1. 6 RATIOnAl expressions

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1 58 CHAPTER 1 Prerequisites learning ObjeCTIveS In this section, ou will: Simplif rational epressions. Multipl rational epressions. Divide rational epressions. Add and subtract rational epressions. Simplif comple rational epressions RATIOnAl epressions A pastr shop has fied costs of $280 per week and variable costs of $9 per bo of pastries. The shop s costs per week in terms of, the number of boes made, is We can divide the costs per week b the number of boes made to determine the cost per bo of pastries Notice that the result is a polnomial epression divided b a second polnomial epression. In this section, we will eplore quotients of polnomial epressions. Simplifing Rational epressions The quotient of two polnomial epressions is called a rational epression. We can appl the properties of fractions to rational epressions, such as simplifing the epressions b canceling common factors from the numerator and the denominator. To do this, we first need to factor both the numerator and denominator. Let s start with the rational epression shown We can factor the numerator and denominator to rewrite the epression. ( + 4) 2 ( + 4)( + 7) Then we can simplif that epression b canceling the common factor ( + 4) How To Given a rational epression, simplif it. 1. Factor the numerator and denominator. 2. Cancel an common factors. Eample 1 Simplifing Rational Epressions Simplif Solution ( + 3)( 3) ( + 3)( + 1) Factor the numerator and the denominator. Cancel common factor ( + 3). Analsis We can cancel the common factor because an epression divided b itself is equal to 1. Q & A Can the 2 term be cancelled in Eample 1? No. A factor is an epression that is multiplied b another epression. The 2 term is not a factor of the numerator or the denominator.

2 SECTION 1.6 rational epressions 5 9 Tr It #1 Simplif multipling Rational epressions Multiplication of rational epressions works the same wa as multiplication of an other fractions. We multipl the numerators to find the numerator of the product, and then multipl the denominators to find the denominator of the product. Before multipling, it is helpful to factor the numerators and denominators just as we did when simplifing rational epressions. We are often able to simplif the product of rational epressions. How To Given two rational epressions, multipl them. 1. Factor the numerator and denominator. 2. Multipl the numerators. 3. Multipl the denominators. 4. Simplif. Eample 2 Multipling Rational Epressions Multipl the rational epressions and show the product in simplest form: Solution ( + 5)( 1) (2 1) 3( + 6) ( + 5) Factor the numerator and denominator. ( + 5)( 1)(2 1) 3( + 6)( + 5) ( + 5) ( 1)(2 1) 3( + 6) ( + 5) Multipl numerators and denominators. Cancel common factors to simplif. ( 1)(2 1) 3( + 6) Tr It #2 Multipl the rational epressions and show the product in simplest form: dividing Rational epressions Division of rational epressions works the same wa as division of other fractions. To divide a rational epression b another rational epression, multipl the first epression b the reciprocal of the second. Using this approach, we would rewrite 1 2 as the product Once the division epression has been rewritten as a multiplication 2 epression, we can multipl as we did before. 1 3 = 3 2 3

3 60 CHAPTER 1 Prerequisites How To Given two rational epressions, divide them. 1. Rewrite as the first rational epression multiplied b the reciprocal of the second. 2. Factor the numerators and denominators. 3. Multipl the numerators. 4. Multipl the denominators. 5. Simplif. Eample 3 Dividing Rational Epressions Divide the rational epressions and epress the quotient in simplest form: Solution Tr It # (2 3)( + 2) ( + 1)( 1) ( + 1)( + 1) ( + 2)( 2) (2 3)( + 2)( + 1)( + 1) ( + 1)( 1)( + 2)( 2) (2 3)( + 2)( + 1)( + 1) ( + 1)( 1)( + 2)( 2) (2 3)( + 1) ( 1)( 2) Divide the rational epressions and epress the quotient in simplest form: Adding and Subtracting Rational epressions Rewrite as multiplication. Factor the numerator and denominator. Multipl numerators and denominators. Cancel common factors to simplif. Adding and subtracting rational epressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let s look at an eample of fraction addition = = = 7 30 We have to rewrite the fractions so the share a common denominator before we are able to add.we must do the same thing when adding or subtracting rational epressions. The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational epressions, we factor the epressions and multipl all of the distinct factors. For instance, if the factored denominators were ( + 3)( + 4) and ( + 4)( + 5), then the LCD would be ( + 3)( + 4)( + 5). Once we find the LCD, we need to multipl each epression b the form of 1 that will change the denominator to the LCD. We would need to multipl the epression with a denominator of ( + 3)( + 4) b and the epression with a denominator of ( + 3)( + 4) b

