Answer
$$\color{blue}{ f(x)= x^4-7x^3+17x^2-x-26 }$$
Work Step by Step
We are given zeros and asked to write a polynomial function.
If $3+2i $ is a zero, then
$n +3+2i =0$
$n +2i = -3 $
$n= -3-2i $
So our factor is $\bf{(x -3-2i )}$
because when $x= 3+2i $, $(x -3-2i )=0$
Since $ 3+2i $ is a complex number, its conjugate, $ 3-2i $ is also a zero
$n +3-2i =0$
$n-2i= -3 $
$n=-3+2i$
So our factor is $\bf{(x-3+2i)}$
because when $x= 3-2i $, $(x-3+2i )=0$
If $ -1 $ is a zero, then
$n -1 =0$
$n = 1 $
So our factor is $\bf{(x +1 )}$
because when $x= -1 $, $(x +1 )=0$
If $ 2 $ is a zero, then
$n +2 =0$
$n= -2 $
So our factor is $\bf{(x -2 )}$
because when $x=2 $, $(x -2 )=0$
So our function is:
$f(x)=(x-3-2i)(x-3+2i)(x +1)(x-2)$
$f(x)=(x^2-6x+9-4i^2)(x +1)(x-2)$
Recall that $i^2=-1$
$f(x)=(x^2-6x+9-4(-1))(x +1)(x-2)$
$f(x)=(x^2-6x+13)(x +1)(x-2)$
$f(x)=(x^2-6x+13)(x +1)(x-2)$
$f(x)=(x^3-5x^2+7x+13)(x-2)$
$$\color{blue}{ f(x)= x^4-7x^3+17x^2-x-26 }$$
