7.3 Adding and Subtracting Rational Expressions

Transcription

1 7.3 Adding and Subtracting Rational Epressions LEARNING OBJECTIVES. Add and subtract rational epressions with common denominators. 2. Add and subtract rational epressions with unlike denominators. 3. Add and subtract rational functions. Adding and Subtracting with Common Denominators Adding and subtracting rational epressions is similar to adding and subtracting fractions. Recall that if the denominators are the same, we can add or subtract the numerators and write the result over the common denominator. When working with rational epressions, the common denominator will be a polynomial. In general, given polynomials P, Q, and R, where Q 0, we have the following: In this section, assume that all variable factors in the denominator are nonzero. Eample : Add: 3 y + 7 y. 27

2 Solution: Add the numerators 3 and 7, and write the result over the common denominator, y. Answer: 0 y Eample 2: Subtract: Solution: Subtract the numerators 5 and, and write the result over the common denominator, 2. Answer: 6 2 Eample 3: Subtract: 2+7 (+5)( 3) +0 (+5)( 3). Solution: We use parentheses to remind us to subtract the entire numerator of the second rational epression. 7.3 Adding and Subtracting Rational Epressions 28

3 Answer: +5 Eample 4: Simplify: Solution: Subtract and add the numerators. Make use of parentheses and write the result over the common denominator, Answer: Adding and Subtracting Rational Epressions 29

4 Try this! Subtract: Answer: 4 Video Solution (click to see video) Adding and Subtracting with Unlike Denominators To add rational epressions with unlike denominators, first find equivalent epressions with common denominators. Do this just as you have with fractions. If the denominators of fractions are relatively prime, then the least common denominator (LCD) is their product. For eample, Multiply each fraction by the appropriate form of to obtain equivalent fractions with a common denominator. The process of adding and subtracting rational epressions is similar. In general, given polynomials P, Q, R, and S, where Q 0 and S 0, we have the following: 7.3 Adding and Subtracting Rational Epressions 30

5 In this section, assume that all variable factors in the denominator are nonzero. Eample 5: Add: + y. Solution: In this eample, the LCD = y. To obtain equivalent terms with this common denominator, multiply the first term by y y and the second term by. Answer: y+ y Eample 6: Subtract: y y 3. Solution: Since the LCD = y(y 3), multiply the first term by in the form of (y 3) (y 3) and the second term by y y. 7.3 Adding and Subtracting Rational Epressions 3

6 Answer: 3 y(y 3) It is not always the case that the LCD is the product of the given denominators. Typically, the denominators are not relatively prime; thus determining the LCD requires some thought. Begin by factoring all denominators. The LCD is the product of all factors with the highest power. For eample, given there are three base factors in the denominator:, ( + 2), and ( 3). The highest powers of these factors are 3, ( + 2) 2, and ( 3). Therefore, The general steps for adding or subtracting rational epressions are illustrated in the following eample. 7.3 Adding and Subtracting Rational Epressions 32

7 Eample 7: Subtract: Solution: Step : Factor all denominators to determine the LCD. The LCD is ( + ) ( + 3) ( 5). Step 2: Multiply by the appropriate factors to obtain equivalent terms with a common denominator. To do this, multiply the first term by ( 5) and the second term by (+3) (+3). ( 5) Step 3: Add or subtract the numerators and place the result over the common denominator. Step 4: Simplify the resulting algebraic fraction. 7.3 Adding and Subtracting Rational Epressions 33

8 Answer: ( 9) (+3)( 5) Eample 8: Subtract: Solution: It is best not to factor the numerator, , because we will most likely need to simplify after we subtract. 7.3 Adding and Subtracting Rational Epressions 34

9 Answer: 8 ( 4)( 9) Eample 9: Subtract: Solution: First, factor the denominators and determine the LCD. Notice how the opposite binomial property is applied to obtain a more workable denominator. The LCD is ( + 2) ( 2). Multiply the second term by in the form of (+2) (+2). 7.3 Adding and Subtracting Rational Epressions 35

10 Now that we have equivalent terms with a common denominator, add the numerators and write the result over the common denominator. Answer: +3 (+2)( 2) Eample 0: Simplify: y y+ y+ y + y2 5 y 2. Solution: Begin by factoring the denominator. We can see that the LCD is (y + ) (y ). Find equivalent fractions with this denominator. 7.3 Adding and Subtracting Rational Epressions 36

11 Net, subtract and add the numerators and place the result over the common denominator. Finish by simplifying the resulting rational epression. Answer: y 5 y Try this! Simplify: ẋ Answer: Adding and Subtracting Rational Epressions 37

12 Video Solution (click to see video) Rational epressions are sometimes epressed using negative eponents. In this case, apply the rules for negative eponents before simplifying the epression. Eample : Simplify: y 2 + (y ). Solution: Recall that n = n. We begin by rewriting the negative eponents as rational epressions. Answer: y 2 +y y 2 (y ) 7.3 Adding and Subtracting Rational Epressions 38

