# What is a fourth degree polynomial function with real coefficients that has 2, -2 and -3i as zeros?

Feb 6, 2016

$f \left(x\right) = {x}^{4} + 5 {x}^{2} - 36$

#### Explanation:

If $f \left(x\right)$ has zeroes at $2 \mathmr{and} - 2$
it will have $\left(x - 2\right) \left(x + 2\right)$ as factors.

If $f \left(x\right)$ has a zero at $- 3 i$
then $\left(x + 3 i\right)$ will be a factor
and we will need to use a fourth factor to "clear" the imaginary component from the coefficients.
(Remember we were told the polynomial was of degree 4 and has no imaginary components).

The most obvious fourth factor would be the complex conjugate of $\left(x + 3 i\right)$, namely $\left(x - 3 i\right)$
and since $\left(x + 3 i\right) \left(x - 3 i\right) = {x}^{2} + 9$

$f \left(x\right) = \left(x - 2\right) \left(x + 2\right) \left({x}^{2} + 9\right)$
which can be expanded to:
$f \left(x\right) = {x}^{4} + 5 {x}^{2} - 36$