# Systems of Equations and Inequalities

## Partial Fractions


### Learning Objectives

1. Find the least common denominator of rational expressions (IA 7.2.3)
2. Solve a system of equations by elimination (IA 4.1.4)

### Objective 1: Find the least common denominator of rational expressions (IA 7.2.3)

A rational expression is an expression of the form  where p and q are polynomials and  .
  are examples of rational expressions.

### Practice Makes Perfect


Find the least common denominator of the following rationals:



### Objective 2: Solve a system of equations by elimination (IA 4.1.4)


### Partial Fraction Decomposition

When we add rational expressions with unlike denominators such as  and , we first need to find the LCD, then rewrite each fraction with the common denominator, and finally add the two numerators.

We want to do the opposite now.

Given a rational expression like,  we would like to rewrite it as an addition of two simpler rational expressions  and  . Our goal is to find the values of A and B such that 



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.

### Decomposing   Where 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 fraction.
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 of 
 are 
 the 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.

### Decomposing    Where 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.

### Decomposing     Where 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 numerators 
   or 
   representing 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 as 
   etc.

### Decomposing      When 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.

### Key Concepts

1. Decompose  
  
 by writing the partial fractions as  
  
 Solve 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 .
2. The decomposition of  
  
 with repeated linear factors must account for the factors of the denominator in increasing powers. See .
3. The decomposition of  
  
 with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in  
  
 See .
4. In the decomposition of  
  
 where  
  
 has 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 .
    

### Section Exercises


### Verbal


### Algebraic

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

### Extensions

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