13 Adding and Subtracting Rational Functions We can simplify sums or differences of rational functions using the techniques learned in this section. The restrictions of the result consist of the restrictions to the domains of each function. Eample 2: Calculate (f + g) (), given f () = the restrictions. +3 and g () =, and state 2 Solution: Here the domain of f consists of all real numbers ecept 3, and the domain of g consists of all real numbers ecept 2. Therefore, the domain of f + g consists of all real numbers ecept 3 and 2. Answer: 2+, where 3, 2 (+3)( 2) 7.3 Adding and Subtracting Rational Epressions 39

14 Eample 3: Calculate (f g) (), given f () = ( ) the restrictions to the domain and g () =, and state 5 Solution: The domain of f consists of all real numbers ecept 5 and 5, and the domain of g consists of all real numbers ecept 5. Therefore, the domain of f g consists of all real numbers ecept 5 and 5. Answer: 3, where ± Adding and Subtracting Rational Epressions 40

15 KEY TAKEAWAYS When adding or subtracting rational epressions with a common denominator, add or subtract the epressions in the numerator and write the result over the common denominator. To find equivalent rational epressions with a common denominator, first factor all denominators and determine the least common multiple. Then multiply numerator and denominator of each term by the appropriate factor to obtain a common denominator. Finally, add or subtract the epressions in the numerator and write the result over the common denominator. The restrictions to the domain of a sum or difference of rational functions consist of the restrictions to the domains of each function. 7.3 Adding and Subtracting Rational Epressions 4

16 TOPIC EXERCISES Part A: Adding and Subtracting with Common Denominators Simplify. (Assume all denominators are nonzero.) y 3 y y+2 2y+3 y+3 2y y 2y 9 3y 3 5y 3y 2. 3y+2 5y 0 + y+7 5y 0 3y+4 5y 0 3. (+)( 3) 3 (+)( 3) (2 )( 6) (2 )( 6) 7.3 Adding and Subtracting Rational Epressions 42

17 Part B: Adding and Subtracting with Unlike Denominators Simplify. (Assume all denominators are nonzero.) y 2 0y y 23. y y y y Adding and Subtracting Rational Epressions 43

18 y+ y + y y y y+4 3y y y + y (+3)( 3) ( ) (3 )(3+) ( 2) 6 3 (2+)( 5) ( 2) (+6)( 6) (+5)( 7) 7.3 Adding and Subtracting Rational Epressions 44

19 (4 ) y+ y 2y 2 +5y 3 4y 2 y y y 2 0y Adding and Subtracting Rational Epressions 45

20 a 4 a + a2 9a+8 a 2 3a+36 3a 2 a+2 a 2 8a+6 4 a a a 2 7a 4 +2a (+) (+5) (+5) y+ + y + 2 y y y+ + y y y 77. (2 ) ( 4) ( + ) 7.3 Adding and Subtracting Rational Epressions 46

21 ( ) (y ) 2 (y ) Part C: Adding and Subtracting Rational Functions Calculate (f + g) () and (f g) () and state the restrictions to the domain. 8. f () = f () = 83. f () = f () = 5 and g () = 2 and g () = +5 and g () = 4 and g () = f () = 2 4 and g () = f () = 5 +2 and g () = Calculate (f + f ) () and state the restrictions to the domain. 87. f () = 88. f () = f () = f () = +2 Part D: Discussion Board 9. Eplain to a classmate why this is incorrect: + 2 = Adding and Subtracting Rational Epressions 47

22 92. Eplain to a classmate how to find the common denominator when adding algebraic epressions. Give an eample. 7.3 Adding and Subtracting Rational Epressions 48

23 ANSWERS : 0 3: 3 y 5: 7: 9: : y y 3: 5: + 6 7: : : 5y+8 60y 3 23: 2y y 25: 2 (+5) +4 27: 4+ (+) 29: 2(+) ( 3)(+5) 7.3 Adding and Subtracting Rational Epressions 49

24 3: ( 2)(+) 33: 2(y 2 +) (y+)(y ) 35: 40 (2+5)(2 5) 37: 3(+2) 8 39: : : ( 2) 45: : (+5)( 5) 49: 5 8 5: 53: 55: 57: 2 ( ) ( 4) (+2)(3 ) ( 5)(+3) 59: y 2 8y 5 (y+5)(y 5) Adding and Subtracting Rational Epressions 50

25 6: : a+5 a 4 65: 6 (+3)( 3) 2 67: : 5 4 7: 2y y(y ) 73: : +y y 77: ( ) 2 2 (2 ) 79: (+2) 8: (f + g) () = 2(2 ) 3( 2) ; (f g) () = 2(+) 3( 2) ; 0, 2 83: (f + g) () = 4 ; (f g) () = + 4 ; 4 85: (f + g) () = 2, 2, 8 ( 5) (+2)( 2)( 8) ; (f g) () = (+2)( 2)( 8) ; 87: (f + f ) () = 2 ; 0 89: (f + f ) () = 2 2 ; Adding and Subtracting Rational Epressions 5