# 11.4 Partial fractions  (Page 5/7)

 Page 5 / 7

## Decomposing a rational function with a repeated irreducible quadratic factor in the denominator

Decompose the given expression that has a repeated irreducible factor in the denominator.

$\frac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\left({x}^{2}+1\right)}^{2}}$

The factors of the denominator are $\text{\hspace{0.17em}}x,\left({x}^{2}+1\right),\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\left({x}^{2}+1\right)}^{2}.\text{\hspace{0.17em}}$ Recall that, when a factor in the denominator is a quadratic that includes at least two terms, the numerator must be of the linear form $\text{\hspace{0.17em}}Ax+B.\text{\hspace{0.17em}}$ So, let’s begin the decomposition.

$\frac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\left({x}^{2}+1\right)}^{2}}=\frac{A}{x}+\frac{Bx+C}{\left({x}^{2}+1\right)}+\frac{Dx+E}{{\left({x}^{2}+1\right)}^{2}}$

We eliminate the denominators by multiplying each term by $\text{\hspace{0.17em}}x{\left({x}^{2}+1\right)}^{2}.\text{\hspace{0.17em}}$ Thus,

${x}^{4}+{x}^{3}+{x}^{2}-x+1=A{\left({x}^{2}+1\right)}^{2}+\left(Bx+C\right)\left(x\right)\left({x}^{2}+1\right)+\left(Dx+E\right)\left(x\right)$

Expand the right side.

Now we will collect like terms.

${x}^{4}+{x}^{3}+{x}^{2}-x+1=\left(A+B\right){x}^{4}+\left(C\right){x}^{3}+\left(2A+B+D\right){x}^{2}+\left(C+E\right)x+A$

Set up the system of equations matching corresponding coefficients on each side of the equal sign.

We can use substitution from this point. Substitute $\text{\hspace{0.17em}}A=1\text{\hspace{0.17em}}$ into the first equation.

Substitute $\text{\hspace{0.17em}}A=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B=0\text{\hspace{0.17em}}$ into the third equation.

Substitute $\text{\hspace{0.17em}}C=1\text{\hspace{0.17em}}$ into the fourth equation.

Now we have solved for all of the unknowns on the right side of the equal sign. We have $\text{\hspace{0.17em}}A=1,\text{\hspace{0.17em}}$ $B=0,\text{\hspace{0.17em}}$ $C=1,\text{\hspace{0.17em}}$ $D=-1,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}E=-2.\text{\hspace{0.17em}}$ We can write the decomposition as follows:

$\frac{{x}^{4}+{x}^{3}+{x}^{2}-x+1}{x{\left({x}^{2}+1\right)}^{2}}=\frac{1}{x}+\frac{1}{\left({x}^{2}+1\right)}-\frac{x+2}{{\left({x}^{2}+1\right)}^{2}}$

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

$\frac{{x}^{3}-4{x}^{2}+9x-5}{{\left({x}^{2}-2x+3\right)}^{2}}$

$\frac{x-2}{{x}^{2}-2x+3}+\frac{2x+1}{{\left({x}^{2}-2x+3\right)}^{2}}$

Access these online resources for additional instruction and practice with partial fractions.

## Key concepts

• Decompose $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ by writing the partial fractions as $\text{\hspace{0.17em}}\frac{A}{{a}_{1}x+{b}_{1}}+\frac{B}{{a}_{2}x+{b}_{2}}.\text{\hspace{0.17em}}$ Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. See [link] .
• The decomposition of $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ with repeated linear factors must account for the factors of the denominator in increasing powers. See [link] .
• The decomposition of $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)}\text{\hspace{0.17em}}$ with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in $\text{\hspace{0.17em}}\frac{A}{x}+\frac{Bx+C}{\left(a{x}^{2}+bx+c\right)}.\text{\hspace{0.17em}}$ See [link] .
• In the decomposition of $\text{\hspace{0.17em}}\frac{P\left(x\right)}{Q\left(x\right)},\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}Q\left(x\right)\text{\hspace{0.17em}}$ has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as
$\frac{Ax+B}{\left(a{x}^{2}+bx+c\right)}+\frac{{A}_{2}x+{B}_{2}}{{\left(a{x}^{2}+bx+c\right)}^{2}}+\cdots \text{+}\frac{{A}_{n}x+{B}_{n}}{{\left(a{x}^{2}+bx+c\right)}^{n}}.$

## Verbal

Can any quotient of polynomials be decomposed into at least two partial fractions? If so, explain why, and if not, give an example of such a fraction

No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example, $\text{\hspace{0.17em}}\frac{1}{{x}^{2}+1}\text{\hspace{0.17em}}$ cannot be decomposed because the denominator cannot be factored.

