Question

How do we write a rational polynomial of degree four, whose zeros are `-3sqrt2` and `4i`? What are the other two zeros?

Answer 1

Other zeros are `3sqrt2` and `-4i` and the polynomial is `x^4-2x^2-288`

Explanation:

As we have one of the irrational zeros as `-3sqrt2`, the other zero is its conjugate `+3sqrt2`. Similarly as one of the imaginary zeros is `4i`, the other zero is its conjugate `-4i`.

Hence, other zeros are `3sqrt2` and `-4i` and the polynomial is

`(x-(-3sqrt2))(x-3sqrt2)(x-4i)(x-(-4i))`

= `(x+3sqrt2)(x-3sqrt2)(x-4i)(x+4i)`

= `(x^2-(3sqrt2)^2)(x^2-(4i)^2)`

= `(x^2-18)(x^2+16)`

= `x^4-2x^2-288`