<< Chapter < Page 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

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask