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

7.4 Partial fractions  (Page 3/7)

<< Chapter < Page
  College algebra     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

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
What is ceteris paribus?
Shukri Reply
other things being equal
AI-Robot
When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Kelo
yes,thank you
Shukri
Can I ask you other question?
Shukri
what is monopoly mean?
Habtamu Reply
What is different between quantity demand and demand?
Shukri Reply
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Lilia Reply
what is the difference between economic growth and development
Fiker Reply
Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
Abdisa Reply
any question about economics?
Awais Reply
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .
Feyisa Reply
Answer
Feyisa
c
Jabir
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
Gsbwnw Reply
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
Abdureman
types of unemployment
Yomi Reply
What is the difference between perfect competition and monopolistic competition?
Mohammed
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