Can you explain why a partial fraction decomposition is unique? (Hint: Think about it as a system of equations.)

Can you explain how to verify a partial fraction decomposition graphically?

Graph both sides and ensure they are equal.

You are unsure if you correctly decomposed the partial fraction correctly. Explain how you could double-check your answer.

Once you have a system of equations generated by the partial fraction decomposition, can you explain another method to solve it? For example if you had $\text{\hspace{0.17em}}\frac{7x+13}{3{x}^{2}+8x+15}=\frac{A}{x+1}+\frac{B}{3x+5},\text{\hspace{0.17em}}$ we eventually simplify to $\text{\hspace{0.17em}}7x+13=A\left(3x+5\right)+B\left(x+1\right).\text{\hspace{0.17em}}$ Explain how you could intelligently choose an $\text{\hspace{0.17em}}x$ -value that will eliminate either $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ and solve for $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B.$

If we choose $\text{\hspace{0.17em}}x=-1,\text{\hspace{0.17em}}$ then the B -term disappears, letting us immediately know that $\text{\hspace{0.17em}}A=3.\text{\hspace{0.17em}}$ We could alternatively plug in $\text{\hspace{0.17em}}x=-\frac{5}{3},\text{\hspace{0.17em}}$ giving us a B -value of $\text{\hspace{0.17em}}-2.$

## Algebraic

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.

$\frac{5x+16}{{x}^{2}+10x+24}$

$\frac{3x-79}{{x}^{2}-5x-24}$

$\frac{8}{x+3}-\frac{5}{x-8}$

$\frac{-x-24}{{x}^{2}-2x-24}$

$\frac{10x+47}{{x}^{2}+7x+10}$

$\frac{1}{x+5}+\frac{9}{x+2}$

$\frac{x}{6{x}^{2}+25x+25}$

$\frac{32x-11}{20{x}^{2}-13x+2}$

$\frac{3}{5x-2}+\frac{4}{4x-1}$

$\frac{x+1}{{x}^{2}+7x+10}$

$\frac{5x}{{x}^{2}-9}$

$\frac{5}{2\left(x+3\right)}+\frac{5}{2\left(x-3\right)}$

$\frac{10x}{{x}^{2}-25}$

$\frac{6x}{{x}^{2}-4}$

$\frac{3}{x+2}+\frac{3}{x-2}$

$\frac{2x-3}{{x}^{2}-6x+5}$

$\frac{4x-1}{{x}^{2}-x-6}$

$\frac{9}{5\left(x+2\right)}+\frac{11}{5\left(x-3\right)}$

$\frac{4x+3}{{x}^{2}+8x+15}$

$\frac{3x-1}{{x}^{2}-5x+6}$

$\frac{8}{x-3}-\frac{5}{x-2}$

For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.

$\frac{-5x-19}{{\left(x+4\right)}^{2}}$

$\frac{x}{{\left(x-2\right)}^{2}}$

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

$\frac{7x+14}{{\left(x+3\right)}^{2}}$

$\frac{-24x-27}{{\left(4x+5\right)}^{2}}$

$-\frac{6}{4x+5}+\frac{3}{{\left(4x+5\right)}^{2}}$

$\frac{-24x-27}{{\left(6x-7\right)}^{2}}$

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

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

$\frac{5x+14}{2{x}^{2}+12x+18}$

$\frac{5{x}^{2}+20x+8}{2x{\left(x+1\right)}^{2}}$

$\frac{4}{x}-\frac{3}{2\left(x+1\right)}+\frac{7}{2{\left(x+1\right)}^{2}}$

$\frac{4{x}^{2}+55x+25}{5x{\left(3x+5\right)}^{2}}$

$\frac{54{x}^{3}+127{x}^{2}+80x+16}{2{x}^{2}{\left(3x+2\right)}^{2}}$

$\frac{4}{x}+\frac{2}{{x}^{2}}-\frac{3}{3x+2}+\frac{7}{2{\left(3x+2\right)}^{2}}$

$\frac{{x}^{3}-5{x}^{2}+12x+144}{{x}^{2}\left({x}^{2}+12x+36\right)}$

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.

