03
SECTION 0.1
.
.
Polynomials and Rational Functions
3
05
4
3
2
1
±
1
±
2
±
3
±
4
±
5
±
±
2
±
3
p
e
FIGURE 0.2
The real line
For real numbers
a
and
b
,
where
a
<
b
,
we deﬁne the
closed interval
[
a
,
b
]tobethe
set of numbers between
a
and
b
,
including
a
and
b
(the
endpoints
), that is,
[
a
,
b
]
={
x
∈
R

a
≤
x
≤
b
}
,
as illustrated in Figure 0.3, where the solid circles indicate that
a
and
b
are included in
[
a
,
b
].
a
b
FIGURE 0.3
A closed interval
a
b
FIGURE 0.4
An open interval
Similarly, the
open interval
(
a
,
b
)is the set of numbers between
a
and
b
,
but
not
including the endpoints
a
and
b
,
that is,
(
a
,
b
)
x
∈
R

a
<
x
<
b
}
,
as illustrated in Figure 0.4, where the open circles indicate that
a
and
b
are not included in
(
a
,
b
).
You should already be very familiar with the following properties of real numbers.
THEOREM 1.1
If
a
and
b
are real numbers and
a
<
b
, then
(i) For any real number
c
,
a
+
c
<
b
+
c
.
(ii) For real numbers
c
and
d
,if
c
<
d
, then
a
+
c
<
b
+
d
.