Systems of Equations and Inequalities

# Partial Fractions

### Learning Objectives

In this section, you will:

- Decompose , where has only nonrepeated linear factors.
- Decompose , where has repeated linear factors.
- Decompose , where has a nonrepeated irreducible quadratic factor.
- Decompose , where has a repeated irreducible quadratic factor.

Earlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilized—the decomposition of rational expressions.

Fractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in this section will help simplify the concept of a rational expression.

### DecomposingWhere *Q(x)* Has Only Nonrepeated Linear Factors

Recall the algebra regarding adding and subtracting rational expressions. These operations depend on finding a common denominator so that we can write the sum or difference as a single, simplified rational expression. In this section, we will look at partial fraction decomposition, which is the undoing of the procedure to add or subtract rational expressions. In other words, it is a return from the single simplified rational expression to the original expressions, called the partial fractions.

For example, suppose we add the following fractions:

We would first need to find a common denominator,

Next, we would write each expression with this common denominator and find the sum of the terms.

Partial fraction decomposition is the reverse of this procedure. We would start with the solution and rewrite (decompose) it as the sum of two fractions.

We will investigate rational expressions with linear factors and quadratic factors in the denominator where the degree of the numerator is less than the degree of the denominator. Regardless of the type of expression we are decomposing, the first and most important thing to do is factor the denominator.

When the denominator of the simplified expression contains distinct linear factors, it is likely that each of the original rational expressions, which were added or subtracted, had one of the linear factors as the denominator. In other words, using the example above, the factors ofarethe denominators of the decomposed rational expression. So we will rewrite the simplified form as the sum of individual fractions and use a variable for each numerator. Then, we will solve for each numerator using one of several methods available for partial fraction decomposition.

### Partial Fraction Decomposition ofHas Nonrepeated Linear Factors

The partial fraction decomposition ofwhenhas nonrepeated linear factors and the degree ofis less than the degree ofis

**Given a rational expression with distinct linear factors in the denominator, decompose it.**

- Use a variable for the original numerators, usuallyordepending on the number of factors, placing each variable over a single factor. For the purpose of this definition, we usefor each numerator
- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.
- Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.

### Decomposing a Rational Function with Distinct Linear Factors

Decompose the given rational expression with distinct linear factors.

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We will separate the denominator factors and give each numerator a symbolic label, likeor

Multiply both sides of the equation by the common denominator to eliminate the fractions:

The resulting equation is

Expand the right side of the equation and collect like terms.

Set up a system of equations associating corresponding coefficients.

Add the two equations and solve for

Substituteinto one of the original equations in the system.

Thus, the partial fraction decomposition is

Another method to use to solve fororis by considering the equation that resulted from eliminating the fractions and substituting a value forthat will make either the *A*– or *B*-term equal 0. If we letthe

Next, either substituteinto the equation and solve foror make the *B*-term 0 by substitutinginto the equation.

We obtain the same values forandusing either method, so the decompositions are the same using either method.

Although this method is not seen very often in textbooks, we present it here as an alternative that may make some partial fraction decompositions easier. It is known as the Heaviside method, named after Charles Heaviside, a pioneer in the study of electronics.

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### Try It

Find the partial fraction decomposition of the following expression.

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### DecomposingWhere *Q(x)* Has Repeated Linear Factors

Some fractions we may come across are special cases that we can decompose into partial fractions with repeated linear factors. We must remember that we account for repeated factors by writing each factor in increasing powers.

### Partial Fraction Decomposition ofHas Repeated Linear Factors

The partial fraction decomposition ofwhenhas a repeated linear factor occurringtimes and the degree ofis less than the degree ofis

Write the denominator powers in increasing order.

**Given a rational expression with repeated linear factors, decompose it.**

- Use a variable likeorfor the numerators and account for increasing powers of the denominators.
- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.
- Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.

### Decomposing with Repeated Linear Factors

Decompose the given rational expression with repeated linear factors.

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The denominator factors areTo allow for the repeated factor ofthe decomposition will include three denominators:andThus,

Next, we multiply both sides by the common denominator.

On the right side of the equation, we expand and collect like terms.

Next, we compare the coefficients of both sides. This will give the system of equations in three variables:

Solving for, we have

Substituteinto equation (1).

Then, to solve forsubstitute the values forandinto equation (2).

Thus,

### Try It

Find the partial fraction decomposition of the expression with repeated linear factors.

