Partial Fractions

Fractions can get very complex, and it gets even harder when we add variables in the denominator. In the methods that we will explain in this note, we will provide a way to decompose or expand a rational function with a complex denominator into a sum of smaller rational functions with a simpler denominator.

Warning

This is a complex topic in Algebra.
We recommend that you are familiar with the following topics.

• Solving Systems of Linear Equations
• Performing Division of Polynomials
• Finding Roots with Rational Root Theorem

Some sections of this text may also require you to be familiar with the following topics:

• Matrix and Matrix Operations
• Solving Systems with Gaussian Elimination
• Solving Systems with Matrix Inverses

Work in Progress!

This note is under construction!

Some sections of this note may still be broken or left incorrectly rendered.

What is a Partial Fraction Decomposition?

Objectives

• Grasp the definition of a partial fraction decomposition.
• Set up a partial fraction decomposition.
• Use long division before setting up a partial fraction decomposition.

Suppose that we have the expression:

We can simplify it by multiplying each term by a least common denominator.
Remember that by doing this, we are simply multiplying by , thus both expressions are equivalent.

Then, we get the identity:

But consider this idea: instead of starting with the individual fractional parts, what if we instead start with a larger fraction and find its individual parts instead?

This is the goal of a partial fraction decomposition.
By decomposing fractions, we can write them as a sum of smaller fractional parts instead.

Definition
Partial Fraction Decomposition

A partial fraction decomposition, or a partial fraction expansion of is a process of writing fractions as a sum of smaller fractional parts.

Suppose that .
If has the irreducible factors , then there are polynomials and such that it can be written as:

where the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.

The definition might seem intimidating but what it says in reality is that if we have a rational function, for example:

We can write it as a sum of the smaller rational functions, where the denominators of each term are factors of the original fraction.

While we do not have methods for finding the numerator of each term, we can should begin by getting a good feel for the definition of a partial fraction expansion.

Worked Example #1
Setting up a Partial Fraction Decomposition

Set up the partial fraction decomposition for .

Solution

We can begin by writing as a product of its linear factors .

Then by definition, its partial fraction decomposition can be expressed as:

A partial fraction expansion can also have two or more terms. As long as we can break our denominator into products of irreducible factors, we can have as many terms as we like.

Worked Example #2
Setting up a Partial Fraction Decomposition with Three or More Factors

Set up the partial fraction decomposition for .

Solution

Then, we can construct a partial fraction decomposition as:

However, we can decompose the second fraction even further.
We can do this until each term has an irreducible denominator.
Therefore:

Now that we know some examples how to set up a partial fraction decomposition, we must first make an important note:

If we have a rational function in the form:

We must first ensure that the degree of the numerator is smaller than the degree of the denominator .

If that is not the case, then we should perform long division first before setting up a partial fraction decomposition.

Worked Example #3
Dividing before Setting up a Partial Fraction Decomposition

Set up a partial fraction decomposition for .

Solution

We can now rewrite our fraction as:

Then, we set up the fractional part of our answer as a partial fraction expansion.
Factoring gives us . Therefore:

Partial Fractions with Non-repeated Linear Factors

Now that we’ve built a foundation for setting up these decompositions, let us figure out a method to actually solve them.

Objectives

• Decompose a rational function with non-repeated linear factors.
• Find a partial fraction expansion using systems of equations.
• Find a partial fraction expansion using substitution.

The simplest case of a partial fraction decomposition is when a fraction is made up of linear factors. To see this in practice, let us say that we want to decompose where:

However, we assume that is made up of linear factors in the form .
In this case, we can decompose as:

Take note here that the degree of the numerator is always less than the degree of the denominator.
Therefore, the numerators must be a constant.

Definition
Partial Fraction Expansions with Non-repeated Linear Factors

Given that:

If contains a linear factor in the form , then it can be decomposed as:

where are constants.

Worked Example #4
Setting up a Partial Fraction Expansion with Non-repeated Linear Factors

Set up the partial fraction expansion for .

Solution

We can then decompose it as a sum of smaller rational functions.
Therefore:

Finding Partial Fractions using Systems of Equations

Now, let us get on to the methods itself on how can we solve for the constants itself.
First, as an example, let us try to solve an example from the previous section.

By multiplying everything by a common denominator, we arrive at a polynomial identity:

We can then expand each polynomial.

Then rewrite it so that we get an expression for each coefficient.

We can see here that the coefficient on the left corresponds to the on the right.
At the same time, the constant term on the left , corresponds to the constant term on the right .
Using this information, we can construct a system of equations.

Solving this system of equations gives us the values and .
Therefore, by substituting it back to our decomposition, we can say that:

Worked Example #5
Finding Partial Fraction Expansions consisting of
Two Linear Factors using Linear Systems

Find the partial fraction expansion of

Solution

Then, we set up a partial fraction decomposition.

