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

Mar 3, 2017

$- 3 - \sqrt{5}$ and $+ i$

Explanation:

Complex zeros always come in pairs:

$x = - 3 \pm \sqrt{5}$ and $x = \pm i$

The pairs form a quadratic equation:

1. $\left(x + 3 - \sqrt{5}\right) \left(x + 3 + \sqrt{5}\right) =$ ${x}^{2} + 3 x + \sqrt{5} x + 3 x + 3 \sqrt{5} - \sqrt{5} x - 3 \sqrt{5} - \sqrt{5} \sqrt{5} =$ ${x}^{2} + 6 x - 5$

2. $\left(x - i\right) \left(x + i\right) = {x}^{2} + x i - x i - {i}^{2} = {x}^{2} + 1$,
Note: ${i}^{2} = - 1$

In summary: The additional zeros are $- 3 - \sqrt{5}$ and $+ i$