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

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
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