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### DecomposingWhere *Q(x)* Has a Nonrepeated Irreducible Quadratic Factor

So far, we have performed partial fraction decomposition with expressions that have had linear factors in the denominator, and we applied numeratorsorrepresenting constants. Now we will look at an example where one of the factors in the denominator is a quadratic expression that does not factor. This is referred to as an irreducible quadratic factor. In cases like this, we use a linear numerator such asetc.

### Decomposition ofHas a Nonrepeated Irreducible Quadratic Factor

The partial fraction decomposition ofsuch thathas a nonrepeated irreducible quadratic factor and the degree ofis less than the degree ofis written as

The decomposition may contain more rational expressions if there are linear factors. Each linear factor will have a different constant numerator:and so on.

**Given a rational expression where the factors of the denominator are distinct, irreducible quadratic factors, decompose it.**

- Use variables such asorfor the constant numerators over linear factors, and linear expressions such asetc., for the numerators of each quadratic factor in the denominator.
- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.
- Set coefficients of like terms from the left side of the equation equal to those on the right side to create a system of equations to solve for the numerators.

### DecomposingWhen *Q(x)* Contains a Nonrepeated Irreducible Quadratic Factor

Find a partial fraction decomposition of the given expression.

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We have one linear factor and one irreducible quadratic factor in the denominator, so one numerator will be a constant and the other numerator will be a linear expression. Thus,

We follow the same steps as in previous problems. First, clear the fractions by multiplying both sides of the equation by the common denominator.

Notice we could easily solve forby choosing a value forthat will make theterm equal 0. Letand substitute it into the equation.

Now that we know the value ofsubstitute it back into the equation. Then expand the right side and collect like terms.

Setting the coefficients of terms on the right side equal to the coefficients of terms on the left side gives the system of equations.

Solve forusing equation (1) and solve forusing equation (3).

Thus, the partial fraction decomposition of the expression is

**Could we have just set up a system of equations to solve (Figure)?**

*Yes, we could have solved it by setting up a system of equations without solving forfirst. The expansion on the right would be:*

*So the system of equations would be:*

### Try It

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

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### DecomposingWhen *Q(x)* Has a Repeated Irreducible Quadratic Factor

Now that we can decompose a simplified rational expression with an irreducible quadratic factor, we will learn how to do partial fraction decomposition when the simplified rational expression has repeated irreducible quadratic factors. The decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented in increasing powers.

### Decomposition ofWhen *Q(x)* Has a Repeated Irreducible Quadratic Factor

The partial fraction decomposition ofwhenhas a repeated irreducible quadratic factor and the degree ofis less than the degree ofis

Write the denominators in increasing powers.

**Given a rational expression that has a repeated irreducible factor, decompose it.**

- Use variables likeorfor the constant numerators over linear factors, and linear expressions such asetc., for the numerators of each quadratic factor in the denominator written in increasing powers, such as
- Multiply both sides of the equation by the common denominator to eliminate fractions.
- Expand the right side of the equation and collect like terms.

### 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.

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The factors of the denominator areandRecall that, when a factor in the denominator is a quadratic that includes at least two terms, the numerator must be of the linear formSo, let’s begin the decomposition.

We eliminate the denominators by multiplying each term byThus,

Expand the right side.

Now we will collect like terms.

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

We can use substitution from this point. Substituteinto the first equation.

Substituteandinto the third equation.

Substituteinto the fourth equation.

Now we have solved for all of the unknowns on the right side of the equal sign. We have[latex]B=0,\,[/latex][latex]D=-1,\,[/latex]andWe can write the decomposition as follows:

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

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

### Key Concepts

- Decomposeby writing the partial fractions asSolve 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 (Figure).
- The decomposition ofwith repeated linear factors must account for the factors of the denominator in increasing powers. See (Figure).
- The decomposition ofwith a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as inSee (Figure).
- In the decomposition ofwherehas a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as
See (Figure).

### Section Exercises

#### 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

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No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example,cannot be decomposed because the denominator cannot be factored.

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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?

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Graph both sides and ensure they are equal.

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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 hadwe eventually simplify toExplain how you could intelligently choose an-value that will eliminate eitherorand solve forand

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If we choosethen the *B*-term disappears, letting us immediately know thatWe could alternatively plug ingiving us a *B*-value of

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#### Algebraic

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

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

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

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

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#### Extensions

For the following exercises, find the partial fraction expansion.

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

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### Glossary

- partial fractions
- the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression

- partial fraction decomposition
- the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions