Question

How do you write the polynomial function given the following zeros: -5i, -i, i, 2i?

Answer 1

It is `P(x)=(x+5i)(x+i)(x-i)(x-2i)=x^4+3ix^3+11x^2+3ix+10`

where `i` is the imaginery unit (`i^2=-1`)

Answer 2

If the polynomial also has Real coefficients, then it also has zeroes `5i` and `-2i`

The simplest such polynomial is:

`f(x) = (x-5i)(x+5i)(x-i)(x+i)(x-2i)(x+2i)`

`=x^6+30x^4+129x^2+100`

Explanation:

The Complex zeroes of polynomials with Real coefficients always occur in conjugate pairs.

So in our example, since `-5i` and `2i` are zeroes, so are `5i` and `-2i`.

Hence:

`f(x) = (x-5i)(x+5i)(x-i)(x+i)(x-2i)(x+2i)`

`= (x^2+25)(x^2+1)(x^2+4)`

`=x^6+(25+1+4)x^4+(1*4+25*4+25*1)x^2+(25*1*4)`

`=x^6+30x^4+129x^2+100`

Any polynomial in `x` with these roots will be a multiple (scalar or polynomial) of this `f(x)`.