2.3
Partial
( fractions
of
fractions
polynomials )
fcpI-q.sn
[
n
1
=
{
=
o
p±
h
ok
,
1
=
≥2
_
_
.
→
→
(F)
(* )
=
du
¥
¥ In
=
{
=p ✗ +
¥
=
43 ,
k
=
pdx
f Inn
du
lulu I
C
+
""
4
=
-
=
(* )
=
=
f- lnlpxtqltc
¥-1 ip¥Ñᵗ
,
÷÷:÷÷:::::÷÷:÷
IRREDUCIBLE
u
=
a ✗ 7-
b✗
t c
die
→
=
Ga ✗
+
b) ok
.
¥
-
-
-
-
-
ok
In
ta f
=
[
h
-
-
-
-
-
-
-
-
-
-
=
22
1 →
→
-
Faln lait bx
zñ
+
el
+
C.
1
b×+cÑ
+
£
OLE
+
bx
+
§ ¥ n°k
IF
=
THEN
9--0
,
5+2
>☒
✗
@
✗ +
c-
b2
Ia
c-
,
9×7 bxtc
> 0
bzʳ
b2
_
.
Ia
because
is
¥
=
-
Gwaii
-
zb_ra)2
U=raxt¥
r2=
a > 0
C
¥a
+
-
du
-
c)
a >0
b2
_
Etc
+ C
the
irreducible
→
bx
=
polynomial
radx
fcaxi-s-xi-cy-ok-fo.hu?-r-yn-dui- i- FiJi#iau- Etan -1¥
-
+
C
-
-
-
h
≥2
-
:
-
y=¥
dy
_
Fda
ku¥pᵈu=↳÷ryndy
=¥f¥ñᵈ7
-4¥ not
SIEH
fy¥÷dy=
f¥
-1*1
dy
fyy¥
=kyñᵈy
%
=
,
-
-
,
{
k÷yndy={I
=
:S End
.
2¥ City
4--1=1*+7*4 [Y¥ñ¥¥
=
-
)
"
t C
-
+
=
¥,-mdy]
=
YE.in#-+f+E.da?-yn-I-
I
ok
IYD
deg (
P( )
x
-
N)
polynomial
I deg (D)
-1
-
linear polynomials
irreducible
quadratic polynomials
deg -2
product
and
deg
of
"
-
☐ (x )
=
pi -1%5412×+9-252
(04×2+4×+4) ( azXT-bzxtcz.IM
(
.
.
.
"
•)¥¥
=
,
p¥+ᵗ(pÉTᵗ
.
-
-
+
.
.
.
±
(
pitot
)
"
¥T =p.fi?-q-.t.--+cp?iTp;.tpE+-q.i-..-tcp?Eijn
.
"
15¥ .EE?ic-+..-+c.FiiiiEim=
,
1¥ E.IE?:-.+--.i::E-?::-.sm
-
-
,
+
E.IE?:-+c.i----+EaEEE?E-m
.
.
IF
oleg
IN )
≥
deg (D)
THEN
,
WE
DO
1¥
THE
=
LONG
a'
,
NH )
deg
=
DIVISION
+5¥
,
QAIDA )
( R)
-_
1-
RE )
deg (D)
;D
?⃝
I
Examinee
deg
N
=
2
>
3
=
4×2+13×-9
É
4×4-13×-9
=
=
1- Btc
A
-
-
deg D
¥ ¥+3 ¥
+
✗k+t¥¥Eii¥%-±
=
A
=
ok
B
3 A
+
=
3C
=
=
A Gts ) (x 1)
-
(At Btc)
2
✗
4
13
-
9
+
→
+
{
B ✗ G-
1)
KA B +3 c)
-
A =3
B
C
=
=
-1
2
+ a-
✗
-
at 3)
3A
¥¥¥→=-¥→+
"
=3 lnkl-lnkt311-2.lu/x- Htc
.lu/X31-lnKt3ItlnCx- 1PtC=lnkfE+-;Y-tC=
4×2+13×-9=1-4-+374-11
+
13×(1--1)
1- C
✗ =o
:
✗
(+1-3)
+ =-3
f✗
=
41012+13101-9=1-(3×-1)
-
9
=
-3A
→
1- =3
1
0
4×2+13×-9
✗
1
=
4.
