MATH 2400 LECTURE NOTES: POLYNOMIAL AND RATIONAL
FUNCTIONS
PETE L. CLARK
Contents
1.
Polynomial Functions
1
2.
Rational Functions
6
1.
Polynomial Functions
Using the basic operations of addition, subtraction, multiplication, division and
composition of functions, we can combine very simple functions to build large and
interesting (and useful!)
classes of functions.
For us, the two simplest kinds of
functions are the following:
Constant functions
: for each
a
∈
R
there is a function
C
a
:
R
→
R
such that for
all
x
∈
R
,
C
a
(
x
) =
a
. In other words, the output of the function does not depend
on the input: whatever we put in, the same value
a
will come out. The graph of
such a function is the horizontal line
y
=
a
. Such functions are called
constant
.
The identity function
I
:
R
→
R
by
I
(
x
) =
x
.
The graph of the identity
function is the straight line
y
=
x
.
Recall that the identity function is socalled because it is an identity element for
the operation of function composition: that is, for any function
f
:
R
→
R
we have
I
◦
f
=
f
◦
I
=
f
.
Example: Let
m, b
∈
R
, and consider the function
L
:
R
→
R
by
x
7→
mx
+
b
.
Then
L
is built up out of constant functions and the identity function by addition
and multiplication:
L
=
C
m
·
I
+
C
b
.
Example: Let
n
∈
Z
+
.
The function
m
n
:
x
7→
x
n
is built up out of the iden
tity function by repreated multiplication:
m
n
=
I
·
I
· · ·
I
(
n I
’s altogether).
The general name for a function
f
:
R
→
R
which is built up out of the identity
function and the constant functions by finitely many additions and multiplications
is a
polynomial
. In other words, every polynomial function is of the form
(1)
f
:
x
7→
a
n
x
n
+
. . .
+
a
1
x
+
a
0
for some constants
a
0
, . . . , a
n
∈
R
.
1
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PETE L. CLARK
However, we also want to take – at least until we prove it doesn’t make a dif
ference – a more algebraic approach to polynomials. Let us define a
polynomial
expression
as an expression of the form
∑
n
i
=0
a
i
x
i
.
Thus, to give a polynomial
expression we need to give for each natural number
i
a constant
a
i
, while requiring
that all but finitely many of these constants are equal to zero: i.e., there exists
some
n
∈
N
such that
a
i
= 0 for all
i > n
.
Then every polynomial expression
f
=
∑
n
i
=0
a
i
x
i
determines a
polynomial func
tion
x
7→
f
(
x
). But it is at least conceivable that two differentlooking polynomial
expressions give rise to the
same function
. To give some rough idea of what I mean
here, consider the two expressions
f
= 2 arcsin
x
+ 2 arccos
x
and
g
=
π
. Now it
turns out for all
x
∈
[
−
1
,
1] (the common domain of the arcsin and arccos func
tions) we have
f
(
x
) =
π
.
(The angle
θ
whose sine is
x
is complementary to the
angle
φ
whose cosine is
x
, so arcsin
x
+ arccos
x
=
θ
+
φ
=
π
2
.) But still
f
and
g
are
given by different “expressions”: if I ask you what the coeﬃcient of arcsin
x
is in
the expression
f
, you will immediately tell me it is 2. If I ask you what the coeﬃ
cient of arcsin
x
is in the expression
π
, you will have no idea what I’m talking about.
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 Fall '11
 Clark
 Calculus, Addition, Division, Multiplication, Subtraction, Rational Functions, Prime number, polynomial division, polynomial expression

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