Partial Fraction Expansion : Sect .
) 5
.
,
S4 & SS of supp
M
T
=
Y((n(x + 2) + (n(x + 3) ~
·
Ex :
Jo +t
=
Al(x-al + B(n(x b) -
· ((
- =
5) - J /
↑ (((x
=
-
3) -
(((x + 3) =
(((
A (x2 + 1)
(x 1)(x +3)(x + 2)
-
* Want to
,
B
,
x+1 =
A(x + 2)(x + 3) + B(x -
1)(x + 2) + ((x -
1)(x + 3)
· x = (2 + 1 =
A(1 + 2)(1 3) + + B .
(0) + C () .
· Ent
(0 = B( 4) 1) 2 5 =
C .
( 3)()
x =3 :
-
· -
( - ·
x
= -
=
B c =
-
Sz
-
Theorem If f(x) : = i -am) ,
where a , -, ..., an are distinct real & P(x) is a polynomial
of begreeh , then :
·
real A Az
, , ....
An St .- P .
a)
degree of P(x)
#s]
·
<
#xe = · a
.....,
an are real
conditions
~
& Can :
a
polynomial acxs us real coefficients always be factures as
:
q(x) = (x1) ... (x-an)
No !
thum
·
Fundamental"of algebra :
Every cron-constant) polynomial (a) w) -crefficients can be factures as
I
ex
=
x3 - 1 =
(x -
k(X -
G)(x -
B) a(x) =
C(x -
d
,
)"(x d)-
...
(x -
d)
& = cus(23) + iSih (24/3)
where I, In are distinct -numbers and hi
, ne are
positive integers
B =
cos(4x/3) + isib (443) ..., ...,
, 2
&
·
Fundamental thro of algebra (For real polynomials) :
Every curn-constant) polynomial (x)
w/real coefficients factors as a product ear
of and iresucabl
quadratic factors
w/ real
- coefficients
=(x
-
n)(x b) -
·
ex : xY -
1 =
((x -) -
12)
x3 -
1 = (x -
1) (x
+ x 1) +
irresucable =
(x -
x(x2 + 1)
+ (x 1)(x 1)(x2 + 1)
427
= +
z)
-
y+ 2 .
-
y + 1 = (x + +
- irreduciblever real 3
-
"Real polynomial = product of linear factors & guadratic
linear quadratic so
use
Et#
PLD 223
Degree of num
segree of denum
base
man exc on
7 /
3x2 + 2 =
A(x"+ 4) +
(bx + c)(x -
1)
Ax 4A By Bx + (x C
I
= +
-
+ -
(n(x 11 +
3x" + 2 (A + B) x 2 (C B)x 4A-
= C
-
=
+ -
+
( (
A+B =
3 A+B =
3
= =
(n(u) + C
matching c -
B =
0 E B= ( 4A B = 2-
Jegrees 4A -
2= 2
SA =
S
= + a =
y+ 4
= (n(x =
+ y) +
~
A = 1
B +A = 3 = B = 2
c= 2
↳ (x
= 26u
=
= fan n + c =
fun " (2) + (
J (x =
(n(x -
+ + ((x2 + 4) +
fan"() +
Ex :·
Unti
+= Scr J bx/
repeating =
factor
Guerence rate
Smartify/use
·
identities to guess integral Y) .
Complex exponential
2 ) Substitution
. S . ) PFE
3 ) /BP
.
· x =
/ 16x =
x +
T
vS .
June b
Ste
-
=
= - Jan
=
[h(n -11 -
E((n + 1 + ) =
&(n) += )
, = · ...
= *
Jegree of
a
<h
=
1 =
A(x + 1) +
B , (x -
1)(x + 1) + Bz(x -
1
=
Ax + 2Ax + A + B x2 ,
-
B, +
Bzx -
Bz
1 =
(A + B ) x
,
+ (2A -
Bi + Bz)x + A -
B, -
Ba
·
A+ B, = 0 Bi = - A
2A B ,
-
+ By = 0 3A + Be =
0 = Br = -
3A
A -
B, -
Bz =
1 =) A -
C A) C 31) 1
-
=
- -
A= 1
) 5
.
