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

Aug 5, 2016

$f \left(x\right) = {x}^{4} - 3 {x}^{3} + 2 {x}^{2} + 2 x - 4$

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:

$f \left(x\right) = \left(x + 1\right) \left(x - 2\right) \left(x - 1 + i\right) \left(x - 1 - i\right)$

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

$= \left({x}^{2} - x - 2\right) \left({x}^{2} - 2 x + 2\right)$

$= {x}^{4} - 3 {x}^{3} + 2 {x}^{2} + 2 x - 4$

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