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.
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