The remainder theorem says that the remainder upon dividing a polynomial [tex]p(x)[/tex] by a linear polynomial [tex]x-a[/tex] is the same as the value of [tex]p(x)[/tex] at [tex]x=a[/tex]. Dividing by any linear polynomial will always result in the following:
[tex]p(x)=(x-a)q(x)+r(x)[/tex]
where [tex]q(x)[/tex] and [tex]r(x)[/tex] are also polynomials. Taking [tex]x=a[/tex], the term involving [tex]q(x)[/tex] vanishes, so that [tex]p(a)=r(a)[/tex] is exactly the remainder upon dividing.
