We are given zeros of the polynomial : 7, -11, and 2 + 8i.
Note: The radical zero always comes with the pair of plus and minus sign.
Therefore, another zero would be 2-8i.
Now, in order to find the polynomial with the zeros 7, -11, 2 + 8i and 2-8i, we need to find the factors of the polynomial.
The factors of the polynomial would be (x-7)(x+11)(x-2-8i)(x-2+8i).
Let us multiply those factors to get the standard form of the polynomial.
[tex]\left(x-2-8i\right)\left(x-2+8i\right)=\left(x-2-8i\right)\left(x-2+8i\right) =x^2-4x+68[/tex]
[tex]\left(x-7\right)\left(x+11\right)=x^2+11x-7x-77[/tex]=[tex]x^2+4x-77[/tex]
[tex]\left(x^2-4x+68\right)\left(x^2+4x-77\right)=x^4-9x^2-16x^2+308x+272x-5236\\[/tex]
[tex]=x^4-25x^2+580x-5236[/tex].
