# 9.4 Partial fractions  (Page 4/7)

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## Decomposing $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ When Q(x) Contains a nonrepeated irreducible quadratic factor

Find a partial fraction decomposition of the given expression.

$\frac{8{x}^{2}+12x-20}{\left(x+3\right)\left({x}^{2}+x+2\right)}$

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,

$\frac{8{x}^{2}+12x-20}{\left(x+3\right)\left({x}^{2}+x+2\right)}=\frac{A}{\left(x+3\right)}+\frac{Bx+C}{\left({x}^{2}+x+2\right)}$

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

Notice we could easily solve for $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ by choosing a value for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that will make the $\text{\hspace{0.17em}}Bx+C\text{\hspace{0.17em}}$ term equal 0. Let $x=-3\text{\hspace{0.17em}}$ and substitute it into the equation.

Now that we know the value of $\text{\hspace{0.17em}}A,\text{\hspace{0.17em}}$ substitute it back into the equation. Then expand the right side and collect like terms.

$\begin{array}{l}\hfill \\ 8{x}^{2}+12x-20=2\left({x}^{2}+x+2\right)+\left(Bx+C\right)\left(x+3\right)\hfill \\ 8{x}^{2}+12x-20=2{x}^{2}+2x+4+B{x}^{2}+3B+Cx+3C\hfill \\ 8{x}^{2}+12x-20=\left(2+B\right){x}^{2}+\left(2+3B+C\right)x+\left(4+3C\right)\hfill \end{array}$

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

Solve for $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ using equation (1) and solve for $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ using equation (3).

Thus, the partial fraction decomposition of the expression is

$\frac{8{x}^{2}+12x-20}{\left(x+3\right)\left({x}^{2}+x+2\right)}=\frac{2}{\left(x+3\right)}+\frac{6x-8}{\left({x}^{2}+x+2\right)}$

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 $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ first. The expansion on the right would be:

$\begin{array}{l}\begin{array}{l}\\ 8{x}^{2}+12x-20=A{x}^{2}+Ax+2A+B{x}^{2}+3B+Cx+3C\end{array}\hfill \\ 8{x}^{2}+12x-20=\left(A+B\right){x}^{2}+\left(A+3B+C\right)x+\left(2A+3C\right)\hfill \end{array}$

So the system of equations would be:

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

$\frac{5{x}^{2}-6x+7}{\left(x-1\right)\left({x}^{2}+1\right)}$

$\frac{3}{x-1}+\frac{2x-4}{{x}^{2}+1}$

## Decomposing $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ When Q(x) Has a repeated irreducible quadratic factor

The partial fraction decomposition of $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)},\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}Q\left(x\right)\text{\hspace{0.17em}}$ has a repeated irreducible quadratic factor and the degree of $\text{\hspace{0.17em}}P\left(x\right)\text{\hspace{0.17em}}$ is less than the degree of $\text{\hspace{0.17em}}Q\left(x\right),\text{\hspace{0.17em}}$ is

$\frac{P\left(x\right)}{{\left(a{x}^{2}+bx+c\right)}^{n}}=\frac{{A}_{1}x+{B}_{1}}{\left(a{x}^{2}+bx+c\right)}+\frac{{A}_{2}x+{B}_{2}}{{\left(a{x}^{2}+bx+c\right)}^{2}}+\frac{{A}_{3}x+{B}_{3}}{{\left(a{x}^{2}+bx+c\right)}^{3}}+\cdot \cdot \cdot +\frac{{A}_{n}x+{B}_{n}}{{\left(a{x}^{2}+bx+c\right)}^{n}}$

Write the denominators in increasing powers.

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

1. Use variables like $\text{\hspace{0.17em}}A,B,\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ for the constant numerators over linear factors, and linear expressions such as $\text{\hspace{0.17em}}{A}_{1}x+{B}_{1},{A}_{2}x+{B}_{2},\text{\hspace{0.17em}}$ etc., for the numerators of each quadratic factor in the denominator written in increasing powers, such as
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 is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply

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