# How do you write a polynomial function of least degree that has real coefficients, the following given zeros 2,-2,-6i and a leading coefficient of 1?

May 27, 2016

$f \left(x\right) = {x}^{4} + 32 {x}^{2} - 144$

#### Explanation:

Since we want Real coefficients, any non-Real zeros must occur in Complex conjugate pairs.

So both $- 6 i$ and $6 i$ are zeros and the simplest polynomial with these zeros is:

$f \left(x\right) = \left(x - 2\right) \left(x + 2\right) \left(x - 6 i\right) \left(x + 6 i\right)$

$= \left({x}^{2} - {2}^{2}\right) \left({x}^{2} - {\left(6 i\right)}^{2}\right)$

$= \left({x}^{2} - 4\right) \left({x}^{2} + 36\right)$

$= {x}^{4} + 32 {x}^{2} - 144$