## 5.2. The division algorithm.

We say that a polynomial divides a polynomial (written
) if there is a polynomial such that . Notice that if
then either or . Notice that if then either
or (or both). This means that we can (only) divide by nonzero
polynomials.

** Problem 5.3.** Let
be two polynomials. Prove that if and then .

In fact, there is an analog of long division for polynomials, as there is for
integers.

** Theorem 5.4.*** Let
be polynomials with . Then
there exist unique polynomials
such that and . *

*Proof.* We ﬁrst prove **existence** of . If then
we set and . Otherwise, .
Let

and

Deﬁne the integer . We will use induction in .

**Base step:** Let , then . We set and
. Notice that
is well-deﬁned
because and that
the coefficient of in vanishes, whence .

**Induction step:** Now we assume that this is true whenever
and let , so that . Let .
Notice that , whence
by induction there exist with and . Therefore

whence deﬁne and the result is true for .

Finally we prove **uniqueness**. Suppose that ,
with and . Rearranging,
we have , whence
divides . Since ,
this can only happen if , whence . In this case,
, which since
implies
that or, equivalently,
that .□

The polynomials and in the statement of
the theorem are called the quotient and **remainder**, respectively.

Given we can ﬁnd by long division, as the following example
shows.

** Example 5.5.** Let and . Then we let . Similarly let ,
whose degree is less than that of . Therefore,

whence and .