Question

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

Answer 1

`f(x)=x^4+5x^2-36`

Explanation:

If `f(x)` has zeroes at `2 and -2`
it will have `(x-2)(x+2)` as factors.

If `f(x)` has a zero at `-3i`
then `(x+3i)` 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 `(x+3i)`, namely `(x-3i)`
and since `(x+3i)(x-3i) = x^2+9`

`f(x)=(x-2)(x+2)(x^2+9)`
which can be expanded to:
`f(x)=x^4+5x^2-36`