,
S4 & SS of supp
M
T
=
Y((n(x + 2) + (n(x + 3) ~
·
Ex :
Jo +t
=
Al(x-al + B(n(x b) -
· ((
- =
5) - J /
↑ (((x
=
-
3) -
(((x + 3) =
(((
A (x2 + 1)
(x 1)(x +3)(x + 2)
-
* Want to
,
B
,
x+1 =
A(x + 2)(x + 3) + B(x -
1)(x + 2) + ((x -
1)(x + 3)
· x = (2 + 1 =
A(1 + 2)(1 3) + + B .
(0) + C () .
· Ent
(0 = B( 4) 1) 2 5 =
C .
( 3)()
x =3 :
-
· -
( - ·
x
= -
=
B c =
-
Sz
-
Theorem If f(x) : = i -am) ,
where a , -, ..., an are distinct real & P(x) is a polynomial
of begreeh , then :
·
real A Az
, , ....
An St .- P .
a)
degree of P(x)
#s]
·
<
#xe = · a
.....,
an are real
conditions
~
& Can :
a
polynomial acxs us real coefficients always be factures as
:
q(x) = (x1) ... (x-an)
No !
thum
·
Fundamental"of algebra :
Every cron-constant) polynomial (a) w) -crefficients can be factures as
I
ex
=
x3 - 1 =
(x -
k(X -
G)(x -
B) a(x) =
C(x -
d
,
)"(x d)-
...
(x -
d)
& = cus(23) + iSih (24/3)
where I, In are distinct -numbers and hi
, ne are
positive integers
B =
cos(4x/3) + isib (443) ..., ...,
, 2
&
·
Fundamental thro of algebra (For real polynomials) :
Every curn-constant) polynomial (x)
w/real coefficients factors as a product ear
of and iresucabl
quadratic factors
w/ real
- coefficients
=(x
-
n)(x b) -
·
ex : xY -
1 =
((x -) -
12)
x3 -
1 = (x -
1) (x
+ x 1) +
irresucable =
(x -
x(x2 + 1)
+ (x 1)(x 1)(x2 + 1)
427
= +
z)
-
y+ 2 .
-
y + 1 = (x + +
- irreduciblever real 3
-
"Real polynomial = product of linear factors & guadratic
linear quadratic so
use
Et#
PLD 223
Degree of num
segree of denum
base
man exc on
7 /
3x2 + 2 =
A(x"+ 4) +
(bx + c)(x -
1)
Ax 4A By Bx + (x C
I
= +
-
+ -
(n(x 11 +
3x" + 2 (A + B) x 2 (C B)x 4A-
= C
-
=
+ -
+
( (
A+B =
3 A+B =
3
= =
(n(u) + C
matching c -
B =
0 E B= ( 4A B = 2-
Jegrees 4A -
2= 2
SA =
S
= + a =
y+ 4
= (n(x =
+ y) +
~
A = 1
B +A = 3 = B = 2
c= 2
↳ (x
= 26u
=
= fan n + c =
fun " (2) + (
J (x =
(n(x -
+ + ((x2 + 4) +
fan"() +
Ex :·
Unti
+= Scr J bx/
repeating =
factor
Guerence rate
Smartify/use
·
identities to guess integral Y) .
Complex exponential
2 ) Substitution
. S . ) PFE
3 ) /BP
.
· x =
/ 16x =
x +
T
vS .
June b
Ste
-
=
= - Jan
=
[h(n -11 -
E((n + 1 + ) =
&(n) += )
, = · ...
= *
Jegree of
a
<h
=
1 =
A(x + 1) +
B , (x -
1)(x + 1) + Bz(x -
1
=
Ax + 2Ax + A + B x2 ,
-
B, +
Bzx -
Bz
1 =
(A + B ) x
,
+ (2A -
Bi + Bz)x + A -
B, -
Ba
·
A+ B, = 0 Bi = - A
2A B ,
-
+ By = 0 3A + Be =
0 = Br = -
3A
A -
B, -
Bz =
1 =) A -
C A) C 31) 1
-
=
- -
A= 1
