If a polynomial function with rational coefficients has the zeros -3+sqrt5, -i, what are the additional zeros?

1 Answer
Mar 3, 2017

-3 - sqrt(5) and +i

Explanation:

Complex zeros always come in pairs:

x = -3 +-sqrt(5) and x = +-i

The pairs form a quadratic equation:

  1. (x+3-sqrt(5))(x+3+sqrt(5)) = x^2 + 3x +sqrt(5)x + 3x + 3sqrt(5) - sqrt(5)x -3 sqrt(5) - sqrt(5) sqrt(5) = x^2+ 6x - 5

  2. (x-i)(x+i) = x^2 + x i -x i -i^2 = x^2 +1,
    Note: i^2 = -1

In summary: The additional zeros are -3 - sqrt(5) and +i