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Example Questions
Example Question #1 : Partial Fractions
Determine the partial fraction decomposition ofÂ
First we need to factor the denominator.
Â
Now we can rewrite it as such
Â
Â
Now we need to get a common denominator.
Â
Â
Now we set up an equation to figure out  andÂ
.
Â
To solve for , we are going to setÂ
.
To find , we need to setÂ
Â
Thus the answer is:
Â
Â
Example Question #2 : Partial Fractions
Determine the partial fraction decomposition ofÂ
 Now we can rewrite it as such
Â
Â
Now we need to get a common denominator.
Â
Â
Now we set up an equation to figure out  andÂ
.
Â
To solve for , we are going to setÂ
.
To find , we need to setÂ
Â
Thus the answer is:
Â
Example Question #2 : Adding And Subtracting Fractions
Add:
To add rational expressions, you must find the common denominator. In this case, it's .
Next, you must change the numerators to offset the new denominator.
becomes
and
becomes
.
Now you can combine the numerators: .
Put that over the denomiator and see if you can simplify/factor further. In this case, you can't.
Therefore, your final answer is:
.
Example Question #1 : Adding And Subtracting Fractions
Subtract:
To subtract rational expressions, you must first find the common denominator, which in this case is . That means we only have to adjust the first fraction since the second fraction has that denominator already.
Therefore, the first fraction now looks like:
.
Now that the denominators are the same, combine numerators:
.
Now, put that over the denominator and see if you can simplify any further.
In this case, you can't, so your final answer is: .
Example Question #3 : Partial Fractions
Add: Â
In order to add the numerators of the fractions, we need to find the least common denominator.
The least common denominator is: Â
We will need to multiply the numerator and denominator by  to match the denominators of both fractions.
Simplify the fraction.
Combine the two fractions.
The answer is: Â
Example Question #3 : Partial Fractions
Add:
Â
Â
Â
The rules for adding fractions containing unknowns  are the same as for fractions containing explicit numbers, so you can guide yourself by recalling how you would proceed adding fractions such as, Â
As you know you need to write them with a common denominator. In this case the least common denominator is . So simply multiply the numerator and denominator of each fraction by the denominator of the other fraciton.
Â
Notice that  andÂ
 are equal to one, this ensures that we are not changing the value of the fractions, we are changing only the representation of the value.Â
Â
Similairily, the procedure for an algebraic expression containing unknowns parallels this idea,Â
Â
Â
Now we can add the numerators directly since we now have both terms expressed with a common denominator, .
Â
Â
Example Question #4 : Partial Fractions
Write the rational function as a sum of terms with linear denominators using a partial fraction decomposition.Â
Â
Â
Not enough information to find ,Â
, orÂ
Â
                                (1.a)
Â
1) First factor the denominator as much as possible; characterize the denominator and write the appropriate expansion:
Â
Â
 The denominator is a product of linear terms, so the partial fraction expansion will have the form,Â
Â
                    (1.b)
Â
2) Write a system of 3-equations and 3-unknowns in order to determine A,B,and C in the partial fraction expansion (equation 1.b).
If we were to take equation (1.b) and add each fraction under a common denominator , the numerator would have the form,Â
Â
        (2.a)
Â
Distribute and multiply the 's,  Â
      (2.b)
Â
Â
3) Find the constants A, B, and C.Â
To find ,Â
, andÂ
 simply expand and collect like terms (there areÂ
,Â
, and constant terms) then compare to the original numerator
.
 For the terms with  we must have,Â
Â
So for  we have,
                                       (3.a)
Â
For the -terms we have,Â
for  we have,Â
                                 (3.b)
Â
For the constant term we have,Â
                                         (3.c)
Â
Right away we can read off the solution for  from equation (3.c) Substitute Â
  into (3.a) and (3.b) Â
Â
4) Solve for the remaining unknown constants B and C,Â
The system:
                                    (4.a)
                                (4.b)
Â
In order to remove the fraction it would be convenient to solve this after multiplying both equations by :Â
Â
                                  (4.c)
                            (4.d)
Â
 In order to make even more simple, multiply equation (4.c) by  and solve for
 in terms ofÂ
 as follows, Â
Â
Â
Substitute into (4.d),
Â
Â
Now we can use this value for  to find thatÂ
.Â
Â
Finally, plug in the values for ,Â
, andÂ
 we obtained into equation (1.b).
Â
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Â
Â
                    Â
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Example Question #4 : Partial Fractions
What is the partial fraction decomposition of
Â
Factor the denominator:Â
Multiply both sides of the equation byÂ
Let :Â
Â
Let :
Â
Â
Example Question #5 : Partial Fractions
What is the partial fraction decomposition of the following:
Factor the denominator:Â
Multiply both sides of the equation byÂ
Let :
Let :
Example Question #5 : Partial Fractions
Find A and B in the expression
Take the expression  and multiply both sides byÂ
. This transforms the equation into the following:
Â
To solve for A we want to get rid of the B. So we set x equal to 1:
Similarly, we can get rid of A by setting x equal to -2:
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