# How do you write a polynomial with zeros i, -3i, 3i and leading coefficient 1?

Oct 8, 2016

The polynomial with zeros $i$, $- 3 i$ and $3 i$

is ${x}^{3} - i {x}^{2} + 9 x - 9 i$

#### Explanation:

A polynomial with zeros $a$, $b$ and $c$ is

$\left(x - a\right) \left(x - b\right) \left(x - c\right)$

Hence a polynomial with zeros $i$, $- 3 i$ and $3 i$ is

$\left(x - i\right) \left(x - \left(- 3 i\right)\right) \left(x - 3 i\right)$

= $\left(x - i\right) \left(x + 3 i\right) \left(x - 3 i\right)$

= $\left(x - i\right) \left({x}^{2} + 9\right)$

= ${x}^{3} - i {x}^{2} + 9 x - 9 i$

Here leading coefficient (of ${x}^{3}$) is already one.

Hence, the polynomial with zeros $i$, $- 3 i$ and $3 i$ is ${x}^{3} - i {x}^{2} + 9 x - 9 i$