Multiplying everything by a common denominator, we get the polynomial identity:

Expanding and rewriting it as a standard form polynomial, we get:

Then we set up our system of linear equations.

Solving this system of equations, we get the values:

Substituting it back to our original decomposition, we get the values:

Oftentimes, we might deal with partial fractions with multiple linear factors.

At this case, while the method of systems of equations might get difficult, the way to solve them is still similar on two-variable linear systems.

Worked Example #6
Finding Partial Fraction Expansions consisting of
Multiple Linear Factors using Linear Systems

Find the partial fraction expansion of .

Solution

Therefore, we can write the original expression as:

We can rewrite the fractional part as a partial fraction expansion.
Setting it up, we get:

Multiplying both sides by the common denominator , we get:

To write our system of equations, we can then rewrite it into standard polynomial form so that we can begin equating coefficients.

By equating coefficients, we get the following system of equations:

Solving the following system of equations gives us the constants:

Therefore, we get the constants: .
Substituting it back to our original decomposition, we get.

Substituting this partial fraction expansion into our full answer, we get the complete decomposition:

Finding Partial Fractions using Substitution

Equating coefficients and using systems of linear equations are not the only way to find partial fraction expansions, we also have an another method that does not utilize systems of equations.

Referencing on the previous example, upon multiplying the common denominator, we get the polynomial identity:

Since this is an identity, these expressions will still be equal regardless of the value of .

By substituting certain values, we can solve coefficients by making other factors equal to zero.

• Let us try this notion by substituting :

• In the same way, by substituting , we can solve for .

Worked Example #7
Finding Partial Fraction Expansions consisting of
Two Linear Factors using Substitution

Find the partial fraction expansion of

on the numerator and the denominator.

Then we set up our partial fraction decomposition.

Multiplying both sides by our common denominator, we get:

Using substitution, we can solve for our coefficients.

• If , then:

• If , then:

Substituting the coefficients in our original decomposition, we get:

Worked Example #8
Finding Partial Fraction Expansions consisting of
Multiple Linear Factors using Substitution

Find the partial fraction decomposition of .

Solution

Then, we can now set up its partial fraction decomposition.

Multiplying both sides by their common denominator, we get the polynomial identity:

Then, we can now perform substitution.

• When :

• When :

• When :

• When :

Finally, we can substitute these coefficients on our original decomposition.

We can write our answer a little neater by using the property

Partial Fractions with Non-repeated Irreducible Non-Linear Factors

Now that we have the required skills to tackle partial fractions, let us explore some of the forms that they might manifest. For instance, we’ve been decomposing fractions that can be factored as products of linear factors, but what if one of the factors is an irreducible quadratic expression? We will tackle that along with generalizing the partial fraction expansion for all polynomials.

Objectives

• Decompose a rational function with non-repeated irreducible quadratic factors.
• Find a partial fraction expansion of a rational function with quadratic factors
• Generalize the partial fraction expansion for any irreducible polynomial factor of degree .
• Decompose a rational function with a non-repeated irreducible polynomial factor of degree .

We now know how to find a partial fraction expansion for any polynomial denominator that contains only linear factors. But what about if it contains an irreducible non-linear factor?

Suppose that we want to find the partial fraction expansion of the function:

By looking into its factors, we can decompose it as:

Focusing our attention on the first fraction, the degree of the numerator must be less than the degree of the denominator . Therefore, is just a constant value.

For the second fraction, the same thing can be said: the degree of the numerator must be lower than the degree of the denominator .

Therefore, must be linear, and so it must contain a linear term and a constant term.
We can write it as:

Worked Example #9
Setting up a Partial Fraction Expansion with an Irreducible Quadratic Factor

Set up a partial fraction decomposition of .

Solution

We can then perform partial fraction decomposition on its fractional part.

By substituting it on our expression from before, we get the full decomposition:

Now, using the techniques we have so far, let us attempt to find a partial fraction expansion that involves an irreducible quadratic factor.

Worked Example #10
Finding a Partial Fraction Expansion with an Irreducible Quadratic Factor

Find the partial fraction decomposition of .

Solution

We can now set up its partial fraction decomposition.

We can rewrite this by multiplying both sides by its common denominator.
Doing so leads us to the polynomial identity:

We have a lot of ways to solve for the coefficients.
For this one, we can use substitution.

• Substituting :

• Substituting :

Since , we can then substitute it back to our equation

• Substituting :

Substituting our previous coefficients, we can solve for :

Substituting our coefficients to our original decomposition, we get:

Analysis

After rewriting our decomposition into the polynomial identity:

We can then expand the expression, then rewrite it in standard form.

We can then write this as a system of equations by equating coefficients:

Solving these system of equations will give us the coefficients to our original decomposition.

Now that we know how to decompose fractions with irreducible quadratic factors, we can ask how can we extend this notion to an irreducible polynomial factor of any degree.

For instance, let us find a partial fraction expansion of the rational function:

We can decompose this by starting with the assumption that the numerator is some polynomial function of .