Shortcut Methods for Partial Fractions

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Offline msu_math

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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
Mohammad Salah Uddin

Lecturer in Mathematics
Department of Natural Sciences
FSIT, DIU

Offline msu_math

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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 »
Mohammad Salah Uddin

Lecturer in Mathematics
Department of Natural Sciences
FSIT, DIU

Offline msu_math

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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 »
Mohammad Salah Uddin

Lecturer in Mathematics
Department of Natural Sciences
FSIT, DIU

Offline proteeti

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

Offline shirin.ns

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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

Offline Mosammat Arifa Akter

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

Offline 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.......
Muhammad Tofazzal Hosain
Lecturer, Natural Sciences
Daffodil International University

Offline 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
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Offline sayma

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