$\frac{4{x}^{2}+6x+11}{\left(x+2\right)\left({x}^{2}+x+3\right)}$

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

$\frac{4{x}^{2}+9x+23}{\left(x-1\right)\left({x}^{2}+6x+11\right)}$

$\frac{-2{x}^{2}+10x+4}{\left(x-1\right)\left({x}^{2}+3x+8\right)}$

$\frac{4-3x}{{x}^{2}+3x+8}+\frac{1}{x-1}$

$\frac{{x}^{2}+3x+1}{\left(x+1\right)\left({x}^{2}+5x-2\right)}$

$\frac{4{x}^{2}+17x-1}{\left(x+3\right)\left({x}^{2}+6x+1\right)}$

$\frac{2x-1}{{x}^{2}+6x+1}+\frac{2}{x+3}$

$\frac{4{x}^{2}}{\left(x+5\right)\left({x}^{2}+7x-5\right)}$

$\frac{4{x}^{2}+5x+3}{{x}^{3}-1}$

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

$\frac{-5{x}^{2}+18x-4}{{x}^{3}+8}$

$\frac{3{x}^{2}-7x+33}{{x}^{3}+27}$

$\frac{2}{{x}^{2}-3x+9}+\frac{3}{x+3}$

$\frac{{x}^{2}+2x+40}{{x}^{3}-125}$

$\frac{4{x}^{2}+4x+12}{8{x}^{3}-27}$

$-\frac{1}{4{x}^{2}+6x+9}+\frac{1}{2x-3}$

$\frac{-50{x}^{2}+5x-3}{125{x}^{3}-1}$

$\frac{-2{x}^{3}-30{x}^{2}+36x+216}{{x}^{4}+216x}$

$\frac{1}{x}+\frac{1}{x+6}-\frac{4x}{{x}^{2}-6x+36}$

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

$\frac{3{x}^{3}+2{x}^{2}+14x+15}{{\left({x}^{2}+4\right)}^{2}}$

$\frac{{x}^{3}+6{x}^{2}+5x+9}{{\left({x}^{2}+1\right)}^{2}}$

$\frac{x+6}{{x}^{2}+1}+\frac{4x+3}{{\left({x}^{2}+1\right)}^{2}}$

$\frac{{x}^{3}-{x}^{2}+x-1}{{\left({x}^{2}-3\right)}^{2}}$

$\frac{{x}^{2}+5x+5}{{\left(x+2\right)}^{2}}$

$\frac{x+1}{x+2}+\frac{2x+3}{{\left(x+2\right)}^{2}}$

$\frac{{x}^{3}+2{x}^{2}+4x}{{\left({x}^{2}+2x+9\right)}^{2}}$

$\frac{{x}^{2}+25}{{\left({x}^{2}+3x+25\right)}^{2}}$

$\frac{1}{{x}^{2}+3x+25}-\frac{3x}{{\left({x}^{2}+3x+25\right)}^{2}}$

$\frac{2{x}^{3}+11x+7x+70}{{\left(2{x}^{2}+x+14\right)}^{2}}$

$\frac{5x+2}{x{\left({x}^{2}+4\right)}^{2}}$

$\frac{1}{8x}-\frac{x}{8\left({x}^{2}+4\right)}+\frac{10-x}{2{\left({x}^{2}+4\right)}^{2}}$

$\frac{{x}^{4}+{x}^{3}+8{x}^{2}+6x+36}{x{\left({x}^{2}+6\right)}^{2}}$

$\frac{2x-9}{{\left({x}^{2}-x\right)}^{2}}$

$-\frac{16}{x}-\frac{9}{{x}^{2}}+\frac{16}{x-1}-\frac{7}{{\left(x-1\right)}^{2}}$

$\frac{5{x}^{3}-2x+1}{{\left({x}^{2}+2x\right)}^{2}}$

## Extensions

For the following exercises, find the partial fraction expansion.

$\frac{{x}^{2}+4}{{\left(x+1\right)}^{3}}$

$\frac{1}{x+1}-\frac{2}{{\left(x+1\right)}^{2}}+\frac{5}{{\left(x+1\right)}^{3}}$

$\frac{{x}^{3}-4{x}^{2}+5x+4}{{\left(x-2\right)}^{3}}$

For the following exercises, perform the operation and then find the partial fraction decomposition.

$\frac{7}{x+8}+\frac{5}{x-2}-\frac{x-1}{{x}^{2}-6x-16}$

$\frac{5}{x-2}-\frac{3}{10\left(x+2\right)}+\frac{7}{x+8}-\frac{7}{10\left(x-8\right)}$

$\frac{1}{x-4}-\frac{3}{x+6}-\frac{2x+7}{{x}^{2}+2x-24}$

$\frac{2x}{{x}^{2}-16}-\frac{1-2x}{{x}^{2}+6x+8}-\frac{x-5}{{x}^{2}-4x}$

$-\frac{5}{4x}-\frac{5}{2\left(x+2\right)}+\frac{11}{2\left(x+4\right)}+\frac{5}{4\left(x+4\right)}$

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given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey