We are given roots of a polynomial function : i, –2, and 2.
And leading coefficient 1 .
We need to find the polynomial function of lowest degree.
<em>Please note: We have one root i, that is a radical root. And a radical always comes in pair of plug and minus sign.</em>
Therefore, there would be another root -i.
So, we got all roots of the polynomial function : i, -i, -2, and 2.
For the given roots, we would have factors of the polynomial (x-i)(x+i)(x+2)(x-2).
Now, we need to multiply those factors to get the polynomial function.
![\mathrm{Expand}\:\left(x-i\right)\left(x+i\right):\quad x^2+1](https://tex.z-dn.net/?f=%5Cmathrm%7BExpand%7D%5C%3A%5Cleft%28x-i%5Cright%29%5Cleft%28x%2Bi%5Cright%29%3A%5Cquad%20x%5E2%2B1)
![\left(x+2\right)\left(x-2\right):\quad x^2-4](https://tex.z-dn.net/?f=%5Cleft%28x%2B2%5Cright%29%5Cleft%28x-2%5Cright%29%3A%5Cquad%20x%5E2-4)
![\left(x-i\right)\left(x+i\right)\left(x+2\right)\left(x-2\right)=\left(x^2+1\right)\left(x^2-4\right)](https://tex.z-dn.net/?f=%5Cleft%28x-i%5Cright%29%5Cleft%28x%2Bi%5Cright%29%5Cleft%28x%2B2%5Cright%29%5Cleft%28x-2%5Cright%29%3D%5Cleft%28x%5E2%2B1%5Cright%29%5Cleft%28x%5E2-4%5Cright%29)
![\mathrm{Expand}\:\left(x^2+1\right)\left(x^2-4\right)=x^4-4x^2+ \:x^2-\:4](https://tex.z-dn.net/?f=%5Cmathrm%7BExpand%7D%5C%3A%5Cleft%28x%5E2%2B1%5Cright%29%5Cleft%28x%5E2-4%5Cright%29%3Dx%5E4-4x%5E2%2B%20%5C%3Ax%5E2-%5C%3A4)
![=x^4-3x^2-4](https://tex.z-dn.net/?f=%3Dx%5E4-3x%5E2-4)
<h3>Therefore, correct option is 2nd option
![f(x)=x^4-3x^2-4](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E4-3x%5E2-4)
.</h3>