Answer
$U(x)=4x^5+6x^4+4x^3 +4x^2 -2$
Work Step by Step
The factor theorem says that if $f(c)=0$, then $(x-c)$ is a factor of $f(x)$ and if $(x-c)$ is a factor of $f(x)$, then $f(c)=0$.
According to the Conjugate Pair Theorem, since $-i$ is a complex zero of the function, $i$ is also a complex zero.
We use the zeros to construct factors, which we multiply to find the original equation:
$U(x)=a(x-i)(x-(-i))(x-(-1))^2(x-0.5)= ax^5 + 1.5a x^4 +a x^3 +a x^2 - 0.5a$
We know the leading coefficient is $4$, so $a=4$. Hence,
$U(x)=4x^5+6x^4+4x^3 +4x^2 -2$
