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## Homework Statement

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Suppose q(z) = z^3 − z^2 + rz + s, is a complex polynomial with 1 + i and i as zeros. Find r and s and the third complex zero.

## The Attempt at a Solution

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(z-(1+i)(z-i) = Z^2-z-1-2iz+i

(Z^2-z-1-2iz+i)(z+d) = Z^3+z^2(d-1-zi)-z(d+1+2di-i)-d(1-i)

Z^2 term

Z^2(d-1-zi)=-z^2

d-1-2i=-1

d=2i

z term

-z(d+1+2di-i)=rz

-d-1-2di+i=r

2i-1-4i^2+i=r

-i+3=r

constant term

-d(1-i)=s

-2i+2i^2=s

-2i-2=s

This is what I have done but I am when I expand the complex zeros I do not get anything close to q(z)