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(a) Solve z² - (3 - 2i)z = (5i - 5) and write each root in the form a + ib. (b) Given that (1 - i) is a root of z³ - 2(2 - i)z² + (4 - 5i)z - 1 + 3i = 0, determine the remaining two roots.

1. (a) z = 2 + i or z = 1 - 3i; (b) other roots are i and 3 - 2i
2. (a) z = 1 + 2i or z = 2 - 3i; (b) other roots are 1 + i and 2 - i
3. (a) z = 2 - i or z = 1 + 3i; (b) other roots are -i and 3 + 2i
4. (a) z = 3 + i or z = -2i; (b) other roots are 2i and 1 - 2i

Correct answer: (a) z = 2 + i or z = 1 - 3i; (b) other roots are i and 3 - 2i

Solution

Solving the quadratic gives 2 + i and 1 - 3i; deflating the cubic by the known root leaves a quadratic whose roots are i and 3 - 2i.

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