Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f.Degree 4; zeros: i, 1 + i

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lemjay | High School Teacher | (Level 3) Senior Educator

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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)`


Note that `i^2=-1` .

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




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