# If a polynomial function with rational coefficients has the zeros -1+i, sqrt5, what are the additional zeros?

Sep 18, 2016

The additional zeros are $\left(- 1 - i\right)$ and $- \sqrt{5}$.

#### Explanation:

Complex zeros always occur in pairs in the form of complex conjugates $a \pm b i$.

The complex conjugate of $\left(- 1 + i\right)$ is $\left(- 1 - i\right)$.

Similarly, zeros containing square roots must also come in pairs. If the polynomial has a zero of $\sqrt{5}$, it must also have a zero of $- \sqrt{5}$.

Think about the quadratic formula $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{21}$.

If the discriminant ${b}^{2} - 4 a c$ is negative, there will be two complex zeros because of the $\pm$ signs before the square root. These are complex conjugates.

If the discriminant is positive but not a perfect square, there will also be two zeros of the form +-sqrt