Intro
Translations in the plane
Examples
HowTo: Gaussian Reduction5.2. The division algorithm.
We say that a polynomial ![g ∈ ℝ[X]](./img/Notes2084x.png)
divides a polynomial ![f ∈ ℝ[X]](./img/Notes2085x.png)
(written
) if there is a polynomial ![q ∈ ℝ[X]](./img/Notes2087x.png)
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 ![f,g ∈ ℝ[X]](./img/Notes2095x.png)
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 ![f,g ∈ ℝ[X]](./img/Notes2099x.png)
be polynomials with 
. Then
there exist unique polynomials ![q,r ∈ ℝ[X]](./img/Notes2101x.png)
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 ![f,g ∈ ℝ[X]](./img/Notes2148x.png)
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 
.
