Copyright © Cengage Learning. All rights reserved. Rational Expressions and Equations; Ratio and Proportion 6 Copyright © Cengage Learning. All rights reserved.

Adding and Subtracting Rational Expressions 6.3 Section 6.3 Copyright © Cengage Learning. All rights reserved.

Objectives Add two rational expressions with like denominators and write the answer in simplest form. Subtract two rational expressions with like denominators and write the answer in simplest form. Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions. 1 2 3

Objectives Add two rational expressions with unlike denominators and write the answer in simplest form. Subtract two rational expressions with unlike denominators and write the answer in simplest form. 4 5

Add two rational expressions with like denominators and write the answer in simplest form 1.

Add two rational expressions with like denominators and write the answer in simplest form To add rational expressions with a common denominator, we follow the same process we use to add arithmetic fractions; add their numerators and keep the common denominator. For example, Add the numerators and keep the common denominator. 2x + 3x = 5x

Add two rational expressions with like denominators and write the answer in simplest form In general, we have the following result. Adding Rational Expressions with Like Denominators If a, b, and d represent polynomials, then provided the denominator is not equal to 0.

Example Solution: In each part, we will add the numerators and keep the common denominator. a. Add the numerators and keep the common denominator. Combine like terms. because

Example 1 – Solution cont’d Add the numerators and keep the common denominator. Combine like terms.

Subtract two rational expressions with like Subtract two rational expressions with like denominators and write the answer in simplest form 2.

Subtract two rational expressions with like denominators and write the answer in simplest form To subtract rational expressions with a common denominator, we subtract their numerators and keep the common denominator. Subtracting Rational Expressions with Like Denominators If a, b, and d represent polynomials, then provided the denominator is not equal to 0.

Example Subtract, assuming no divisions by zero. a. b.

Example – Solution In each part, the rational expressions have the same denominator. To subtract them, we subtract their numerators and keep the common denominator. a. Subtract the numerators and keep the common denominator. Combine like terms. Divide out the common factor.

Example – Solution cont’d Subtract the numerators and keep the common denominator. Use the distributive property to remove parentheses. Combine like terms.

Find the least common denominator (LCD) of Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions 3.

Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions Since the denominators of the fractions in the addition are different, we cannot add the fractions in their present form. four-sevenths + three-fifths To add these fractions, we need to find a common denominator. The smallest common denominator (called the least or lowest common denominator) is the easiest one to use. Different denominators

Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions Least Common Denominator The least common denominator (LCD) for a set of fractions is the smallest number that each denominator will divide exactly. We now review the method of writing two fractions using the LCD. In the addition , the denominators are 7 and 5. The smallest number that 7 and 5 will divide exactly is 35. This is the LCD.

Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions We now build each fraction into a fraction with a denominator of 35. Now that the fractions have a common denominator, we can add them. Multiply numerator and denominator of by 5, and multiply numerator and denominator of by 7. Do the multiplications.

Example Write each rational expression as an equivalent expression with a denominator of 30y (y  0). a. b. c.

Example – Solution To build each rational expression into an expression with a denominator of 30y, we multiply the numerator and denominator by what it takes to make the denominator 30y y  (0). a. b. c.

Find the least common denominator (LCD) of two or more polynomials and use it to write equivalent rational expressions There is a process that we can use to find the least common denominator of several rational expressions. Finding the Least Common Denominator (LCD) 1. List the different denominators that appear in the rational expressions. 2. Completely factor each denominator. 3. Form a product using each different factor obtained in Step 2. Use each different factor the greatest number of times it appears in any one factorization. The product formed by multiplying these factors is the LCD.

Add two rational expressions with unlike Add two rational expressions with unlike denominators and write the answer in simplest form 4.

Add two rational expressions with unlike denominators and write the answer in simplest form The process for adding and subtracting rational expressions with different denominators is the same as the process for adding and subtracting expressions with different numerical denominators. For example, to add and , we first find the LCD of 7 and 5, which is 35. We then build the rational expressions so that each one has a denominator of 35.

Finally, we add the results. Add two rational expressions with unlike denominators and write the answer in simplest form Finally, we add the results. Multiply numerator and denominator of by 5, and numerator and denominator of by 7. Do the multiplications Add the numerators and keep the common denominator.

Add two rational expressions with unlike denominators and write the answer in simplest form The following steps summarize how to add rational expressions that have unlike denominators. Adding Rational Expressions with Unlike Denominators To add rational expressions with unlike denominators: 1. Find the LCD. 2. Write each rational expression as an equivalent expression with a denominator that is the LCD. 3. Add the resulting fractions. 4. Simplify the result, if possible.

Example Add: , , and (b  0). Solution: The LCD of these rational expressions is 2  2  2  3  3  b = 72b. To add the rational expressions, we first factor each denominator:

Example – Solution cont’d In each resulting expression, we multiply the numerator and the denominator by whatever it takes to build the denominator to the lowest common denominator of 2  2  2  3  3  b. Do the multiplications. Add the fractions. Simplify.

Subtract two rational expressions with Subtract two rational expressions with unlike denominators and write the answer in simplest form 5.

Subtract two rational expressions with unlike denominators and write the answer in simplest form To subtract rational expressions with unlike denominators, we first write them as expressions with the same denominator and then subtract the numerators.

Example Subtract: (x ≠ 0, –1) Solution: Because x and x + 1 represent different values and have no common factors, the least common denominator (LCD) is their product, (x + 1)x. Build the fractions to obtain the common denominator.

Example – Solution cont’d Subtract the numerators and keep the common denominator. Do the multiplication in the numerator.