Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 7 Rational Expressions and Equations

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 Rational Expressions and Equations 7.1Simplifying, Multiplying, and Dividing Rational Expressions 7.2Adding and Subtracting Rational Expressions 7.3Simplifying Complex Rational Expressions 7.4Solving Equations Containing Rational Expressions CHAPTER 7

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 Adding and Subtracting Rational Expressions 1.Add or subtract rational expressions with the same denominator. 2.Find the least common denominator (LCD). 3.Write equivalent rational expressions with the LCD as the denominator. 4.Add or subtract rational expressions with unlike denominators. 7.2

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Adding or Subtracting Rational Expressions with the Same Denominator To add or subtract rational expressions with the same denominator, 1. Add or subtract the numerators and keep the same denominator. 2. Simplify to lowest terms.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Example Subtract. Solution Divide out the common factor, x – 4. Note: The numerator can be factored, so we may be able to simplify.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 Example Add. Solution Combine like terms in the numerator. Factor the numerator and the denominator. Divide out the common factor, b.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Example Subtract. Solution Note: To write an equivalent addition, change the operation symbol from a minus sign to a plus sign and change all the signs in the subtrahend (second) polynomial.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 Example Add. Solution Combine like terms in the numerator. Factor the numerator and the denominator.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 continued Divide out the common factors, 2 and x + 2.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 Remember that when adding or subtracting fractions with different denominators, we must first find a common denominator. It is helpful to use the least common denominator (LCD), which is the smallest number that is evenly divisible by all the denominators.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 Finding the LCD To find the LCD of two or more rational expressions, 1. Find the prime factorization of each denominator. 2. Write the product that contains each unique prime factor the greatest number of times that factor occurs in any factorization. Or, if you prefer to use exponents, write the product that contains each unique prime factor raised to the greatest exponent that occurs on that factor in any factorization. 3.Simplify the product found in step 2.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 Example Find the LCD. Solution We first factor the denominators 12y 2 and 8y 3 by writing their prime factorizations. The unique factors are 2, 3, and y. To generate the LCD, include 2, 3, and y the greatest number of times each appears in any of the factorizations.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Example Find the LCD. Solution Find the factors. The unique factors are x – 3, x, and x + 6, and the highest power of each is 1.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 Example Find the LCD. Solution Factor the denominators x 2 – 25 and 2x – 10. The unique factors are 2, (x + 5), and (x – 5). The greatest number of times that 2 appears is once. The greatest number of times that (x + 5) appears is once. The greatest number of times that (x – 5) appears is once.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 Example Find the LCD. Solution Factor the denominators. The unique factors are 3, 4, x, (x + 1), and (x + 5). The highest power of 3, 4, and x + 5 is 1, and the highest power of x and x + 1 is 2.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16 Example Write the fractions with the LCD as the denominator. Solution In example 2(a), we found that the LCD was 24y 3 Convert each to a fraction whose denominator is 24y 3.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17 Example Write the fractions with the LCD as the denominator. Solution In example 2(c), we found that the LCD was 2(x + 5)(x − 5). Convert each to a fraction whose denominator is 2(x + 5)(x − 5).

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 18 Adding or Subtracting Rational Expressions with Different Denominators To add or subtract rational expressions with different denominators, 1. Find the LCD. 2. Write each rational expression as an equivalent expression with the LCD. 3. Add or subtract the numerators and keep the LCD as the denominator. 4. Simplify.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 19 Example Add. Solution The LCD is 24x 2. Write equivalent rational expressions with the LCD, 24x 2. Subtract numerators. Note: Remember that to add polynomials, we combine like terms.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 20 Example Add. Solution Since x – 6 and 6 – x are additive inverses, we obtain the LCD by multiplying the numerator and denominator of one of the rational expressions by –1. We chose the second rational expression.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 21 Example Solution Find the LCD.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 22 continued

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 23 Example Solution Subtract: