Complex Numbers - Finding Roots
Hi! I'm stuck on this question!
(a) For β = 1 - i√3, write the product z - β and z - conjugateβ as a quadratic expression in z, with real coefficients.
I got z^2 - 2z + 4
(b) (i) Express β in mod-arg form. Ans: 2cis(-π/3)
(ii) Find β^2 and β^3. Ans: 4cis(-2π/3) and 8cis(π)
(iii) Hence show that β is a root of z^3 - z^2 + 2z + 4 = 0. Stuck on this part, "hence"?
(c) Let the three roots be a, b, c. Let a be the point in the 1st quadrant, B the point on the real axis. Let c be the other root.
(i) Find the lengths AB and CB.
(ii) Describe the triangle ABC