4 How To Given two rational epressions, add or subtract them. SECTION 1.6 rational epressions Factor the numerator and denominator. 2. Find the LCD of the epressions. 3. Multipl the epressions b a form of 1 that changes the denominators to the LCD. 4. Add or subtract the numerators. 5. Simplif. Eample 4 Add the rational epressions: Adding Rational Epressions Solution First, we have to find the LCD. In this case, the LCD will be. We then multipl each epression b the appropriate form of 1 to obtain as the denominator for each fraction Now that the epressions have the same denominator, we simpl add the numerators to find the sum Analsis Multipling b or does not change the value of the original epression because an number divided b itself is 1, and multipling an epression b 1 gives the original epression. Eample 5 Subtract the rational epressions: Solution Subtracting Rational Epressions 6 ( + 2) 2 2 ( + 2)( 2) Factor. 6 ( + 2) ( + 2)( 2) ( 2) ( + 2) 2 ( 2) 2( + 2) ( + 2) 2 ( 2) 6 12 (2 + 4) ( + 2) 2 ( 2) 4 16 ( + 2) 2 ( 2) 4( 4) ( + 2) 2 ( 2) Multipl each fraction to get LCD as denominator. Multipl. Appl distributive propert. Subtract. Simplif. Q & A Do we have to use the LCD to add or subtract rational epressions? No. An common denominator will work, but it is easiest to use the LCD.

5 62 CHAPTER 1 Prerequisites Tr It #4 Subtract the rational epressions: Simplifing Comple Rational epressions A comple rational epression is a rational epression that contains additional rational epressions in the numerator, the denominator, or both. We can simplif comple rational epressions b rewriting the numerator and denominator as single rational epressions and dividing. The comple rational epression a can be simplified b rewriting the 1 b + c numerator as the fraction a and combining the epressions in the denominator as 1 + bc. We can then rewrite the 1 b epression as a multiplication problem using the reciprocal of the denominator. We get a b, which is equal bc to ab 1 + bc. How To Given a comple rational epression, simplif it. 1. Combine the epressions in the numerator into a single rational epression b adding or subtracting. 2. Combine the epressions in the denominator into a single rational epression b adding or subtracting. 3. Rewrite as the numerator divided b the denominator. 4. Rewrite as multiplication. 5. Multipl. 6. Simplif. Eample 6 Simplifing Comple Rational Epressions Simplif: + 1. Solution Begin b combining the epressions in the numerator into one epression Multipl b to get LCD as denominator. Add numerators. Now the numerator is a single rational epression and the denominator is a single rational epression. We can rewrite this as division, and then multiplication Rewrite as multiplication. + 1 ( + 1) 2 Multipl.

6 SECTION 1.6 rational epressions 6 3 Tr It #5 Simplif: Q & A Can a comple rational epression alwas be simplified? Yes. We can alwas rewrite a comple rational epression as a simplified rational epression. Access these online resources for additional instruction and practice with rational epressions. Simplif Rational epressions ( multipl and divide Rational epressions ( Add and Subtract Rational epressions ( Simplif a Comple Fraction (

7 64 CHAPTER 1 Prerequisites 1.6 SeCTIOn eercises verbal 1. How can ou use factoring to simplif rational epressions? 3. Tell whether the following statement is true or false and eplain wh: You onl need to find the LCD when adding or subtracting rational epressions. 2. How do ou use the LCD to combine two rational epressions? AlgebRAIC For the following eercises, simplif the rational epressions a2 24a a b2 + 18b + 9 3b m 12 m a2 + 9a + 18 a 2 + 3a c2 + 25c 18 3c 2 23c n2 29n 8 28n 2 5n 3 For the following eercises, multipl the rational epressions and epress the product in simplest form c2 + 2c 24 c c + 36 c2 10c + 24 c 2 8c d 2 + 9d 35 d d d 2 + 2d 21 3d d b2 + 13b + 6 4b 2 9 6b2 + 31b 30 18b 2 3b n2 n 15 6n n 5 12n2 13n + 3 4n 2 15n h2 9h 9 2h 2 19h + 24 h2 16h h 2 37h d d d d 2 15d d 2 1 t t 2 + 4t + 3 t 2 + 2t 15 t 2 4t For the following eercises, divide the rational epressions p 2 + p 12 8p p + 9 6p 2 11p + 4 2p p 6 q q 2 + 6q + 9 q2 2q 3 q 2 + 2q a2 24a + 9 4a a 15 16a 2 9 4a a d d 18 27d 2 15d + 2 3d d 44 9d 2 15d b b 2 6b 10 18b2 21b b 2 18b

8 SECTION 1.6 section eercises 65 For the following eercises, add and subtract the rational epressions, and then simplif q 6 3p a a c c 4 4 3z 39. z z + 5 z p 40. p + 1 p + 1 4p For the following eercises, simplif the rational epression a + 7 b b p 8 p a + b 6 2b 3a a b b a a + b ab c 3 7 c c 1 c c + 1 c ReAl-WORld APPlICATIOnS 51. Brenda is placing tile on her bathroom floor. The area of the floor is ft 2. The area of one tile is ft 2. To find the number of tiles needed, simplif the rational epression: Area = The area of Sand s ard is ft 2. A patch of sod has an area of ft 2. Divide the two areas and simplif to find how man pieces of sod Sand needs to cover her ard. 53. Aaron wants to mulch his garden. His garden is ft 2. One bag of mulch covers 2 81 ft 2. Divide the epressions and simplif to find how man bags of mulch Aaron needs to mulch his garden. etensions For the following eercises, perform the given operations and simplif a + 1 2a 3 + 2a 3 2a + 3 4a a