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.
Via synthetic division, we have
... | 2 -9 7 -5 11
4 | 8 -4 12 28
- - - - - - - - - - - - - - - - - -
... | 2 -1 3 7 39
which translates to
[tex]\dfrac{2x^4-9x^3+7x^2-5x+11}{x-4}=2x^3-x^2+3x+7+\dfrac{39}{x-4}[/tex]
that is, we're left with a remainder of 39.
Via the remainder theorem, we have
[tex]p(4)=2\times4^4-9\times4^3+7\times4^2-5\times4+11=39[/tex]
as expected.
