# How do you write a polynomial function of least degree that has real coefficients, the following given zeros 2, -1+i, -1-i?

Aug 9, 2016

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

#### Explanation:

Each zero corresponds to a linear factor.

So we can write:

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

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

$= \left(x - 2\right) \left({\left(x + 1\right)}^{2} - {i}^{2}\right)$

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

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

Any polynomial in $x$ with these zeros will be a multiple (scalar or polynomial) of this $f \left(x\right)$.