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12.6: Addition and Subtraction of Rational Expressions

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

At the end of this lesson, students will be able to:

  • Add and subtract rational expressions with the same denominator.
  • Find the least common denominator of rational expressions.
  • Add and subtract rational expressions with different denominators.
  • Solve real-world problems involving addition and subtraction of rational expressions.


Terms introduced in this lesson:

least common denominator (LCD)
least common multiple (LCM)
prime factorization
factor completely
equivalent fraction

Teaching Strategies and Tips

Use the ordinary fractions in Examples 1 and 5 to motivate adding and subtracting variable rational expressions with and without common denominators, respectively.

Draw the analogy between finding the LCM of polynomials and the LCM of integers.

  • Use prime factorization of numbers and polynomials.
  • In general, the LCM is found by taking each factor to the highest power that it appears in each expression.

In Examples 6, 8, and 9, remind students to distribute the minus to each term in the second rational expression. See also Review Questions 4, 5, 8, 17, 18, 21, 23, 27, and 30.

Draw the analogy between the formula: part of the task completed \begin{align*}=\end{align*} rate of work \begin{align*}\cdot\end{align*} time spent on the task and the formula: distance \begin{align*}=\end{align*} rate \begin{align*}\cdot\end{align*} time.

In Review Questions 34-36, encourage students to set up a table similar to the one in Example 12.

  • Emphasize that all known and unknown variables for each person or machine can be listed, which makes organizing the given information easy.
  • Combining parts of the task completed by each person or machine is a matter of reading across the rows or down the columns depending on how the table is set up.
  • Many students find tables useful for work problems; others rely heavily upon it.

In Review Questions 34-36,

  • Suggest that students begin by looking at the part of the task completed by each person or machine separately.
  • Encourage students to check the reasonableness of their answers. Ask: What kind of answer should we expect based upon the given information?

Error Troubleshooting

In Review Questions 7 and 8, suggest that students factor out a negative from the second rational expression first.

  • In Example 7, the LCM of \begin{align*}x-4\end{align*} and \begin{align*}4-x=-(x-4)\end{align*} is \begin{align*}x-4\end{align*}.

General Tip: Remind students to find the LCD of rational expressions by factoring. Students needlessly use larger common multiples when expressions are not completely factored.

General Tip: Have students leave the LCM in factored form. This makes simplifying and determining excluded values easier.

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