# College Algebra : Partial Fractions

## Example Questions

### Example Question #1 : Partial Fractions

Determine the partial fraction decomposition of

Explanation:

First we need to factor the denominator.

Now we can rewrite it as such

Now we need to get a common denominator.

Now we set up an equation to figure out  and .

To solve for , we are going to set .

To find , we need to set

Thus the answer is:

### Example Question #2 : Partial Fractions

Determine the partial fraction decomposition of

Explanation:

Now we can rewrite it as such

Now we need to get a common denominator.

Now we set up an equation to figure out  and .

To solve for , we are going to set .

To find , we need to set

Thus the answer is:

### Example Question #3 : Partial Fractions

Explanation:

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.

.

### Example Question #4 : Partial Fractions

Subtract:

Explanation:

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 #5 : Partial Fractions

Explanation:

In order to add the numerators of the fractions, we need to find the least common denominator.

The least common denominator is:

We will need to multiply the numerator and denominator by  to match the denominators of both fractions.

Simplify the fraction.

Combine the two fractions.

### Example Question #6 : Partial Fractions

Explanation:

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,

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.

Notice that  and  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.

Similairily, the procedure for an algebraic expression containing unknowns parallels this idea,

Now we can add the numerators directly since we now have both terms expressed with a common denominator, .

### Example Question #1 : Partial Fractions

Write the rational function as a sum of terms with linear denominators using a partial fraction decomposition.

Not enough information to find , or

Explanation:

(1.a)

1) First factor the denominator as much as possible; characterize the denominator and write the appropriate expansion:

The denominator is a product of linear terms, so the partial fraction expansion will have the form,

(1.b)

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,

(2.a)

Distribute and multiply the 's,

(2.b)

3) Find the constants A, B, and C.

To find , and  simply expand and collect like terms (there are , and constant terms) then compare to the original numerator .

For the terms with  we must have,

So for  we have,

(3.a)

For the -terms we have,

for  we have,

(3.b)

For the constant term we have,

(3.c)

Right away we can read off the solution for  from equation (3.c) Substitute    into (3.a) and (3.b)

4) Solve for the remaining unknown constants B and C,

The system:

(4.a)

(4.b)

In order to remove the fraction it would be convenient to solve this after multiplying both equations by

(4.c)

(4.d)

In order to make even more simple, multiply equation (4.c) by  and solve for  in terms of  as follows,

Substitute into (4.d),

Now we can use this value for  to find that

Finally, plug in the values for , and  we obtained into equation (1.b).

### Example Question #1 : Partial Fractions

What is the partial fraction decomposition of

Explanation:

Factor the denominator:

Multiply both sides of the equation by

Let

Let :

### Example Question #1 : Partial Fractions

What is the partial fraction decomposition of the following:

Explanation:

Factor the denominator:

Multiply both sides of the equation by

Let :

Let :

### Example Question #101 : Polynomial Functions

Find A and B in the expression