1. Home
  2. College algebra
  3. Systems of equations and
  4. Partial fractions

7.4 Partial fractions  (Page 2/7)

<< Chapter < Page
  College algebra     Page 2 / 7
Chapter >> Page >

Given a rational expression with distinct linear factors in the denominator, decompose it.

  1. Use a variable for the original numerators, usually   A , B ,  or   C ,   depending on the number of factors, placing each variable over a single factor. For the purpose of this definition, we use   A n   for each numerator
    P ( x ) Q ( x ) = A 1 ( a 1 x + b 1 ) + A 2 ( a 2 x + b 2 ) + ⋯ + A n ( a n x + b n )
  2. Multiply both sides of the equation by the common denominator to eliminate fractions.
  3. Expand the right side of the equation and collect like terms.
  4. Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.

Decomposing a rational function with distinct linear factors

Decompose the given rational expression    with distinct linear factors.

3 x ( x + 2 ) ( x −1 )

We will separate the denominator factors and give each numerator a symbolic label, like   A , B  , or   C .

3 x ( x + 2 ) ( x −1 ) = A ( x + 2 ) + B ( x −1 )

Multiply both sides of the equation by the common denominator to eliminate the fractions:

( x + 2 ) ( x −1 ) [ 3 x ( x + 2 ) ( x −1 ) ] = ( x + 2 ) ( x −1 ) [ A ( x + 2 ) ] + ( x + 2 ) ( x −1 ) [ B ( x −1 ) ]

The resulting equation is

3 x = A ( x −1 ) + B ( x + 2 )

Expand the right side of the equation and collect like terms.

3 x = A x − A + B x + 2 B 3 x = ( A + B ) x − A + 2 B

Set up a system of equations associating corresponding coefficients.

3 =         A + B 0 = − A + 2 B

Add the two equations and solve for   B .

3 =           A + B 0 = − A + 2 B ¯ 3   =           0     + 3 B 1 = B

Substitute   B = 1   into one of the original equations in the system.

3 = A + 1 2 = A

Thus, the partial fraction decomposition is

3 x ( x + 2 ) ( x −1 ) = 2 ( x + 2 ) + 1 ( x −1 )

Another method to use to solve for   A   or   B   is by considering the equation that resulted from eliminating the fractions and substituting a value for   x   that will make either the A - or B -term equal 0. If we let   x = 1 ,   the
A - term becomes 0 and we can simply solve for   B .

       3 x = A ( x − 1 ) + B ( x + 2 )   3 ( 1 ) = A [ ( 1 ) − 1 ] + B [ ( 1 ) + 2 ]          3 = 0 + 3 B          1 = B

Next, either substitute   B = 1   into the equation and solve for   A ,   or make the B -term 0 by substituting   x = −2   into the equation.

          3 x = A ( x − 1 ) + B ( x + 2 )        3 ( − 2 ) = A [ ( − 2 ) − 1 ] + B [ ( − 2 ) + 2 ]           − 6 = − 3 A + 0           − 6 − 3 = A            2 = A

We obtain the same values for   A   and   B   using either method, so the decompositions are the same using either method.

3 x ( x + 2 ) ( x −1 ) = 2 ( x + 2 ) + 1 ( x −1 )

Although this method is not seen very often in textbooks, we present it here as an alternative that may make some partial fraction decompositions easier. It is known as the Heaviside method , named after Charles Heaviside, a pioneer in the study of electronics.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the partial fraction decomposition of the following expression.

x ( x −3 ) ( x −2 )

3 x −3 − 2 x −2

Got questions? Get instant answers now!

Decomposing   P ( x ) Q ( x )   Where Q(x) Has repeated linear factors

Some fractions we may come across are special cases that we can decompose into partial fractions with repeated linear factors. We must remember that we account for repeated factors by writing each factor in increasing powers.

Partial fraction decomposition of   P ( x ) Q ( x ) : Q ( x )   Has repeated linear factors

The partial fraction decomposition of   P ( x ) Q ( x ) ,   when   Q ( x )   has a repeated linear factor occurring   n   times and the degree of   P ( x )   is less than the degree of   Q ( x ) ,   is

P ( x ) Q ( x ) = A 1 ( a x + b ) + A 2 ( a x + b ) 2 + A 3 ( a x + b ) 3 + ⋅ ⋅ ⋅ + A n ( a x + b ) n

Write the denominator powers in increasing order.

Questions & Answers

if three forces F1.f2 .f3 act at a point on a Cartesian plane in the daigram .....so if the question says write down the x and y components ..... I really don't understand
Syamthanda Reply
hey , can you please explain oxidation reaction & redox ?
Boitumelo Reply
hey , can you please explain oxidation reaction and redox ?
Boitumelo
for grade 12 or grade 11?
Sibulele
the value of V1 and V2
Tumelo Reply
advantages of electrons in a circuit
Rethabile Reply
we're do you find electromagnetism past papers
Ntombifuthi
what a normal force
Tholulwazi Reply
it is the force or component of the force that the surface exert on an object incontact with it and which acts perpendicular to the surface
Sihle
what is physics?
Petrus Reply
what is the half reaction of Potassium and chlorine
Anna Reply
how to calculate coefficient of static friction
Lisa Reply
how to calculate static friction
Lisa
How to calculate a current
Tumelo
how to calculate the magnitude of horizontal component of the applied force
Mogano
How to calculate force
Monambi
a structure of a thermocouple used to measure inner temperature
Anna Reply
a fixed gas of a mass is held at standard pressure temperature of 15 degrees Celsius .Calculate the temperature of the gas in Celsius if the pressure is changed to 2×10 to the power 4
Amahle Reply
How is energy being used in bonding?
Raymond Reply
what is acceleration
Syamthanda Reply
a rate of change in velocity of an object whith respect to time
Khuthadzo
how can we find the moment of torque of a circular object
Kidist
Acceleration is a rate of change in velocity.
Justice
t =r×f
Khuthadzo
how to calculate tension by substitution
Precious Reply
hi
Shongi
hi
Leago
use fnet method. how many obects are being calculated ?
Khuthadzo
khuthadzo hii
Hulisani
how to calculate acceleration and tension force
Lungile Reply
you use Fnet equals ma , newtoms second law formula
Masego
please help me with vectors in two dimensions
Mulaudzi Reply
how to calculate normal force
Mulaudzi
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 2

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask