1. 1 3.1.1: Adding and Subtracting Rational Expressions Introduction Expressions come in a variety of types, including rational expressions. A rational expression is a ratio of two polynomials, in which a variable appears in the denominator; for example, is a rational expression. Working with rational expressions can often be made easier by analyzing them to uncover more familiar (and sometimes less complex) structures within them. In this section, you will explore some of those structures.

2. Introduction, continued Where one rational expression exists, another may as well. Extracting meaning from the context may require these expressions to be combined in order to determine a sum or difference. Combining rational expressions through addition or subtraction is not complex, though it does demand attention to detail. Rewriting equivalent fractions can often simplify the solution process. 2 3.1.1: Adding and Subtracting Rational Expressions

3. Key Concepts Before adding or subtracting rational expressions, you must find a common denominator. A common denominator is a quantity that is a shared multiple of the denominators of two or more fractions. 3 3.1.1: Adding and Subtracting Rational Expressions

4. Key Concepts, continued A common denominator can be determined by finding the product of the denominators. For example, a common denominator of the rational expression is found by multiplying the two denominators, x – 1 and x 2 + x – 2: (x – 1)(x 2 + x – 2) = x 3 – 3x + 2. 4 3.1.1: Adding and Subtracting Rational Expressions

5. Key Concepts, continued Once a common denominator has been found, it can be used to write equivalent rational expressions for each term of the sum (or difference, if subtracting). Using common denominators, the same rational expression as before,, can be rewritten in an equivalent form, as shown on the next slide. 5 3.1.1: Adding and Subtracting Rational Expressions

6. Key Concepts, continued 6 3.1.1: Adding and Subtracting Rational Expressions

7. Key Concepts, continued Expressed with a common denominator, the sum of rational expressions is the sum of the numerators:. Thus, we can rewrite the expressions over a single denominator, as shown. 7 3.1.1: Adding and Subtracting Rational Expressions

8. Key Concepts, continued Expressed with a common denominator, the difference of rational expressions is the difference of the numerators: 8 3.1.1: Adding and Subtracting Rational Expressions

9. Key Concepts, continued The least common denominator (LCD) is the least common multiple of the denominators of two or more fractions. In other words, it’s the smallest possible common denominator. The LCD can be determined by finding the product of all the unique factors of the denominator. 9 3.1.1: Adding and Subtracting Rational Expressions

10. Key Concepts, continued Our sample rational expression, is equivalent to the rational expression. The LCD is (x – 1)(x + 2), or x 2 – x – 2. Recall that a rational expression cannot include a value in the denominator that causes it to equal 0, since 0 is undefined in the denominator. 10 3.1.1: Adding and Subtracting Rational Expressions

11. Common Errors/Misconceptions canceling individual terms of the numerator or denominator, rather than factors distributing values incorrectly, especially negative numbers forgetting to check for invalid values in the denominator neglecting to keep track of signs (particularly when subtracting terms that must first be properly distributed) 11 3.1.1: Adding and Subtracting Rational Expressions

12. Guided Practice Example 3 Simplify the rational expression. 12 3.1.1: Adding and Subtracting Rational Expressions

13. Guided Practice: Example 3, continued 1.Identify any invalid values of the expression. In the first rational term, x 2 ≠ 0. Taking the square root of each side results in x ≠ 0. This same value also makes the other two denominators invalid. We only need to avoid this one value, so the domain of the expression is x ≠ 0. 13 3.1.1: Adding and Subtracting Rational Expressions

14. Guided Practice: Example 3, continued 2.Find a common denominator. As in previous examples, we can determine a common denominator by finding the product of the denominators. That gives us x 2 x 4x = 4x 4. This would work just fine (it is most certainly a common denominator), however, it does seem large. We can likely determine a smaller common denominator. 14 3.1.1: Adding and Subtracting Rational Expressions

