Recently I've started to study polynomials, when I found out about the remainder and factor theorems as a way to avoid long polynomial division I couldn't understand the reason for every linear factor to start with:
$$x-$$
For example, when the actual x-inteceptor is $-2$, the factor becomes:
$$x+2$$
Why is the $-$ after the $x$?
Maybe its a stupid quastion because if the factor is $x-2$ for example:
$$g(x) = x-2$$
So moving the $-2$ to the left will produce $x=2$(the actual x-interceptor), but I think there is more than that or maybe I just completely misunderstood the all idea.
The factor theorem states that $\,x-a\,$ of a factor of the polynomial $\,f(x)\iff f(a) = 0.$ Negating $\,a\,$ this is instead is $\,x+a\,$ of a factor of the polynomial $\,f(x)\iff f(-a) = 0.$
Notice that when $\,f(x) = g(x)h(x)\,$ then any root of $\,g(x)\,$ is also root of $\,f(x).\,$ Therefore, $ $ when $\,f\,$ has a factor $\,x-a\,$ then its root $\,x = a\,$ is also a root of $\,f,\,$ i.e. $\,f(a) = 0.\,$ And, similarly when $\,f\,$ has a factor $\,x+a,\,$ then its root $\,x = -a\,$ is also a root of $\,f,\,$ i.e. $\,f(-a) = 0$.