THE CHINESE UNIVERSITY OF HONG KONG
Department of Mathematics
MATH1020 General Mathematics (Fall 2017)
uReply set 4
1. Let
f
(
x
) = ln(2
x
+ 3)

1
.
The vertical asymptote of
f
(
x
) is

3
2
.
False.

3
2
is a number, not a line. The vertical asymptote of
f
(
x
) should be
x
=

3
2
.
2. The polynomial
x
5
+
x
can be factorized as
x
5
+
x
=
x
(
x
4
+ 1) but not any further.
False.
Every polynomial can be factorized as a product of linear and irreducible
quadratic polynomials. Indeed,
x
5
+
x
=
x
(
x
4
+ 1) =
x
(
x
2
+
√
2
x
+ 1)(
x
2

√
2
x
+ 1)
.
3. By partial fraction decomposition, there exists real constants
A, B
and
C
such that
x
3

1
(
x
2
+ 4)(
x

3)
=
Ax
+
B
x
2
+ 4
+
C
x

3
.
False.
The rational function
x
3
+ 1
(
x
2
+ 4)(
x

3)
is improper.
To decompose it, we
should do long division to obtain a proper rational function first:
x
3

1
(
x
2
+ 4)(
x

3)
= 1 +
3
x
2

4
x
+ 11
(
x
2
+ 4)(
x

3)
= 1 +
x

1
x
2
+ 4
+
2
x

3
.
4. By partial fraction decomposition,
x
4

x
2
+ 1
x
3
(
x
2

1)(
x
2
+
x
+ 1)
=
A
x
3
+
B
x
2
+
C
x
+
D
x

1
+
E
x
+ 1
+
Fx
+
G
x
2
+
x
+ 1
for some constants
A, B, C, D, E, F, G
.
True, as
x
3
should be regarded as (
x

0)
3
and
x
2
+
x
+ 1 can’t be factorized any
further (Δ
<
0).
5. The geometric series
+
∞
X
k
=1
r
k
diverges for

r
 ≥
1.
True. The series has partial sum