Partial Fraction Expansion  Quadratic Factors (Single and Repeating)
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Okay, now we look at how to handle quadratic factors when they cannot be factored (in the real number system).
Terms with quadratic factors are of the form \(ax^2+bx+c\) where the highest power on the variable (\(x\) in this case) is two and \(a \neq 0\). We do not have the same requirement on \(b\) and \(c\), so \(ax^2\) and \(ax^2+c\) are also considered quadratic factors. However, the term
\(ax^2+bx\) is NOT considered a quadratic factor, since it can be factored into two simple factors, i.e.
\(ax^2 + bx = x(ax+b)\). In this case, we follow the techniques associated with simple factors.
When we have a quadratic factor, the numerator must be of the form \(Ax+B\). Notice that we now have an \(x\) in the numerator, not just constants. Also, notice that the highest power of \(x\) in the numerator is one less than the highest power in the denominator.
Quadratic Single Factors Example
\(\displaystyle{ \frac{1}{x(x^2+3)} = \frac{A}{x} + \frac{Bx+C}{x^2+3} }\) 

Repeated Factors
If you understand how repeated factors work for linear terms discussed on the previous page, you should be able to anticipate how repeated factors work for quadratic factors.
factor in the denominator 
partial fraction terms  

\(ax^2+bx+c\) 
\(\displaystyle{ \frac{A_1x+B_1}{ax^2+bx+c} }\)  
\((ax^2+bx+c)^2\) 
\(\displaystyle{ \frac{A_1x+B_1}{ax^2+bx+c} + \frac{A_2x+B_2}{(ax^2+bx+c)^2} }\)  
\((ax^2+bx+c)^3\) 
\(\displaystyle{ \frac{A_1x+B_1}{ax^2+bx+c} + \frac{A_2x+B_2}{(ax^2+bx+c)^2} + }\) \(\displaystyle{ + \frac{A_3x+B_3}{(ax^2+bx+c)^3} }\) 
Do you see the pattern? For a denominator with the term
\((ax^2+bx+c)^k\), we would have the factors
\(\displaystyle{ \frac{A_1x+B_1}{ax^2+bx+c} + \frac{A_2x+B_2}{(ax^2+bx+c)^2} + }\)
\(\displaystyle{ \frac{A_3x+B_3}{(ax^2+bx+c)^3} + }\)
\(\displaystyle{ . . . + \frac{A_kx+B_k}{(ax^2+bx+c)^k} }\)
Once these expansions are set up, the steps to find the constants are the same as with linear factors.
Okay, let's work the practice problems.
Practice
Unless otherwise instructed, expand the given fraction using partial fraction expansion. Give your answer in exact terms.
Basic
\(\displaystyle{\frac{5x^2+10+7}{(x^2+2)(x+3)}}\)
Problem Statement
Expand \(\displaystyle{\frac{5x^2+10+7}{(x^2+2)(x+3)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{x^4+3x^3+7x1}{x(x^2+1)^2}}\)
Problem Statement
Expand \(\displaystyle{\frac{x^4+3x^3+7x1}{x(x^2+1)^2}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{6x^2+21x+11}{(x^2+3)(x+5)}}\)
Problem Statement
Expand \(\displaystyle{\frac{6x^2+21x+11}{(x^2+3)(x+5)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{3x^2+5x4}{(x^27)(x+1)}}\)
Problem Statement
Expand \(\displaystyle{\frac{3x^2+5x4}{(x^27)(x+1)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{3x^42x^3+6x^23x+3}{(x^2+2)^2(x+3)}}\)
Problem Statement
Expand \(\displaystyle{\frac{3x^42x^3+6x^23x+3}{(x^2+2)^2(x+3)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{x^2  1}{x(x^2+1)}}\)
Problem Statement
Expand \(\displaystyle{\frac{x^2  1}{x(x^2+1)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{2x^211x+17}{x^37x^2+15x9}}\)
Problem Statement
Expand \(\displaystyle{\frac{2x^211x+17}{x^37x^2+15x9}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{7x^27x+18}{x^3+8}}\)
Problem Statement
Expand \(\displaystyle{\frac{7x^27x+18}{x^3+8}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{x^4+3x^3+6x^2+11x+20}{(x^2+2)^2(x+3)}}\)
Problem Statement
Expand \(\displaystyle{\frac{x^4+3x^3+6x^2+11x+20}{(x^2+2)^2(x+3)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{5x^2+7x+8}{(x+1)(x^2+2x+3)}}\)
Problem Statement
Expand \(\displaystyle{\frac{5x^2+7x+8}{(x+1)(x^2+2x+3)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{2x^2x+4}{x^2+4x}}\)
Problem Statement
Expand \(\displaystyle{\frac{2x^2x+4}{x^2+4x}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{5x1}{(3x^22)(x1)}}\)
Problem Statement
Expand \(\displaystyle{\frac{5x1}{(3x^22)(x1)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{x^2+4x2}{x^3+4x}}\)
Problem Statement
Expand \(\displaystyle{\frac{x^2+4x2}{x^3+4x}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{1}{x(x^2+1)}}\)
Problem Statement
Expand \(\displaystyle{\frac{1}{x(x^2+1)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{10}{(x1)(x^2+9)}}\)
Problem Statement
Expand \(\displaystyle{\frac{10}{(x1)(x^2+9)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{x^2+x+1}{(2x+1)(x^2+1)}}\)
Problem Statement
Expand \(\displaystyle{\frac{x^2+x+1}{(2x+1)(x^2+1)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
There is more going on in this problem than partial fraction expansion, which you can ignore. You will get a chance to do this in calculus.
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\(\displaystyle{\frac{x^4+1}{x(x^2+1)^2}}\)
Problem Statement
Expand \(\displaystyle{\frac{x^4+1}{x(x^2+1)^2}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
video by Krista King Math 

