Example 1

Let ${\displaystyle f(x)=x^{3}-12x^{2}-42}$. Polynomial division of ${\displaystyle f(x)}$ by ${\displaystyle (x-3)}$ gives the quotient ${\displaystyle x^{2}-9x-27}$ and the remainder ${\displaystyle -123}$. Therefore, ${\displaystyle f(3)=-123}$.

Example 2

Proof that the polynomial remainder theorem holds for an arbitrary second degree polynomial ${\displaystyle f(x)=ax^{2}+bx+c}$ by using algebraic manipulation:

$${\displaystyle {\begin{aligned}f(x)-f(r)&=ax^{2}+bx+c-(ar^{2}+br+c)\\&=a(x^{2}-r^{2})+b(x-r)\\&=a(x-r)(x+r)+b(x-r)\\&=(x-r)(ax+ar+b)\end{aligned}}}$$

So,

$${\displaystyle f(x)=(x-r)(ax+ar+b)+f(r),}$$

which is exactly the formula of Euclidean division.

The generalization of this proof to any degree is given below in § Direct proof.

Using Euclidean division

The polynomial remainder theorem follows from the theorem of Euclidean division, which, given two polynomials f(x) (the dividend) and g(x) (the divisor), asserts the existence (and the uniqueness) of a quotient Q(x) and a remainder R(x) such that

${\displaystyle f(x)=Q(x)g(x)+R(x)\quad {\text{and}}\quad R(x)=0\ {\text{ or }}\deg(R)<\deg(g).}$

If the divisor is ${\displaystyle g(x)=x-r,}$ where r is a constant, then either R(x) = 0 or its degree is zero; in both cases, R(x) is a constant that is independent of x; that is

${\displaystyle f(x)=Q(x)(x-r)+R.}$

Setting ${\displaystyle x=r}$ in this formula, we obtain:

${\displaystyle f(r)=R.}$

Direct proof

A constructive proof—that does not involve the existence theorem of Euclidean division—uses the identity

${\displaystyle x^{k}-r^{k}=(x-r)(x^{k-1}+x^{k-2}r+\dots +xr^{k-2}+r^{k-1}).}$

If ${\displaystyle S_{k}}$ denotes the large factor in the right-hand side of this identity, and

${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0},}$

one has

${\displaystyle f(x)-f(r)=(x-r)(a_{n}S_{n}+\cdots +a_{2}S_{2}+a_{1}),}$

(since ${\displaystyle S_{1}=1}$).

Adding ${\displaystyle f(r)}$ to both sides of this equation, one gets simultaneously the polynomial remainder theorem and the existence part of the theorem of Euclidean division for this specific case.