Note that if the given zeros of a polynomial function has non-real complex numbers then the other root is the conjugate of the complex number.

So if the zeros are `i` and `1+i` , then the other zeros are `-i` and `1-i` .

To determine the factors of the polynomial,...

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Note that if the given zeros of a polynomial function has non-real complex numbers then the other root is the conjugate of the complex number.

So if the zeros are `i` and `1+i` , then the other zeros are `-i` and `1-i` .

To determine the factors of the polynomial, set the variable of the polynomial equal to the zeros of f(x).

`x=i` `x=-i` `x=1+i` `x=1-i`

Perform opposite operation to make the right side zero.

`x-i=0` `x+i=0` `x-1-i=0` `x-1+i=0`

So the factors of f(x) are `x-i` , `x+i` , `x-1-i` and `x-1+i` .

Then, multiply the factors.

`(x-i)(x+i)(x-1-i)(x-1+i) `

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

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

Note that `i^2=-1` .

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

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

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

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

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