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

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Answer 1 · 2017-03-03T21:23:31

$-3 - sqrt(5)$ and $+i$

Complex zeros always come in pairs:

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

The pairs form a quadratic equation:

$(x+3-sqrt(5))(x+3+sqrt(5)) =$ $x^2 + 3x +sqrt(5)x + 3x + 3sqrt(5) - sqrt(5)x -3 sqrt(5) - sqrt(5) sqrt(5) =$ $x^2+ 6x - 5$
$(x-i)(x+i) = x^2 + x i -x i -i^2 = x^2 +1$ ,
Note: $i^2 = -1$

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

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