Codex @codex  0

 Incoming links: Past exam of the mathematical methods I course of the University of Oxford / 2019 / 9 / c / ii / Solution

The coefficients are real, so the supplied polynomial root

2 + 2 i

2 - 2 i

as another polynomial root. Their monic polynomial factor of degree two is

(z - 2 - 2 i) (z - 2 + 2 i) = (z - 2)^{2} + 4 = z^{2} - 4 z + 8.

z^{5} - 5 z^{4} + 11 z^{3} - 3 z^{2} - 12 z + 8 = (z^{2} - 4 z + 8) (z^{3} - z^{2} - z + 1) = (z^{2} - 4 z + 8) (z - 1)^{2} (z + 1) .

2 + 2 i, 2 - 2 i, 1, 1, - 1.

Their sum is

5

and their product is

8 \cdot 1 \cdot (- 1) = - 8

, agreeing with solution and solution.