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Chapter 8 Rational Expressions

8.1 Simplifying Rational Expressions 8.2 Multiplying and Dividing Rational Expressions 8.3 Finding the Least Common Denominator 8.4Adding and Subtracting Rational Expressions Putting It All Together 8.5Simplifying Complex Fractions 8.6Solving Rational Equations 8.7Applications of Rational Equations 8 Rational Expressions

Finding the Least Common Denominator for a Group of Rational Expressions8.3 Find the Least Common Denominator for a Group of Rational Expressions In order to add or subtract fractions, they must have a common denominator. In the same way rational expressions must have common denominators in order to be added or subtracted. Find the Least Common Denominator for the following fractions: First write each of the denominators, 8 and 12, as the product of their prime factors: The LCD will contain each factor the greatest number of times it appears in any single factorization. This means that for we will choose because it appears the greatest number of times, in this case three. The LCD will also contain 3 because 3 appears one time in the factorization of 12.

Example 1 Solution Step 1: Factor the denominators. Step 2: The LCD will contain each unique factor the greatest number of times it appears in any single factorization. Step 3: The LCD is the product of the factors identified in step 2.

Example 2 Solution Step 1: Factor the denominators. Step 2: The LCD will contain each unique factor the greatest number of times it appears in any single factorization. Step 3: The LCD is the product of the factors identified in step 2.

Example 3 Solution Step 1: Factor the denominators. Step 2: The LCD will contain each unique factor the greatest number of times it appears in any single factorization. Step 3: The LCD is the product of the factors identified in step 2.

Example 4 Solution

Example 5 Solution Step 1: Factor the denominators. Step 2: The LCD will contain each unique factor the greatest number of times it appears in any single factorization. Step 3: The LCD is the product of the factors identified in step 2.

Example 6 Solution Step 1: Factor the denominators. Step 2: The LCD will contain each unique factor the greatest number of times it appears in any single factorization. The unique factors are 4, d and (d-3), with (d-3) appearing at most twice. Step 3: The LCD is the product of the factors identified in step 2.

Example 7 Solution Since, we can rewrite. Therefore, we can now think of our task as finding the LCD of The least common denominator is.

Rewrite Rational Expressions Using Their LCD In the previous slides, we learned how to find the LCD of a group of rational expressions. Now that we know how to find the LCD then we must rewrite those rational expressions as equivalent expressions with the LCD so that they can be added or subtracted. Let’s practice the procedure on two fractions first, before applying it to rational expressions. On slide 5, we found that the LCD of these two fractions is 90. Next, we must rewrite each fraction with a denominator of 90. By what number should we multiply 15 to get 90?6 By what number should we multiply 18 to get 90?5

Let’s look at some examples for writing rational expressions as equivalent Expressions with the Least Common Denominator.

Example 8 Solution Step 1: Identify and write down the LCD. Step 2: Compare denominators and ask yourself, “What’s missing?” or “What do I need to multiply the denominator to get LCD?” Step 3: Multiply the numerator and denominator by the missing factors.

Example 9 Solution Step 1: Identify and write down the LCD. Step 2: Compare denominators and ask yourself, “What’s missing?” or “What do I need to multiply the denominator to get LCD?” Step 3: Multiply the numerator and denominator by the missing factors.

Example 10 Solution Step 1: Identify and write down the LCD. Step 2: Compare denominators and ask yourself, “What’s missing?” or “What do I need to multiply the denominator to get LCD?” Step 3: Multiply the numerator and denominator by the missing factors.

Example 11 Solution