Shortcut Methods for Partial Fractions

### Author Topic: Shortcut Methods for Partial Fractions  (Read 12042 times)

#### msu_math

• Jr. Member
• Posts: 81
##### Shortcut Methods for Partial Fractions
« on: November 21, 2012, 10:55:57 AM »
In algebra, the partial fraction decomposition or partial fraction expansion is a procedure used to reduce the degree of either the numerator or the denominator of a rational function. The main motivation to decompose a rational function into a sum of simpler fractions is that it makes it simpler to perform linear operations on it.

The problem of computing derivatives, anti-derivatives, integrals, inverse Laplace transforms, power series expansions, Fourier series, residues, and linear functional transformations of rational functions can be reduced, via partial fraction decomposition, to making the computation on each single element used in the decomposition.

Standard Form 1:

p                   p           1            1
----------------  =  ------- ( --------  -  ------- )
(x+a)(x+b)          b-a       x+a         x+b

Example:

5                   5             1             1               5       1             1
----------------  =  ---------  ( --------  -  ------ )   =   ---- ( -------  -  -------- )
(x-1)(x+3)          3-(-1)        x-1         x+3             4      x-1         x+3

Standard Form 2:

p                      p           1               1
--------------------  =  ------- ( ---------  -  --------- )
(xn+a)(xn+b)          b-a       xn+a         xn+b

Example:

3                       3             1              1                 3        1              1
------------------  =  ---------  ( ---------  -  --------- )   =   ---- ( --------  -  --------- )
(x2-2)(x2+3)         3-(-2)        x2-2         x2+3              5      x2-2         x2+3

Standard Form 3:

p                     p            1             x-a
----------------  =  ---------- ( --------  -  ---------- )
(x+a)(x2+b)         b+a2       x+a          x2+b

Example:

7                        7              1            x+2               1           x+2
-----------------  =  ------------  ( -------  -  --------- )   =  -------  -  ---------
(x-2)(x2+3)          3+(-2)2        x-2         x2+3             x-2         x2+3

Lecturer in Mathematics
Department of Natural Sciences
FSIT, DIU

#### msu_math

• Jr. Member
• Posts: 81
##### Re: Shortcut Methods for Partial Fractions
« Reply #1 on: November 21, 2012, 10:58:14 AM »
Useful Example:

4               1        4        1         1               4        1             1                 4          4      1          1
------------  =  ---- . ------- ( ----  -  ------ )   =   ---- ( ------  -  ----------- )  =  ------  -  ---- ( ----  -  ------- )
x2(x+3)          x      3-0       x       x+3             3       x2         x(x+3)           3x2        3       x        x+3

4          4              4
=  ------  -  -----  +  ----------
3x2        3x        3(x+3)
« Last Edit: May 20, 2013, 08:33:22 AM by msu_math »

Lecturer in Mathematics
Department of Natural Sciences
FSIT, DIU

#### msu_math

• Jr. Member
• Posts: 81
##### Re: Shortcut Methods for Partial Fractions
« Reply #2 on: December 10, 2012, 11:02:55 AM »
Important Example:

4               1        4        1         1               4        1             1                 4          4      1          1
------------  =  ---- . ------- ( ----  -  ------ )   =   ---- ( ------  -  ----------- )  =  ------  -  ---- ( ----  -  ------- )
x3(x2+3)        x      3-0       x2      x2+3           3       x3        x(x2+3)          3x3        3      x2      x2+3

4           4                4
=  ------  -  ------  +  ------------
3x3        3x2         3(x2+3)
« Last Edit: December 10, 2012, 11:07:41 AM by msu_math »

Lecturer in Mathematics
Department of Natural Sciences
FSIT, DIU

#### proteeti

• Full Member
• Posts: 102
##### Re: Shortcut Methods for Partial Fractions
« Reply #3 on: March 28, 2014, 10:49:01 PM »
oh great!

#### shirin.ns

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• Posts: 343
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##### Re: Shortcut Methods for Partial Fractions
« Reply #4 on: March 31, 2014, 02:33:20 PM »
Awesome!!!
Shirin Sultana
Lecturer (Mathematics)
Dept. of General Educational Development (GED)
Daffodil International university

#### Mosammat Arifa Akter

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##### Re: Shortcut Methods for Partial Fractions
« Reply #5 on: May 24, 2014, 12:52:59 PM »
Mosammat Arifa Akter
Senior Lecturer(Mathematics)
General Educational Development
Daffodil International University

#### Tofazzal.ns

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##### Re: Shortcut Methods for Partial Fractions
« Reply #6 on: May 27, 2015, 06:43:40 PM »
Excellent....Thanks to share.....helpful for every students.......
Lecturer, Natural Sciences
Daffodil International University

#### Anuz

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##### Re: Shortcut Methods for Partial Fractions
« Reply #7 on: November 29, 2015, 12:19:49 PM »
Useful for the students...........
Anuz Kumar Chakrabarty
Assistant Professor
Department of General Educational Development
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#### sayma

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##### Re: Shortcut Methods for Partial Fractions
« Reply #8 on: November 30, 2015, 03:26:39 PM »