15. Guided Practice: Example 3, continued Each of the three denominators in the original expression has x as a factor, so we only need to include it once. The first term has an additional factor of x, and the last term has an additional term of 4. The product of these three values—the common x, another x from the first term, and the 4 from the last term—is 4x 2. This is a smaller and perhaps easier to handle denominator. Thus, we can use 4x 2 as the common denominator. 15 3.1.1: Adding and Subtracting Rational Expressions

16. Guided Practice: Example 3, continued 3.Rewrite each term of the expression using the new denominator. Original expression Multiply by, by, and by. 16 3.1.1: Adding and Subtracting Rational Expressions

17. Guided Practice: Example 3, continued Simplify each term. Rewrite the expression by adding the numerators over the common denominator. Combine like terms. The rewritten expression is 17 3.1.1: Adding and Subtracting Rational Expressions

18. Guided Practice: Example 3, continued 4.Check to see if the result can be written in a simpler form. There are no common factors of the numerator or denominator. This expression cannot be written in any other format that will simplify the result. Therefore,, where x ≠ 0, is the simplest way to rewrite the original expression. 18 3.1.1: Adding and Subtracting Rational Expressions

19. Guided Practice: Example 3, continued You can use this strategy to combine many different rational expressions, including those that appear to be more complex. The expressions may involve more terms, or perhaps subtraction here and there. Yet the process outlined in this example remains the same. Carefully following these same steps will yield successful results. 19 3.1.1: Adding and Subtracting Rational Expressions ✔

20. Guided Practice: Example 3, continued 20 3.1.1: Adding and Subtracting Rational Expressions

21. Guided Practice Example 4 Simplify the rational expression. 21 3.1.1: Adding and Subtracting Rational Expressions

22. Guided Practice: Example 4, continued 1.Identify any invalid values of the expression. The first rational term has a denominator of 2x + 1. Set the denominator equal to 0 and then solve the resulting equation for x to determine values that make this expression invalid. 2x + 1 = 0 2x = –1 22 3.1.1: Adding and Subtracting Rational Expressions

23. Guided Practice: Example 4, continued An x-value of invalidates the expression, so. The second rational term has a denominator of x – 3. Again, set the denominator equal to 0 and then solve the resulting equation for x to determine values that make this expression invalid. 23 3.1.1: Adding and Subtracting Rational Expressions

24. Guided Practice: Example 4, continued x – 3 = 0 x = 3 An x-value of 3 invalidates the expression, so x ≠ 3. The rational expression is only valid when and x ≠ 3. 24 3.1.1: Adding and Subtracting Rational Expressions

25. Guided Practice: Example 4, continued 2.Find a common denominator. The two denominators are 2x + 1 and x – 3. They share no factors, so their product will be a good common denominator. (2x + 1)(x – 3) = 2x(x – 3) + 1(x – 3) = 2x 2 – 6x + x – 3 = 2x 2 – 5x – 3 The common denominator is 2x 2 – 5x – 3. 25 3.1.1: Adding and Subtracting Rational Expressions

26. Guided Practice: Example 4, continued 3.Rewrite each term of the expression using the new denominator. Original expression Multiply by and by. 26 3.1.1: Adding and Subtracting Rational Expressions

27. Guided Practice: Example 4, continued Simplify each term. Multiply. Rewrite the expression by adding the numerators over the common denominator. Combine like terms. 27 3.1.1: Adding and Subtracting Rational Expressions

28. Guided Practice: Example 4, continued 4.Check to see if the result can be written in a simpler form. We already know the denominator has x – 3 and 2x + 1 as factors. The numerator, however, has no integer factors. This expression cannot be written in any other format that will simplify the result. Thus, where and x = ̸ 3, is the simplest way to rewrite the original expression, 28 3.1.1: Adding and Subtracting Rational Expressions ✔

29. Guided Practice: Example 4, continued 29 3.1.1: Adding and Subtracting Rational Expressions