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8.4 Add and subtract rational expressions with unlike denominators

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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 ) .

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

( x − 4 ) ( x + 4 ) ( x + 3 )

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

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

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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. .

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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 )

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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 )

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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.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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