 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/12/c/solution

Divide the complex polynomial by its known factor

z - (2 + i)

p (z) = (z - 2 - i) [z^{2} - (2 + i) z + (1 + i)] .

The quadratic factors further as

(z - 1) (z - 1 - i)

, because its two proposed roots sum to

2 + i

and multiply to

1 + i

. Therefore the remaining solutions are

z = 1, z = 1 + i .

All three roots are distinct. Complex coefficients do not require nonreal roots to occur in conjugate pairs.

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