Adding and Subtracting Rational Expressions

 

Learning Objective(s)

·         Add rational expressions and simplify.

·         Subtract rational expressions and simplify.

·         Find the Least Common Multiple of several algebraic expressions.

·         Simplify problems that combine both adding and subtracting.

 

Introduction

 

In beginning math, students usually learn how to add and subtract whole numbers before they are 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 not as easy to perform as multiplication because, as with numeric fractions, the process involves finding common denominators. By working carefully and writing down the steps along the way, you can keep track of all of the numbers and variables and perform the operations accurately.

 

Adding and Subtracting Rational Expressions with Like Denominators

 

Adding rational expressions with the same denominator is the simplest place to start, so let’s begin there.

 

To add fractions with like denominators, add the numerators and keep the same denominator. Then simplify the sum. You know how to do this with numeric fractions.

 

 

 

Follow the same process to add rational expressions with like denominators. Let's try one.

 

 

Example

Problem

Add. State the sum in simplest form.

 

 

Since the denominators are the same, add the numerators. Remember that x cannot be -4 because the denominators would be 0.

 

 

 

Factor the numerator.

 

 

Rewrite the common factor as multiplication by 1 and simplify.

Answer

 

 

 

Remember that you should also 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. In the problem above, the domain is all real numbers except −4, since a value of x = −4 will create a denominator of 0. Sometimes, when we simplify an expression, the reader looking only at the simplified answer would not realize that there are excluded values.  In the example above, just looking at the simplified form of 2x as a replacement for the original , the reader would have no way of knowing that a value of −4 can’t be used for x.  So when we claim that 2x is the equivalent of , we need to state that −4 is an excluded value.

 

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

 

 

Example

Problem

Subtract. State the difference in simplest form.

 

Subtract the second numerator from the first and keep the denominator the same. Remember that x cannot be -6 because the denominators would be 0.

 

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

 

Combine like terms. This rational expression cannot be simplified any further.

Answer

 

 

 

Subtract, and state the difference in simplest form. , x ≠ 5

 

 

A)

 

B) x + 5

 

C) x – 5

 

D) 5

 

Show/Hide Answer

A)

Incorrect. You performed the subtraction properly, 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.

 

B) x + 5

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 numerator and denominator. .

 

C) x – 5

Incorrect. The common factor present in the numerator and denominator is x – 5, not x + 5. After factoring you get: . The correct answer is x + 5.

 

D) 5

Incorrect. To find the difference, subtract the numerator of the second fraction from the numerator of the first, like this: . Then factor the numerator and simplify. The correct answer is x + 5.

 

 

 

Adding and Subtracting Rational Expressions with Unlike Denominators

 

Before adding and subtracting rational expressions with unlike denominators, you need to find a common denominator. Once again, this process is 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, you 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. You 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 least 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

68

 

 

 

 

 

10

20

30

40

50

60

 

 

 

 

 

 

 

 

 

 

 

 

 

The other method is to use prime factorization, the process of finding the prime 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 factorization of each denominator.

 

6 = 3 • 2

10 = 5 • 2

4 = 2 • 2

 

 

 

 

 

 

 

LCM = 3 • 5 • 2 • 2

The LCM will contain factors of 2, 3, and 5. Multiply each number the maximum number of times it appears in a single factorization.

 

In this case, 3 appears once, 5 appears once, and 2 is used twice because it appears twice in the prime factorization of 4.

 

Therefore, the LCM of 6, 10 and 4 is 3 • 5 • 2 • 2, or 60.

Answer

The least common multiple of 6, 10, and 4 is 60.

 

 

Look at that—you found the same least common multiple using both methods. Prime factorization was faster, though, because you did not have to make a chart full of multiples to find a common multiple.

 

Now that you have found the least common multiple, you can use that number as the least common denominator of the fractions. Multiply each fraction by the fractional form of 1 that will produce a denominator of 60:

 

 

 

 

Now that you have like denominators, add the fractions:

 

 

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

 

 

Example

Problem

Add. State the sum in simplest form.

 

15m2 = 3 • 5 • m • m

 

21m = 3 • 7 • m

 

Find the prime factorization of each denominator.

 

15m2 = 3 • 5 • m • m

 

21m = 3 • 7 • m

 

 LCM: 3 • 5 • 7 • m • m

 

LCM: 105m2

 

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

 

Use the least common multiple for your new common denominator, it will be the LCD.

 

 

 

 

 

 

 

 

 

 

 

 

 

Compare each original denominator and the new common denominator. Now rewrite the rational expressions to each have the common denominator of 105m2. Remember that m cannot be 0 because the denominators would be 0.

 

The first denominator is 15m2 and the LCD is 105m2. You need to multiply 15m2 by 7 to get the LCD, so multiply the entire rational expression by .

