1. Home
  2. Algebra and trigonometry
  3. Systems of equations and
  4. Partial fractions

11.4 Partial fractions  (Page 3/7)

<< Chapter < Page
  Algebra and trigonometry     Page 3 / 7
Chapter >> Page >

Given a rational expression with repeated linear factors, decompose it.

  1. Use a variable like   A , B ,   or   C   for the numerators and account for increasing powers of the denominators.
    P ( x ) Q ( x ) = A 1 ( a x + b ) + A 2 ( a x + b ) 2 +   .  .  . +  A n ( a 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 with repeated linear factors

Decompose the given rational expression with repeated linear factors.

− x 2 + 2 x + 4 x 3 −4 x 2 + 4 x

The denominator factors are   x ( x −2 ) 2 .   To allow for the repeated factor of   ( x −2 ) ,   the decomposition will include three denominators:   x , ( x −2 ) ,   and   ( x −2 ) 2 .   Thus,

− x 2 + 2 x + 4 x 3 −4 x 2 + 4 x = A x + B ( x −2 ) + C ( x −2 ) 2

Next, we multiply both sides by the common denominator.

x ( x −2 ) 2 [ − x 2 + 2 x + 4 x ( x −2 ) 2 ] = [ A x + B ( x −2 ) + C ( x −2 ) 2 ] x ( x −2 ) 2                  − x 2 + 2 x + 4 = A ( x −2 ) 2 + B x ( x −2 ) + C x

On the right side of the equation, we expand and collect like terms.

− x 2 + 2 x + 4 = A ( x 2 − 4 x + 4 ) + B ( x 2 − 2 x ) + C x                        = A x 2 − 4 A x + 4 A + B x 2 − 2 B x + C x                        = ( A + B ) x 2 + ( − 4 A − 2 B + C ) x + 4 A

Next, we compare the coefficients of both sides. This will give the system of equations in three variables:

− x 2 + 2 x + 4 = ( A + B ) x 2 + ( −4 A −2 B + C ) x + 4 A
A + B = −1 (1) −4 A −2 B + C = 2         (2) 4 A = 4         (3)

Solving for   A , we have

4 A = 4    A = 1

Substitute   A = 1   into equation (1).

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

Then, to solve for   C ,   substitute the values for   A   and   B   into equation (2).

       −4 A −2 B + C = 2 −4 ( 1 ) −2 ( −2 ) + C = 2              −4 + 4 + C = 2                             C = 2

Thus,

− x 2 + 2 x + 4 x 3 −4 x 2 + 4 x = 1 x − 2 ( x −2 ) + 2 ( x −2 ) 2
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the partial fraction decomposition of the expression with repeated linear factors.

6 x −11 ( x −1 ) 2

6 x −1 − 5 ( x −1 ) 2

Got questions? Get instant answers now!

Decomposing   P ( x ) Q ( x ) ,   Where Q(x) Has a nonrepeated irreducible quadratic factor

So far, we have performed partial fraction decomposition with expressions that have had linear factors in the denominator, and we applied numerators   A , B ,   or   C   representing constants. Now we will look at an example where one of the factors in the denominator is a quadratic expression that does not factor. This is referred to as an irreducible quadratic factor. In cases like this, we use a linear numerator such as   A x + B , B x + C ,   etc.

Decomposition of   P ( x ) Q ( x ) : Q ( x )   Has a nonrepeated irreducible quadratic factor

The partial fraction decomposition of   P ( x ) Q ( x )   such that   Q ( x )   has a nonrepeated irreducible quadratic factor and the degree of   P ( x )   is less than the degree of   Q ( x )   is written as

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

The decomposition may contain more rational expressions if there are linear factors. Each linear factor will have a different constant numerator:   A , B , C ,   and so on.

Given a rational expression where the factors of the denominator are distinct, irreducible quadratic factors, decompose it.

  1. Use variables such as   A , B ,   or   C   for the constant numerators over linear factors, and linear expressions such as   A 1 x + B 1 , A 2 x + B 2 ,   etc., for the numerators of each quadratic factor in the denominator.
    P ( x ) Q ( x ) = A a x + b + A 1 x + B 1 ( a 1 x 2 + b 1 x + c 1 ) + A 2 x + B 2 ( a 2 x 2 + b 2 x + c 2 ) + ⋅ ⋅ ⋅ + A n x + B n ( a n x 2 + b n x + c 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.

Questions & Answers

what does mean opportunity cost?
Aster Reply
what is poetive effect of population growth
Solomon Reply
what is inflation
Nasir Reply
what is demand
Eleni
what is economics
IMLAN Reply
economics theory describes individual behavior as the result of a process of optimization under constraints the objective to be reached being determined by
Kalkidan
Economics is a branch of social science that deal with How to wise use of resource ,s
Kassie
need
WARKISA
Economic Needs: In economics, needs are goods or services that are necessary for maintaining a certain standard of living. This includes things like healthcare, education, and transportation.
Kalkidan
What is demand and supply
EMPEROR Reply
deman means?
Alex
what is supply?
Alex
ex play supply?
Alex
Money market is a branch or segment of financial market where short-term debt instruments are traded upon. The instruments in this market includes Treasury bills, Bonds, Commercial Papers, Call money among other.
murana Reply
good
Kayode
what is money market
umar Reply
Examine the distinction between theory of comparative cost Advantage and theory of factor proportion
Fatima Reply
What is inflation
Bright Reply
a general and ongoing rise in the level of prices in an economy
AI-Robot
What are the factors that affect demand for a commodity
Florence Reply
price
Kenu
differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
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, 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