12+13.1-9
8
=
-3
=
-
.
_
.
3) ( -11+13×1×-1 )
×
1-
C
:
41-13
✗
Aight
=
9
-
4C
:
=
=
→
C. 1.
=
=
(+1-3)
4C
4C
C
=
41-312+131-31
B
4
✗
-1
.
2
-9=13 (-311-4)
Examiner
deg D=
4
3
>
=
f3¥¥¥%%
oleg
ok
N
=¥i÷i←÷
+
¥-2,5
tBtÉ-4+DG
A
=
3
Ft
✗ 3-
18×2
1)
+
(x
-
2)
3
29×-4
=
A 1*-213+131×+114 -44-4×+4 G- 2) 1- DG
=
-1
+
=
3
C- 1) 3-
A
:
18
=
(-112+291-1)
2
-
4
=
A f- 3)
3
+
1)
✗
2
=
:
3 (2) 3- 18 (2)2+29 (2) -4
D. (3)
=
D
✗
=
0
:
4=21-431-131111-47
-
10=413
✗
=
1
-2C
→5=2B
:
31113-181112+29111-4
{
=
213
-
21-113+131-112 (2)
2B -2C
a- s
B- c-
-
=
+
→
→
+
CHI (2) +214
4=
{¥
?⃝
=
8
call-4+241
1- =L
S"¥¥; *
=
1¥ ¥ ¥ EH
+
+
.
=
=
=
2%1×+11
In
+
,
+
ln
.
"
It
-21-3.114--4
# 44×-4]
+
¥2
-
+2.12k¥
¥2
tc
+C
Examines
f2¥;¥¥¥¥•k
deg N=4 3=degD
>
2×3
'
-
✗
←
+8+-42×1×3 +9×2-5×-21
F¥÷÷÷¥*f¥+¥→
"
"
{
2+3
<
=
✗
'
-
✗
(2×-1)
+
+
2
✗
8×-4
4
=
(2×-1)=(+4-476×-1)
S
*
¥¥E÷
¥
+
=
,
=-É+I
§ 4- 4) (2×-1)
✗ 2-
✗
-21
=
=
✗
±
,
=
-
{
{
C
(A
(
2 A +c)
(2 A
:
2
1
-
=
-
=
B) (2×-1)
+ +
+
<
✗
5) ✗
+
2
+
( E+¥)
-
-
-
-
7
a- 4-
( Z B A) x
-
(2 B- A) x
t
C
A
5
t c
B.
(&
=
=
3
1
+
-
-
B
B
e)
4)
+
4C
1- 20
J
if
☒
35
=
ok
+
+
fÉ+a
✗
=
(1)
(2)
(* I
→
die
0th
%
Zz f
=
Iz
=
u
(3)
4- 4
=
-
=
tan
-
'
2×-1
-
}
=
¥
→
link4- 4)
+
ok
,
%±↑¥
(3)
,
ok
2✗
=
ok
%
f
§
{
"
(2)
(1)
U
-5¥
ok
ln 141
+
Ci
=
{ that 4) +4
Cz
du
=
+
=
-
{
2 ok
Ezln 12+-11+5
tan
-
'
E- { bike it
-
-
mPf5?¥¥
ok
degN=3=4=degD
5¥¥
.fi?-3+ii-+i - %EEEiji1- a-+D=
5×3 -3×4-7×-3=11-+1-1371×72)tGtD
=A✗3tBx2+(1-+0×+13+9
A- 5
*
→
131-5=-3
:{
I
{
13=-3
13=-3
→
-31-5=-3
you
D=o
Ss
=
◦k=*÷i⇐¥¥
51¥
ok
3f¥ik +21¥.pk
-
,
(2)
(1)
h
(1)
(2)
(3)
(* )
=
=
✗
2+1
5) % ok
=-3 tan
=
=
du
→
/%
^
×
du
1-
=
4
Zxdx
5 In Kkr )
=
-
=
C })
+
C
,
Cz
-
5hr47 1) -3 tan
I
-
+4
"
✗
-
=
¥-1 +9
¥
,
+
C