Adding and Subtracting Rational Expressions

 

Learning Objective(s)

·         Add and subtract rational expressions and simplify.

 

Introduction

 

In beginning math, students usually learn how to add and subtract whole numbers before they're taught multiplication and division. However, with fractions and rational expressions, multiplication and division are sometimes taught first because these operations are easier to perform than addition and subtraction. Addition and subtraction of rational expressions are trickier than multiplication because, like with numeric fractions, the process involves finding common denominators. By working carefully and writing down the steps along the way, we can keep track of all of the numbers and variables and perform the operations accurately.

 

Adding and Subtracting Rational Expressions with Like Denominators

 

We follow the same process for adding rational expressions that we do to combine numeric fractions. To add fractions with like denominators, we add the numerators and keep the same denominator. After adding, we express the fraction in simplest terms:

 

 

 

We follow the same process to add rational expressions with like denominators, but we also have to describe the domain, the set of all possible values for the variables. The excluded values of the domain are any values of the variable(s) that result in any denominator being equal to 0.

 

Let's try one:

 

Example

Problem

Add, simplify, and state the domain of

 

 

x + 4 = 0

 

x = -4

 

Determine the excluded values by setting the denominator equal to 0 and solving for x.

 

 

Since the denominators are the same, add the numerators.

 

 

 

 

Factor the numerator

 

 

Rewrite the common factor as multiplication by 1

Answer

 

2x, x -4

 

 

 

To subtract rational expressions with like denominators, we follow the same process we use to subtract fractions with like denominators. The process is just like addition of rational expressions, except that we subtract.

 

Example

Problem

Subtract, simplify, and state the domain of

 

 

x + 6 = 0

x = -6

-6 is an excluded value.

 

Determine the excluded values by setting the denominator equal to 0 and solving for x.

 

 

 

Subtract the second numerator from the first and keep the denominator the same.

 

 

 

Be careful to distribute the negative to both terms of the second numerator.

 

 

 

Note that -7 – (-8) = -7 + 8 = 1

 

This is the final answer because this rational expression cannot be simplified.

Answer

 , x

 

 

 

 

Simplify

 

A)  ,  x5

B) x + 5, x5

C) x – 5, x5

D) x + 5, x -5 or 5

 

Show/Hide Answer

A) Incorrect. You performed the subtraction properly and found the correct excluded value, but this rational expression can be simplified because the numerator and denominator have a common factor of (x – 5). The correct answer is x + 5, x5.

 

B) Correct. Since there is a common denominator, subtract the numerators to get . The numerator can be factored and a common factor of (x – 5) is present in the numerator and denominator. . The value 5 is excluded because this value makes the denominator equal to 0.

 

C) Incorrect. The common factor present in the numerator and denominator is x – 5, not x + 5. . The correct answer is x + 5, x5.

 

D) Incorrect. The only excluded value is x = 5 because this value of x makes the denominator 0. -5 is not an excluded value because -5 – 5 = -10, which is an acceptable value of the denominator. -5 will make the numerator equal to 0, but that is not a problem. (0 in the numerator makes the value of the rational expression also equal  to 0.) The correct answer is x + 5, x5.

 

 

Adding and Subtracting Rational Expressions with Unlike Denominators

 

Before adding and subtracting rational expressions with unlike denominators, we need to find a common denominator. This process is once again similar to the one used for adding and subtracting numeric fractions with unlike denominators. Let’s look at a numeric example to start.

 

 

Since the denominators are 6, 10, and 4, we want to find the least common denominator and express each fraction with this denominator before adding. (By the way, you can add fractions by finding any common denominator; it does not have to be the least. We focus on using the least because then there is less simplifying to do. But either way works.)

 

Finding the least common denominator is the same as finding the least common multiple of 4, 6 and 10. There are a couple of ways to do this.  The first is to list the multiples of each number and determine which multiples they have in common. The smallest of these numbers will be the least common denominator.

 

Number

Multiples

4

8

12

16

20

24

28

32

36

40

44

48

52

56

60

64

 

6

12

18

24

30

36

42

48

54

60

66

72

 

 

 

 

 

10

20

30

40

50

60

 

 

 

 

 

 

 

 

 

 

 

 

 

The other method is to use prime factorization, the process of finding the prime number factors of a number. This is how the method works with numbers:

 

Example

Problem

Use prime factorization to find the least common multiple of 6, 10, and 4

 

 

6 = 3 • 2

10 = 5 • 2

4 = 2 • 2

 

First, find the prime factors of each denominator

 

 

3 • 5 • 2 • 2

 

 

Multiply all of the prime factors. Use each number the maximum number of times it appears in a single factorization.

 

In this case, 2 is used twice because it appears twice in the prime factorization of 4.

Answer

60

 

 

 

 

We found the same least common multiple with both methods. Prime factorization was faster though, because we didn't have to make a chart full of multiples.

