Question

How do you write a polynomial in standard form given zeros -1, 2, and 1 - i?

Answer 1

`f(x) = x^4-3x^3+2x^2+2x-4`

Explanation:

Assuming that we want the polynomial to also have Real coefficients, any non-Real zeros will occur in Complex conjugate pairs. So if `1-i` is a zero then `1+i` is a zero too.

The simplest polynomial in `x` with zeros `-1`, `2`, `1-i` and `1+i` is:

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

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

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

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

graph{x^4-3x^3+2x^2+2x-4 [-10, 10, -5, 5]}