1. Copyright © 2013, 2009, 2006 Pearson Education, Inc.
Section 6.2
Adding and
Subtracting
Rational
Expressions
Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1

3. Adding Rational Expressions With Common Denominators
If are rational expressions, then
To add rational expressions with the same denominator, add numerators and place the sum over the common denominator. If possible, factor and simplify the result.

4. Adding Rational Expressions
EXAMPLE
Add:
SOLUTION
This is the original expression.
Add numerators. Place this sum over the common denominator.
Combine like terms.
Factor.

5. Adding Rational Expressions
CONTINUED
Factor and simplify by dividing out the common factor, x.
Simplify.

6. Objective #1: Example

7. Objective #1: Example

9. Subtracting Rational Expressions With Common Denominators
If are rational expressions, then
To subtract rational expressions with the same denominator, subtract numerators and place the difference over the common denominator. If possible, factor and simplify the result.

10. Subtracting Rational Expressions
EXAMPLE
Subtract:
SOLUTION
This is the original expression.
Subtract numerators. Place this difference over the common denominator.
Remove the parentheses and distribute.

11. Subtracting Rational Expressions
CONTINUED
Combine like terms.
Factor.
Factor and simplify by dividing out the common factor, x + 3.
Simplify.

12. Objective #2: Example

13. Objective #2: Example

15. Finding the Least Common Denominator (LCD)
Least Common Denominators
Finding the Least Common Denominator (LCD)
1) Factor each denominator completely.
2) List the factors of the first denominator.
3) Add to the list in step 2 any factors of the second denominator that do not appear in the list.
4) Form the product of each different factor from the list in step 3. This product is the least common denominator.

16. Least Common Denominators
EXAMPLE
Find the LCD of:
SOLUTION
1) Factor each denominator completely.
2) List the factors of the first denominator.

17. Least Common Denominators
CONTINUED
3) Add any unlisted factors from the second denominator. The second denominator is (2y – 1)(y + 4). One factor of y + 4 is already in our list, but the factor 2y – 1 is not. We add the factor 2y – 1 to our list.
4) The least common denominator is the product of all factors in the final list. Thus,
is the least common denominator.

18. Objective #3: Example

19. Objective #3: Example

20. Objective #3: Example
CONTINUED

21. Objective #3: Example

22. Objective #3: Example

23. Objective #3: Example
CONTINUED

25. Add & Subtract Fractions
Adding and Subtracting Rational Expressions That Have Different Denominators
1) Find the LCD of the rational expressions.
2) Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD.
3) Add or subtract numerators, placing the resulting expression over the LCD.
4) If possible, simplify the resulting rational expressions.

26. Adding Fractions
EXAMPLE
Add:
SOLUTION
1) Find the least common denominator. Begin by factoring the denominators.
The factors of the first denominator are x + 4 and x – 2. The only factor from the second denominator that is unlisted is x – 1. Thus, the least common denominator is,

27. Adding Fractions
CONTINUED
2) Write equivalent expressions with the LCD as denominators.
This is the original expression.
Factored denominators.
Multiply each numerator and denominator by the extra factor required to form the LCD.

28. Adding Fractions
CONTINUED
3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible.
Add numerators.
Perform the multiplications using the distributive property.
Combine like terms.

29. Adding Fractions
CONTINUED
Since the numerator does not factor, there are clearly no common factors in the numerator and the denominator. Therefore, the final solution is,

30. Subtracting Fractions
EXAMPLE
Subtract:
SOLUTION
1) Find the least common denominator. Begin by factoring the denominators.

31. Subtracting Fractions
CONTINUED
2) Write equivalent expressions with the LCD as denominators.
This is the original expression.
Factored denominators.
Multiply each numerator and denominator by the extra factor required to form the LCD.

32. Subtracting Fractions
CONTINUED
3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible.
Subtract numerators.
Perform the multiplications using the distributive property and FOIL.
Remove parentheses.

33. Subtracting Fractions
CONTINUED
Combine like terms in the numerator.
Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,

34. Addition of Fractions Perform the indicated operations:
EXAMPLE
Perform the indicated operations:
SOLUTION
1) Find the least common denominator. Begin by factoring the denominators.
The factors of the first denominator are 1 and x – 3. The only factor from the second denominator that is unlisted is x + 1. We have already listed all factors from the third denominator. Thus, the least common denominator is,

35. Addition of Fractions
CONTINUED
2) Write equivalent expressions with the LCD as denominator.
This is the original expression.
Factor the second denominator.
Multiply each numerator and denominator by the extra factor required to form the LCD.

36. Addition of Fractions
CONTINUED
3) & 4) Add and subtract numerators, putting this result over the LCD. Simplify if possible.
Add and subtract numerators.
Perform the multiplications using the distributive property.
Combine like terms in the numerator.

37. Addition of Fractions
CONTINUED
Since the numerator does not factor, there are clearly no common factors in the numerator and the denominator. Therefore, the final solution is,

38. Objective #4: Example

39. Objective #4: Example

40. Objective #4: Example

41. Objective #4: Example

42. Objective #4: Example

43. Objective #4: Example

44. Objective #4: Example
CONTINUED

45. Objective #4: Example
CONTINUED

46. Objective #4: Example

47. Objective #4: Example

48. Objective #4: Example
CONTINUED

49. Objective #4: Example
CONTINUED

51. Addition of Fractions Add: This is the original expression.
EXAMPLE
Add:
SOLUTION
This is the original expression.
Factor the first denominator.

52. Addition of Fractions Rewrite –y + x as x – y.
CONTINUED
Rewrite –y + x as x – y.
Notice the LCD is (x + y)(x – y).
Multiply the second numerator and denominator by the extra factor required to form the LCD.
Perform the multiplications using the distributive property.

53. Addition of Fractions Add and subtract numerators.
CONTINUED
Add and subtract numerators.
Remove parentheses and distribute.
Combine like terms in the numerator.
Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,

54. Objective #5: Example

55. Objective #5: Example