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

11.4 Partial fractions  (Page 4/7)

<< Chapter < Page
  Algebra and trigonometry     Page 4 / 7
Chapter >> Page >

Decomposing   P ( x ) Q ( x )   When Q(x) Contains a nonrepeated irreducible quadratic factor

Find a partial fraction decomposition of the given expression.

8 x 2 + 12 x −20 ( x + 3 ) ( x 2 + x + 2 )

We have one linear factor and one irreducible quadratic factor in the denominator, so one numerator will be a constant and the other numerator will be a linear expression. Thus,

8 x 2 + 12 x −20 ( x + 3 ) ( x 2 + x + 2 ) = A ( x + 3 ) + B x + C ( x 2 + x + 2 )

We follow the same steps as in previous problems. First, clear the fractions by multiplying both sides of the equation by the common denominator.

( x + 3 ) ( x 2 + x + 2 ) [ 8 x 2 + 12 x − 20 ( x + 3 ) ( x 2 + x + 2 ) ] = [ A ( x + 3 ) + B x + C ( x 2 + x + 2 ) ] ( x + 3 ) ( x 2 + x + 2 )                                         8 x 2 + 12 x − 20 = A ( x 2 + x + 2 ) + ( B x + C ) ( x + 3 )

Notice we could easily solve for   A   by choosing a value for   x   that will make the   B x + C   term equal 0. Let x = −3   and substitute it into the equation.

                         8 x 2 + 12 x − 20 = A ( x 2 + x + 2 ) + ( B x + C ) ( x + 3 )     8 ( − 3 ) 2 + 12 ( − 3 ) − 20 = A ( ( − 3 ) 2 + ( − 3 ) + 2 ) + ( B ( − 3 ) + C ) ( ( − 3 ) + 3 )                                                 16 = 8 A                                                  A = 2

Now that we know the value of   A ,   substitute it back into the equation. Then expand the right side and collect like terms.

8 x 2 + 12 x −20 = 2 ( x 2 + x + 2 ) + ( B x + C ) ( x + 3 ) 8 x 2 + 12 x −20 = 2 x 2 + 2 x + 4 + B x 2 + 3 B + C x + 3 C 8 x 2 + 12 x −20 = ( 2 + B ) x 2 + ( 2 + 3 B + C ) x + ( 4 + 3 C )

Setting the coefficients of terms on the right side equal to the coefficients of terms on the left side gives the system of equations.

          2 + B = 8 (1) 2 + 3 B + C = 12 (2)         4 + 3 C = −20 (3)

Solve for   B   using equation (1) and solve for   C   using equation (3).

    2 + B = 8 (1)           B = 6 4 + 3 C = −20 (3)         3 C = −24           C = −8

Thus, the partial fraction decomposition of the expression is

8 x 2 + 12 x −20 ( x + 3 ) ( x 2 + x + 2 ) = 2 ( x + 3 ) + 6 x −8 ( x 2 + x + 2 )
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Could we have just set up a system of equations to solve [link] ?

Yes, we could have solved it by setting up a system of equations without solving for   A   first. The expansion on the right would be:

8 x 2 + 12 x −20 = A x 2 + A x + 2 A + B x 2 + 3 B + C x + 3 C 8 x 2 + 12 x −20 = ( A + B ) x 2 + ( A + 3 B + C ) x + ( 2 A + 3 C )

So the system of equations would be:

          A + B = 8 A + 3 B + C = 12       2 A + 3 C = −20

Find the partial fraction decomposition of the expression with a nonrepeating irreducible quadratic factor.

5 x 2 −6 x + 7 ( x −1 ) ( x 2 + 1 )

3 x −1 + 2 x −4 x 2 + 1

Got questions? Get instant answers now!

Decomposing   P ( x ) Q ( x )   When Q(x) Has a repeated irreducible quadratic factor

Now that we can decompose a simplified rational expression    with an irreducible quadratic factor, we will learn how to do partial fraction decomposition when the simplified rational expression has repeated irreducible quadratic factors. The decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented in increasing powers.

Decomposition of   P ( x ) Q ( x )   When Q(x) Has a repeated irreducible quadratic factor

The partial fraction decomposition of   P ( x ) Q ( x ) ,   when   Q ( x )   has a repeated irreducible quadratic factor and the degree of   P ( x )   is less than the degree of   Q ( x ) ,   is

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

Write the denominators in increasing powers.

Given a rational expression that has a repeated irreducible factor, decompose it.

  1. Use variables like   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 written in increasing powers, such as
    P ( x ) Q ( x ) = A a x + b + A 1 x + B 1 ( a x 2 + b x + c ) + A 2 x + B 2 ( a x 2 + b x + c ) 2 + ⋯ +   A n + B n ( a x 2 + b 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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