Answer
$x^6-12x^5+55x^4-120x^3+139x^2-108x+85$
Work Step by Step
The Conjugate Pairs Theorem states that when a polynomial has real coefficients, then any complex zeros occur in conjugate pairs. This means that, when $(p +i \ q)$ is a zero of a polynomial function with a real number of the coefficients, then its conjugate $(p –i q)$, is also a zero of the function.
We are given that the zeros of the function are: $2 \pm i, 4 \pm i, \pm i$.
We write the factors of the function as $(x-zero)$ and multiply them to get the polynomial.
Therefore, we can write the equation of the function as:
$f(x)=[x-(2+i)][x-(2-i)](x-i)(x-i)(x-(-i))[(x-(4-i)][x-(4+i)] \\=(x-2-i)(x-2+i)(x-i)(x+i)(x-4+i)(x-4-i) \\=x^6-12x^5+55x^4-120x^3+139x^2-108x+85$
