# How do you write a polynomial in standard form given zeros 4i,4, and -4i?

Mar 16, 2016

Start from factored form to find that one such polynomial is

${x}^{3} - 4 {x}^{2} + 16 x - 64$

#### Explanation:

To create a polynomial with a given set of zeros, we can use the fact that $a$ is a zero of a polynomial if and only if $\left(x - a\right)$ is a factor of the polynomial. Then, starting from the zeros, we can simply take the product of all such factors.

$\left(x - 4\right) \left(x - 4 i\right) \left(x - \left(- 4 i\right)\right) = \left(x - 4\right) \left(x - 4 i\right) \left(x + 4 i\right)$

$= \left(x - 4\right) \left({x}^{2} + 16\right)$

$= {x}^{3} - 4 {x}^{2} + 16 x - 64$