Given that α= 1+3i is a root of the equation z^3 - pz^2 + 18z - q = 0 where p and q are real, find the other roots, then p and q.

All coefficients of z are real, therefore one root must be the complex conjugate so β = 1-3i.It is known that Σαβ = 18 (the coefficient of z), so we can get an equation in the third root, γ, as follows: Σαβ = αβ+αγ+βγ = (1+3i) (1-3i) + (1+3i)γ + (1-3i)γ = 18. Rearranging this we get γ = 4.To find p we use Σα = α+β+γ = 1+3i +1-3i + 4 = -p. Rearranging this we get that p=6. To find q we use Σαβγ = αβγ = (1+3i) (1-3i) (4) = -q. Rearranging this we get that q=-40.

ZG
Answered by Zachary G. Further Mathematics tutor

3922 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

A complex number z has argument θ and modulus 1. Show that (z^n)-(z^-n)=2iSin(nθ).


How do I use proof by induction?


Find the values of x where x+3>2/(x-4), what about x+3>2/mod(x-4)?


A=[5k,3k-1;-3,k+1] where k is a real constant. Given that A is singular, find all the possible values of k.


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences