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By the end of this section, you will be able to:
  • Add rational expressions with a common denominator
  • Subtract rational expressions with a common denominator
  • Add and subtract rational expressions whose denominators are opposites

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: y 3 + 9 3 .
    If you missed this problem, review [link] .
  2. Subtract: 10 x 2 x .
    If you missed this problem, review [link] .
  3. Factor completely: 8 n 5 20 n 3 .
    If you missed this problem, review [link] .
  4. Factor completely: 45 a 3 5 a b 2 .
    If you missed this problem, review [link] .

Add rational expressions with a common denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

Rational expression addition

If p , q , and r are polynomials where r 0 , then

p r + q r = p + q r

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.

We will add two numerical fractions first, to remind us of how this is done.

Add: 5 18 + 7 18 .

Solution

5 18 + 7 18 The fractions have a common denominator, so add the numerators and place the sum over the common denominator. 5 + 7 18 Add in the numerator. 12 18 Factor the numerator and denominator to show the common factors. 6 · 2 6 · 3 Remove common factors. 6 · 2 6 · 3 Simplify. 2 3

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Remember, we do not allow values that would make the denominator zero. What value of y should be excluded in the next example?

Add: 3 y 4 y 3 + 7 4 y 3 .

Solution

3 y 4 y 3 + 7 4 y 3 The fractions have a common denominator, so add the numerators and place the sum over the common denominator. 3 y + 7 4 y 3

The numerator and denominator cannot be factored. The fraction is simplified.

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Add: 5 x 2 x + 3 + 2 2 x + 3 .

5 x + 2 2 x + 3 .

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Add: x x 2 + 1 x 2 .

x + 1 x 2

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Add: 7 x + 12 x + 3 + x 2 x + 3 .

Solution

7 x + 12 x + 3 + x 2 x + 3 The fractions have a common denominator, so add the numerators and place the sum over the common denominator. 7 x + 12 + x 2 x + 3 Write the degrees in descending order. x 2 + 7 x + 12 x + 3 Factor the numerator. ( x + 3 ) ( x + 4 ) x + 3 Simplify by removing common factors. ( x + 3 ) ( x + 4 ) x + 3 Simplify. x + 4

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Add: 9 x + 14 x + 7 + x 2 x + 7 .

x + 2

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Add: x 2 + 8 x x + 5 + 15 x + 5 .

x + 3

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Subtract rational expressions with a common denominator

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.

Rational expression subtraction

If p , q , and r are polynomials where r 0 , then

p r q r = p q r

To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

Subtract: n 2 n 10 100 n 10 .

Solution

n 2 n 10 100 n 10 The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator. n 2 100 n 10 Factor the numerator. ( n 10 ) ( n + 10 ) n 10 Simplify by removing common factors. ( n 10 ) ( n + 10 ) n 10 Simplify. n + 10

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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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