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\(\displaystyle{\frac{2x^2x+9}{x(x^2+9)}}\)
Problem Statement
Expand \(\displaystyle{\frac{2x^2x+9}{x(x^2+9)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
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\(\displaystyle{\frac{5x^28x+5}{(x2)(x^2x+1)}}\)
Problem Statement
Expand \(\displaystyle{\frac{5x^28x+5}{(x2)(x^2x+1)}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
video by MIP4U 

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\(\displaystyle{\frac{3x^2x+4}{x^3+2x^2+6x}}\)
Problem Statement
Expand \(\displaystyle{\frac{3x^2x+4}{x^3+2x^2+6x}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
video by PatrickJMT 

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\(\displaystyle{\frac{8x^3+13x}{(x^2+2)^2}}\)
Problem Statement
Expand \(\displaystyle{\frac{8x^3+13x}{(x^2+2)^2}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
video by MIP4U 

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\(\displaystyle{\frac{10x^2+12x+20}{x^38}}\)
Problem Statement
Expand \(\displaystyle{\frac{10x^2+12x+20}{x^38}}\) using partial fraction expansion. Give your answer in exact terms.
Solution
video by Khan Academy 

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Intermediate
\(\displaystyle{ \frac{4x^2}{4x^4+1} }\)
Problem Statement 

Expand \(\displaystyle{ \frac{4x^2}{4x^4+1} }\) using partial fraction expansion. Give your answer in exact terms.
Hint 

To factor \( 4x^4+1 \), start by adding and subtracting \( 4x^2 \).
Problem Statement 

Expand \(\displaystyle{ \frac{4x^2}{4x^4+1} }\) using partial fraction expansion. Give your answer in exact terms.
Final Answer 

\(\displaystyle{ \frac{4x^2}{4x^4+1} }\) \(\displaystyle{ = \frac{x}{2x^22x+1}  }\) \(\displaystyle{ \frac{x}{2x^2+2x+1} }\)
Problem Statement
Expand \(\displaystyle{ \frac{4x^2}{4x^4+1} }\) using partial fraction expansion. Give your answer in exact terms.
Hint
To factor \( 4x^4+1 \), start by adding and subtracting \( 4x^2 \).
Solution
This is part of a improper integral problem. You do not need to know how to work improper integrals to understand how to solve this problem.
video by Michael Penn 

Final Answer
\(\displaystyle{ \frac{4x^2}{4x^4+1} }\) \(\displaystyle{ = \frac{x}{2x^22x+1}  }\) \(\displaystyle{ \frac{x}{2x^2+2x+1} }\)
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