# How do find all zeros of the function?

## $g \left(x\right) = {x}^{4} - 3 {x}^{3} - {x}^{2} - 12 x - 20$ $x = 2 i$

Jan 2, 2018

$x = \pm 2 i , x = \frac{3 \pm \sqrt{29}}{2}$

#### Explanation:

We have been given that $x = 2 i$ is a zero. Since complex solutions to polynomials with real coefficients always come in conjugate pairs (the imaginary parts need to cancel), we know that $x = - 2 i$ is also a solution.

This means that $\left(x - 2 i\right) \left(x + 2 i\right) = {x}^{2} + 4$ is a factor of the polynomial, and we can find out the remainder using polynomial long division:
$\left({x}^{2} + 4\right) \left({x}^{2} - 3 x - 5\right)$

We can solve for when the quadratic factor equals zero using the quadratic formula:
$x = \frac{3 \pm \sqrt{29}}{2}$