 

The second denominator is 21m and the LCD is 105m2. You need to multiply 21m by 5m to get the LCD, so multiply the entire rational expression by .

 

 

Add the numerators and keep the denominator the same.

 

 

If possible, simplify by finding common factors in the numerator and denominator. This rational expression is already in simplest form because the numerator and denominator have no factors in common.

 

Answer

 

 

 

 

 

That took a while, but you got through it. Adding rational expressions can be a lengthy process, but taken one step at a time, it can be done.

 

Now let’s try subtracting rational expressions. You'll use the same basic technique of finding the least common denominator and rewriting each rational expression to have that denominator.

 

 

Example

Problem

Subtract. State the difference in simplest form.

 

 

Find the prime factorization of each denominator. t + 1 cannot be factored any further, but  can be. Remember that t cannot be -1 or 2 because the denominators would be 0.

 

 

 

 

 

 

 

LCM: (t + 1)(t – 2)

Find the least common multiple. t + 1 appears exactly once in both of the expressions, so it will appear once in the least common denominator. t – 2 also appears once.

 

This means that (t - 2)(t + 1) is the least common multiple. In this case, it is easier to leave the common multiple in terms of the factors, so you will not multiply it out.

Use the least common multiple for your new common denominator, it will be the LCD.

 

 

 

 

 

 

 

 

 

 

 

Compare each original denominator and the new common denominator. Now rewrite the rational expressions to each have the common denominator of (t + 1)(t – 2).

 

You need to multiply t + 1 by t – 2 to get the LCD, so multiply the entire rational expression by .

 

The second expression already has a denominator of (t + 1)(t – 2), so you do not need to multiply it by anything.

 

Then rewrite the subtraction problem with the common denominator.

 

 

 

 

Subtract the numerators and simplify. Remember that parentheses need to be included around the second (t – 2) in the numerator because the whole quantity is subtracted. Otherwise you would be subtracting just the t.”

 

 

 

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

Answer

 

 

 

 

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

 

 

Example

Problem

Subtract. State the difference in simplest form.

 

LCM = (2y - 1)(y - 5)

Neither 2y – 1 nor y – 5 can be factored. Because they have no common factors, the least common multiple, which will become the least common denominator, is the product of these denominators. Remember that y cannot be ½ or 5 because the denominators would be 0.

 

 

 

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

 

Then rewrite the subtraction problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.

 

 

 

Subtract and simplify.

Answer

  

 

 

 

Add. State the sum in simplest form.

 

A)

 

B)

 

C)

 

D)

 

 

Show/Hide Answer

A)

Incorrect. The approach is correct, but the answer has not been simplified. The numerator of the rational expression can be simplified by multiplying and combining like terms. The correct answer is .

 

B)

Incorrect. To add rational expressions with unlike denominators, you must first find a common denominator. The common denominator for these rational expressions is  because the denominators do not have any common factors. Write both addends with a common denominator, , and then simplify. The correct answer is .

 

C)

Incorrect. You can only simplify numerator and denominator when there are like factors, not like terms. You cannot cancel the x2 terms and 12’s. The correct answer is .

 

D)

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

 

 

 

Combining Multiple Rational Expressions

 

You may need to combine more than two rational expressions. While this may seem pretty straightforward if they all have the same denominator, what happens if they do not?

 

In the example below, notice how a common denominator is found for three rational expressions. Once that is done, the addition and subtraction of the terms looks the same as earlier, when you were only dealing with two terms.

 

 

Example

Problem

Simplify. State the result in simplest form.

 

x2 – 4 = (x + 2)(x – 2)

 

x – 2 = x – 2

 

x + 2 = x + 2

 

 

 

 

 

 

LCM = (x + 2)(x – 2)

Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization. Remember that x cannot be 2 or -2 because the denominators would be 0.

 

(x + 2) appears a maximum of one time, as does (x – 2). This means the LCM is (x + 2)(x – 2).

 

 

 

 

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

 

 

 

 

Rewrite the original problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.

 

 

 

Combine the numerators.

 

Check for simplest form. Since neither  nor  is a factor of , this expression is in simplest form.

Answer

 

 

 

 

Example

Problem

Simplify. State the result in simplest form.

 

3y = 3 • y

 

x = x

 

9 = 3 • 3

 

LCM = 3 • 3 • x • y

 

LCM = 9xy

Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization. Remember that x and y cannot be 0 because the denominators would be 0.

 

 

 

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

 

 

 

Rewrite the original problem with the common denominator.

 

Combine the numerators.

 

 

 

 

Check for simplest form.

Answer

 

 

 

Summary

 

To add and subtract rational expressions, apply the same ideas you use to add and subtract numeric 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. Then use that common denominator to rewrite the fractions so that you can add or subtract them.