1. 6-1 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Addition, Subtraction, and Least Common Denominators Addition When Denominators Are the Same Subtraction When Denominators Are the Same Least Common Multiples and Denominators 6.3

2. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. The Sum of Two Rational Expressions To add when the denominators are the same, add the numerators and keep the common denominator:

3. 6-3 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example

4. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. The Difference of Two Rational Expressions To subtract when the denominators are the same, subtract the second numerator from the first and keep the common denominator:

5. 6-5 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Subtract and, if possible, simplify: a)b) Solution

6. 6-6 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Least Common Multiples and Denominators To add or subtract rational expressions that have different denominators, we must first find equivalent rational expressions that do have a common denominator. The least common multiple must include the factors of each number, so it must include each prime factor the greatest number of times that it appears in any factorizations.

7. Copyright © 2014, 2010, and 2006 Pearson Education, Inc. To Find the Least Common Denominator (LCD) 1.Write the prime factorization of each denominator. 2.Select one of the factorizations and inspect it to see if it completely contains the other factorization. a)If it does, it represents the LCM of the denominators. b)If it does not, multiply that factorization by any factors of the other denominator that it lacks. The final product is the LCM of the denominators. The LCD is the LCM of the denominators. It should contain each factor the greatest number of times that it occurs in any of the individual factorizations.

8. 6-8 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find the LCD of Solution

9. 6-9 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example For each pair of polynomials, find the least common multiple. a) 16a and 24b b) 24x 4 y 4 and 6x 6 y 2 c) x 2  4 and x 2  2x  8 Solution

10. 6-10 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example For each group of polynomials, find the least common multiple. a) 15x, 30y, 25xyzb) x 2 + 3, x + 2, 7 Solution

11. 6-11 Copyright © 2014, 2010, and 2006 Pearson Education, Inc. Example Find equivalent expressions that have the LCD: Solution