Answer
$T(x)=6x^4-12x^3+18x^2-12x+12$
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 $1+i$ is a complex zero, $1-i$ is also a complex zero. Similarly, $i$ and $-i$ are complex zero pairs.
We use the zeros to construct factors, which we multiply to find the original equation:
$T(x)=a(x-i)(x-(-i))(x-(1+i))(x-(1-i))=ax^4-2ax^3+3ax^2-2ax+2a$
But we know the constant term is $12$, so $a=6$, and hence
$T(x)=6x^4-12x^3+18x^2-12x+12$
