II
Polynomials and Rational Functions
II.1
II
Polynomials and Rational Functions
Polynomials are perhaps the most important functions we will deal with. They are easy to manipulate or to
evaluate, but finding roots of polynomials is one of the oldest and more difficult tasks in mathematics. We
review the definitions and important properties; most of them you have probably seen before.
II.A
Polynomials
A
real polynomial
is an expression of the form
P
(
x
) =
a
n
x
n
+
a
n

1
x
n

1
+
· · ·
+
a
0
n
≥
0
, n
∈
N
with
a
i
∈
R
and
x
a variable. The
a
i
are called the
coefficients
,
a
0
is the
constant coefficient
. If
a
n
6
= 0 we
say
P
(
x
) has
degree
n
, deg
P
=
n
, and
a
n
is called the
leading coefficient
The
zero polynomial
P
(
x
) = 0,
i.e.
a
n
=
a
n

1
=
· · ·
=
a
0
= 0 poses a bit of a problem: It has
no
degree, thus
no
leading coefficient, but
constant coefficient zero! To exclude this pesty zero polynomial, we often speak only of “polynomials that
have a degree” or of “nonzero polynomials”.
If
P
and
Q
are nonzero polynomials, then
deg (
PQ
) = deg
P
+ deg
Q,
deg (
P
+
Q
)
= max(deg
P,
deg
Q
)
if deg
P
6
= deg
Q,
≤
max(deg
P,
deg
Q
)
if deg
P
= deg
Q.
If
a
m
is the leading coefficient of
P
and
b
n
the leading coefficient of
Q
, then
a
m
b
n
is the leading coefficient
of
P
·
Q
. The constant coefficient of
P
·
Q
is
a
0
b
0
if
a
0
is that of
P
and
b
0
that of
Q
.
Examples
.
1. The polynomial
P
(
x
) = 2
x
+ 1 has degree 1, it is a
linear
polynomial. Its leading coefficient is 2, its
constant coefficient is 1.
2.
P
(
x
) =
x
2

√
5
x
+ 3 is of degree 2, it is a
quadratic
polynomial.
3.
P
(
x
) =
πx
3

x
2
+ 2
/
9 is of degree 3, it is a
cubic
polynomial.
4.
P
(
x
) = 1124 is of degree 0. It is a
constant
polynomial. The leading coefficient is 1124 which is also
the constant coefficient.
5. If
P
(
x
) =
x
2
+ 1,
Q
(
x
) =

x
2
+ 1, then deg
P
= deg
Q
= 2, and (
P
·
Q
)(
x
) =

x
4
+ 1 has degree 4,
whereas
P
(
x
) +
Q
(
x
) = 2 has degree 0, less than 2 = max(deg
P,
deg
Q
).
Polynomials are not always given in
expanded
form as above. For example,
P
(
x
) = (
x

1)(
x
+ 1)(
x
2
+ 1) is
also a polynomial, of degree 4 with leading coefficient 1 and constant coefficient

1. Its expanded form is
P
(
x
) =
x
4
+ 0
·
x
3
+ 0
·
x
2
+ 0
·
x

1 =
x
4

1, as you can check easily.
A basic result on polynomials is that one can perform
polynomial
or
long division
:
Theorem I (Polynomial Division):
Let
P
(
x
)
, Q
(
x
) be polynomials with deg
Q
(
x
) =
m
, (in particular,
Q
(
x
) is not the zero polynomial!). There are then
unique
polynomials
S
(
x
) and
R
(
x
) such that
(i)
P
(
x
) =
S
(
x
)
Q
(
x
) +
R
(
x
)