 

Let's move on. Now that we have found the least common multiple, we'll use that number as the least common denominator of our fractions. We'll multiply each fraction by the fractional form of 1 that will produce a denominator of 60:

 

 

 

 

Now that we have like denominators, we can easily add the fractions:

 

 

We can find least common denominators for rational expressions, too, and use them to allow us to add rational expressions with unlike denominators:

 

Example

 

Problem

Add

 

 

15m2n3 = 0

m = 0 or n = 0

 

21mn2 = 0

m = 0 or n = 0

 

Find excluded values

 

15m2n3 = 3 • 5 • m • m • n • n • n

 

21mn2 = 3 • 7 • m • n • n

 

 

Find the prime factors of each denominator

 

 

3 • 5 • 7 • m • m • n • n • n

 

 105m2n3

 

 

Find the least common multiple. 3 appears exactly once in both of the expressions, so it will appear once in the least common denominator. Both 5 and 7 appear at most once. For the variables, the most m appears is twice, and the most n appears is three times.

 

15m2n3 = 3 • 5 • m • m • n • n • n

105m2n3 = 3 • 5 • 7 • m • m • n • n • n

 

 =

 

 

21mn2 = 3 • 7 • m • n • n

105m2n3 = 3 • 5 • 7 • m • m • n • n • n

 

 =  

 

 

Rewrite the rational expressions to each have a denominator of 105m2n3

 

Compare the prime factors of each denominator and the common denominator; to get the common denominator, multiply the original denominator by whatever factors are missing.

 

 

 

 

 

Add the numerators and keep the denominator the same.

 

 

 

 

 

 

Simplify by finding common factors in the numerator and denominator.

Answer

 

, m 0 and n 0.

 

 

 

 

 

That took a while, but we got through it. Adding rational expressions can be a lengthy process, but if we take it one step at a time, we'll get there.

 

Ready to try subtracting rational expressions? We'll use the same basic technique of finding the least common denominator and rewriting each rational expression to have that denominator:

 

Example

Problem

Subtract  and state any excluded values

 

 

 

t + 1 = 0

t = -1

 

t2 – t – 2 = (t – 2)(t + 1) = 0

t  = -1, 2

 

 

Determine the excluded values by setting each denominator equal to 0 and solving

 

t + 1 = t + 1

 

t2 – t – 2 = (t -2)(t + 1)

 

 

Find a least common multiple by factoring each denominator.

 

Since t + 1 is already a factor of

t2 – t – 2, the least common denominator is (t + 1)(t -2).

 

 

 

 

 

Multiply the first expression by the equivalent of 1 to give it the common denominator.

 

Then rewrite the rational expressions with the common denominator.

 

 

 

 

 

 

 

Subtract the numerators and simplify.

 

 

 

 

The numerator and denominator have a common factor of t – 2, and so the rational expression can be simplified.

Answer

, t -1 or 2

 

 

 

 

So far all the rational expressions we've added and subtracted have shared some factors. What happens when they don't have factors in common?

 

Example

Problem

Subtract   and state the excluded values

 

 

 

2y – 1 = 0

y =

 

y – 5 = 0

y = 5

 and 5 are excluded values.

 

Find the excluded values.

 

 

common denominator = (2y - 1)(y - 5)

 

Find a least common multiple by factoring each denominator.

 

Neither 2y – 1 or y – 5 can be factored. Because they have no common factors, the least common denominator is the product of these denominators.

 

 

 –

 

 

 

Multiply each expression by the equivalent of 1 that will  give it the common denominator.

 

Then rewrite the rational expressions with the common denominator.

 

 

Subtract the numerators.

 

 

 

Simplify.

Answer

, y  , 5

 

 

 

 

Simplify  and state any excluded values.

 

A)  

 

B) , x -

 

C) , x -4, 0, 3

 

D) , x -4, 3

 

Show/Hide Answer

A) Incorrect. The approach is correct, but the answer is incomplete. The numerator of the rational expression can be simplified by multiplying and combining like terms. Also, the answer should state any excluded values. The correct answer is , x -4, 3.

 

B) Incorrect. To add rational expressions with unlike denominators, you must first find a common denominator. The common denominator for these rational expressions is (x + 4)(x – 3) because the denominators do not have any common factors. Write both addends with a common denominator, , and then simplify. To find the excluded values, look at the original expression as well as the denominators along the way. The correct answer is , x -4, 3.

 

C) Incorrect. You can only simplify numerator and denominator when there are like factors, not like terms. You cannot cancel the x2 terms and 12s.  The correct answer is, x -4, 3.

 

D) Correct. First find a common denominator, (x + 4)(x – 3), and rewrite each addend using that denominator: . Multiply and add the numerators: . The correct answer is , x -4, 3.

 

 

Summary

 

To add and subtract rational expressions, we apply the same idea used for adding and subtracting fractions: first find a common denominator. The least common denominator is the same as the least common multiple and can be found by listing multiples of each denominator or through prime factorization.

 

When working with rational expressions, it is always important to include the excluded values of the domain with the answer.