Adding and Subtractracting Rational Expressions Part II
Learning Outcomes
Add and subtract rational expressions that share no common factors
Add and subtract more than two rational expressions
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?
No Common Factors
Add and Subtract Rational Expressions with No Common Factor
In the next example, we show how to find a common denominator when there are no common factors in the expressions.
Example
Subtract 2y−13y−y−54, and give the domain.
State the difference in simplest form.
Answer:
Neither 2y–1 nor y–5 can be factored. Because theyhave no common factors, the least common multiple, which will become the least common denominator, is the product of these denominators.
LCM=(2y−1)(y−5)
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.
The domain is found by setting the original denominators equal to zero.
2y−1=0 and y−5=0y=21 and y=5
The domain is y=21,y=5
Answer
2y−13y−y−54=2y2−11y+53y2−23y+4,y=21,5
In the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors.
https://www.youtube.com/watch?v=CKGpiTE5vIg&feature=youtu.be
Try It
[ohm_question]40252[/ohm_question]
Add and Subtract More Than Two 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
Simplifyx2−42x2+x−2x−x+21, and give the domain.
State the result in simplest form.
Answer:
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).
x2−4=(x+2)(x−2)x−2=x−2x+2=x+2LCM=(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.
In the video that follows we present an example of subtracting 3 rational expressions with unlike denominators. One of the terms being subtracted is a number, so the denominator is 1.
https://www.youtube.com/watch?v=c-8xQyU0ch0&feature=youtu.be
Example
Simplify3yy2−x2−915, and give the domain.
State the result in simplest form.
Answer:
Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization.
3y=3⋅yx=x9=3⋅3LCM=3⋅3⋅x⋅yLCM=9xy
The LCM becomes the common denominator. Multiply each expression by the equivalent of 1 that will give it the common denominator.
The domain is found by setting the denominators equal to zero. 9=0 is nonsense, so we don't need to worry about that denominator.
3y=0 and x=0y=0 and x=0
The domain is y=0,x=0
Answer
3yy2−x2−915=3xxy−5x−6,y=0,x=0
In this last video, we present another example of adding and subtracting three rational expressions with unlike denominators.
https://www.youtube.com/watch?v=43xPStLm39A&feature=youtu.be
Add and Subtract
Try It
[ohm_question]189261[/ohm_question]
Summary
The methods shown here will help you when you are solving rational equations later on. To add and subtract rational expressions that share common factors, you first identify which factors are missing from each expression, and build the LCD with them. To add and subtract rational expressions with no common factors, the LCD will be the product of all the factors of the denominators.
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Licenses & Attributions
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Image: No common factors..Provided by: Lumen LearningLicense: CC BY: Attribution.
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Revision and Adaptation.Provided by: Lumen LearningLicense: CC BY: Attribution.
Subtract Rational Expressions with UnLike Denominators - 3 Expressions.Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.
Add and Subtract Rational Expressions with UnLike Denominators - 3 Expressions.Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.
CC licensed content, Shared previously
Unit 15: Rational Expressions, from Developmental Math: An Open Program.Provided by: Monterey Institute of Technology and EducationLocated at: https://www.nroc.org/.License: CC BY: Attribution.
Ex: Add Rational Expressions with Unlike Denominators.Authored by: James Sousa (Mathispower4u.com) .License: CC BY: Attribution.