1. Home
  2. Elementary algebra
  3. Rational expressions and equations
  4. Add and subtract rational

8.4 Add and subtract rational expressions with unlike denominators

<< Chapter < Page
  Elementary algebra     Page 1 / 4
Chapter >> Page >
By the end of this section, you will be able to:
  • Find the least common denominator of rational expressions
  • Find equivalent rational expressions
  • Add rational expressions with different denominators
  • Subtract rational expressions with different denominators

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

  1. Add: 7 10 + 8 15 .
    If you missed this problem, review [link] .
  2. Subtract: 6 ( 2 x + 1 ) − 4 ( x − 5 ) .
    If you missed this problem, review [link] .
  3. Find the Greatest Common Factor of 9 x 2 y 3 and 12 x y 5 .
    If you missed this problem, review [link] .
  4. Factor completely −48 n − 12 .
    If you missed this problem, review [link] .

Find the least common denominator of rational expressions

When we add or subtract rational expressions with unlike denominators we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.

Let’s look at the example 7 12 + 5 18 from Foundations . Since the denominators are not the same, the first step was to find the least common denominator (LCD). Remember, the LCD is the least common multiple of the denominators. It is the smallest number we can use as a common denominator.

To find the LCD of 12 and 18, we factored each number into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.

12 = 2 · 2 · 3 18 = 2 · 3 · 3 LCD = 2 · 2 · 3 · 3 LCD = 36

We do the same thing for rational expressions. However, we leave the LCD in factored form.

Find the least common denominator of rational expressions.

  1. Factor each expression completely.
  2. List the factors of each expression. Match factors vertically when possible.
  3. Bring down the columns.
  4. Multiply the factors.

Remember, we always exclude values that would make the denominator zero. What values of x should we exclude in this next example?

Find the LCD for 8 x 2 − 2 x − 3 , 3 x x 2 + 4 x + 3 .

Solution

Find the LCD for 8 x 2 − 2 x − 3 , 3 x x 2 + 4 x + 3 . Factor each expression completely, lining up common factors. Bring down the columns. x 2 − 2 x − 3 = ( x + 1 ) ( x − 2 ) x 2 + 4 x + 3 = ( x + 1 ) ( x + 3 ) LCD = ( x + 1 ) ( x − 2 ) ( x + 3 ) Multiply the factors. The LCD is ( x + 1 ) ( x − 3 ) ( x + 3 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the LCD for 2 x 2 − x − 12 , 1 x 2 − 16 .

( x − 4 ) ( x + 4 ) ( x + 3 )

Got questions? Get instant answers now!

Find the LCD for x x 2 + 8 x + 15 , 5 x 2 + 9 x + 18 .

( x + 3 ) ( x + 6 ) ( x + 5 )

Got questions? Get instant answers now!

Find equivalent rational expressions

When we add numerical fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD.

The above image shows how to find the LCD (least common denominator) when adding numerical fractions in the example seven-twelfths plus five-eighteenths. The image shows 7 times 3 divided by 12 times 3 plus 5 times 2 plus 18 times 2. Below this is 21 divided by 36 plus 10 divided by 36. The image next to this shows that 12 equals 2 times 2 times 3. Below this shows 18 equals 2 times 3 times 3. A line is drawn. Below it is LCD equals 2 times 2 times 3 times 3. The line below this shows that the LCD equals 36.

We will do the same thing for rational expressions.

Rewrite as equivalent rational expressions with denominator ( x + 1 ) ( x − 3 ) ( x + 3 ) : 8 x 2 − 2 x − 3 , 3 x x 2 + 4 x + 3 .

Solution

.
Factor each denominator. .
Find the LCD.   .
Multiply each denominator by the 'missing' factor and multiply each numerator by the same factor. .
Simplify the numerators. .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Rewrite as equivalent rational expressions with denominator ( x + 3 ) ( x − 4 ) ( x + 4 ) :
2 x 2 − x − 12 , 1 x 2 − 16 .

2 x + 8 ( x − 4 ) ( x + 3 ) ( x + 4 ) ,
x + 3 ( x − 4 ) ( x + 3 ) ( x + 4 )

Got questions? Get instant answers now!

Rewrite as equivalent rational expressions with denominator ( x + 3 ) ( x + 5 ) ( x + 6 ) :
x x 2 + 8 x + 15 , 5 x 2 + 9 x + 18 .

x 2 + 6 x ( x + 3 ) ( x + 5 ) ( x + 6 ) ,
x + 3 ( x + 3 ) ( x + 5 ) ( x + 6 )

Got questions? Get instant answers now!

Add rational expressions with different denominators

Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first.

Questions & Answers

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask