Numbers -- Finding a Polynomial

Originally Posted by Fratricide
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?
the key to (d) is to realize that if you want real coefficients then complex roots must appear in conjugate pairs.

so if you have 1-2i as a root then 1+2i must be a root, and similarly with 2+i, 2-i must be a root.

so you just take the product of all the (x-root) terms and you get

$$(x-3)(x-1+2i)(x-1-2i)(x-2-i)(x-2+i) = x^5-9 x^4+36 x^3-84 x^2+115 x-75$$