Question

How do you create a polynomial with Degree 4, 2 Positive real zeros, 0 negative real zeros, 2 complex zeros?

Answer 1

Any polynomial which factors to the form
`c(x-a)(x-b)(x-di)(x+di)` with `a, b, c, d in RR^+`
will satisfy the given conditions.

Explanation:

A simple way to creative a polynomial with certain desired zeros is to write it first in its factored form. For example, in this case, let's make a polynomial with the zeros `1, 2, i, -i`.

Note that the solutions to
`(x-1)(x-2)(x-i)(x+i)= 0`
are `1, 2, i, -i`

Thus we know that `(x-1)(x-2)(x-i)(x+i)` is a polynomial with the desired properties. Then, all that remains is to multiply it out.

`(x-1)(x-2)(x-i)(x+i) = (x-1)(x-2)(x^2+1)`

`= (x-1)(x^3 -2x^2 + x -2)`

`=x^4 -3x^3 +3x^2 -3x +2`

While the above works, `1, 2, i, -i` were chosen arbitrarily. Any polynomial which factors to the form
`c(x-a)(x-b)(x-di)(x+di)` with `a, b, c, d in RR^+` will satisfy the given conditions.