The roots of the polynomial $f(x) = - x^4$ are $0$ with a multiplicity of $4$.
How can I apply the factor theorem:
$$f(x) = (x-0)(x-0)(x-0)(x-0)$$
correctly as to account for the minus sign?
I know the specific example is a bit nonsensical because of all the zeros, but it is about the theorem, and the apparent difficulty in going from $-x^4$ to $(x - a)$ factors.
The factor theorem (over a field) says that $x=c$ is a root of $f(x)$ if and only if $f(x) = (x-c)g(x)$ for some polynomial $g(x)$. Your minus sign is inside $g(x)$.