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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 )
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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

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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

how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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