 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/12/c/solution

Real coefficients imply that the conjugate

- i

is also a polynomial root. Write the remaining root as

r

. Then

p (z) = 2 (z - 2) (z - r) (z^{2} + 1) .

z

in this product is

- 2 (2 + r)

. That coefficient is zero in the specified polynomial, so

r = - 2

. Hence the two other roots are

- i, - 2.

As a check, the full polynomial is

2 (z^{2} - 4) (z^{2} + 1) = 2 z^{4} - 6 z^{2} - 8

, corresponding to

a = 0

b = - 6

and

c = - 8

 New to topics? Read the docs here!