Multiplying and Dividing Rational Expressions

Transcription

1 6.3 Multiplying and Dividing Rational Epressions Essential Question How can you determine the ecluded values in a product or quotient of two rational epressions? You can multiply and divide rational epressions in much the same way that you multiply and divide fractions. Values that make the denominator of an epression zero are ecluded values. + = +, 0 + = + = + 2, Product of rational epressions Quotient of rational epressions Multiplying and Dividing Rational Epressions REASONING ABSTRACTLY To be proficient in math, you need to know and fleibly use different properties of operations and objects. Work with a partner. Find the product or quotient of the two rational epressions. Then match the product or quotient with its ecluded values. Eplain your reasoning. Product or Quotient Ecluded Values a. b. c d e. f. g h. + 2 = = A., 0, and 2 = B. 2 and = C. 2, 0, and = D. and 2 = E., 0, and = F. and + = H. G. 2 and Writing a Product or Quotient Work with a partner. Write a product or quotient of rational epressions that has the given ecluded values. Justify your answer. a. b. and 3 c., 0, and 3 Communicate Your Answer 3. How can you determine the ecluded values in a product or quotient of two rational epressions?. Is it possible for the product or quotient of two rational epressions to have no ecluded values? Eplain your reasoning. If it is possible, give an eample. Section 6.3 Multiplying and Dividing Rational Epressions 323

2 6.3 Lesson What You Will Learn Core Vocabulary rational epression, p. 32 simplified form of a rational epression, p. 32 Previous fractions polynomials domain equivalent epressions reciprocal Simplify rational epressions. Multiply rational epressions. Divide rational epressions. Simplifying Rational Epressions A rational epression is a fraction whose numerator and denominator are nonzero polynomials. The domain of a rational epression ecludes values that make the denominator zero. A rational epression is in simplified form when its numerator and denominator have no common factors (other than ±). Core Concept Simplifying Rational Epressions Let a, b, and c be epressions with b 0 and c 0. Property ac bc = a b Divide out common factor c. STUDY TIP Notice that you can divide out common factors in the second epression at the right. You cannot, however, divide out like terms in the third epression. Eamples 5 65 = = 3 3 ( + 3) ( + 3)( + 3) = + 3 Simplifying a rational epression usually requires two steps. First, factor the numerator and denominator. Then, divide out any factors that are common to both the numerator and denominator. Here is an eample: = ( + 7) Divide out common factor 5. Divide out common factor + 3. = + 7 Simplifying a Rational Epression Simplify COMMON ERROR Do not divide out variable terms that are not factors ( + 2)( 6) = ( + 2)( 2) ( + 2)( 6) = ( + 2)( 2) Factor numerator and denominator. Divide out common factor. = 6, 2 Simplified form 2 The original epression is undefined when = 2. To make the original and simplified epressions equivalent, restrict the domain of the simplified epression by ecluding = 2. Both epressions are undefined when = 2, so it is not necessary to list it. 32 Chapter 6 Rational Functions Monitoring Progress Simplify the rational epression, if possible.. 2( + ) ( + )( + 3) Help in English and Spanish at BigIdeasMath.com 3. ( + 2)

3 ANOTHER WAY In Eample 2, you can first simplify each rational epression, then multiply, and finally simplify the result. 8 3 y 2y 7 y 3 2 y = 2 y 7 y 2 = 7 6 y y y = 7 6 y, 0, y 0 Multiplying Rational Epressions The rule for multiplying rational epressions is the same as the rule for multiplying numerical fractions: multiply numerators, multiply denominators, and write the new fraction in simplified form. Similar to rational numbers, rational epressions are closed under multiplication. Core Concept Multiplying Rational Epressions Let a, b, c, and d be epressions with b 0 and d 0. Property Eample Find the product 83 y 2y 7 y 3 2 y. a b c d = ac bd 5 2 2y 6y3 2 0y = 303 y y 2y 7 y 3 2 y = 567 y 8y 3 Multiplying Rational Epressions = y 3 y 8 y Simplify ac if possible. bd 20y 3 = y y 3 = 32 Multiply numerators and denominators. 3 Factor and divide out common factors. = 7 6 y, 0, y 0 Simplified form 2, 0, y 0 Multiplying Rational Epressions 3 3 Find the product ( ) ( + 5)( ) = ( )( + 5) 3 Factor numerators and denominators. 3( )( + 5)( ) = ( )( + 5)(3) Multiply numerators and denominators. 3( )( )( + 5)( ) = ( )( + 5)(3) Rewrite as ( )( ). Check X X=- Y ERROR ERROR ERROR Y2 3( )( )( + 5)( ) = ( )( + 5)(3) Divide out common factors. = +, 5, 0, Simplified form Check the simplified epression. Enter the original epression as y and the simplified epression as y 2 in a graphing calculator. Then use the table feature to compare the values of the two epressions. The values of y and y 2 are the same, ecept when = 5, = 0, and =. So, when these values are ecluded from the domain of the simplified epression, it is equivalent to the original epression. Section 6.3 Multiplying and Dividing Rational Epressions 325

4 Multiplying a Rational Epression by a Polynomial STUDY TIP Notice that does not equal zero for any real value of. So, no values must be ecluded from the domain to make the simplified form equivalent to the original. Find the product ( ) ( ) = = ( + 2)( ) ( 3)( ) = ( + 2)( ) ( 3)( ) = Write polynomial as a rational epression. Multiply. Factor denominator. Divide out common factor. Simplified form Monitoring Progress Find the product y 2 8y 6y2 9 3 y Help in English and Spanish at BigIdeasMath.com (2 + + ) Dividing Rational Epressions To divide one rational epression by another, multiply the first rational epression by the reciprocal of the second rational epression. Rational epressions are closed under nonzero division. Core Concept Dividing Rational Epressions Let a, b, c, and d be epressions with b 0, c 0, and d 0. Property a b c d = a b d c = ad bc Simplify ad if possible. bc Eample = = 7(2 3) ( + )( + 2), 3 2 Dividing Rational Epressions 7 Find the quotient = ( 5)( 6) = 2( 5) ( 6) 7( 5)( 6) = 2( 5)()( 6) Multiply by reciprocal. Factor. Multiply. Divide out common factors. 326 Chapter 6 Rational Functions = 7, 0, 5, 6 Simplified form 2

5 Dividing a Rational Epression by a Polynomial Find the quotient (32 + 5) (32 + 5) = (3 + 5)(2 3) = 2 (3 + 5) Factor. (3 + 5)(2 3) = 2 ()(3 + 5) = 2 3 3, 5 3 Solving a Real-Life Problem Multiply by reciprocal. Multiply. Divide out common factor. Simplified form The total annual amount I (in millions of dollars) of personal income earned in Alabama and its annual population P (in millions) can be modeled by 6922t + 06,97 I = t + and P = 0.033t +.32 where t represents the year, with t = corresponding to 200. Find a model M for the annual per capita income. (Per capita means per person.) Estimate the per capita income in 200. (Assume t > 0.) To find a model M for the annual per capita income, divide the total amount I by the population P. 6922t + 06,97 M = (0.033t +.32) Divide I by P t t + 06,97 = t t t + 06,97 = (0.0063t + )(0.033t +.32) Multiply by reciprocal. Multiply. To estimate Alabama s per capita income in 200, let t = 0 in the model M = ,97 ( )( ) 3,707 Substitute 0 for t. Use a calculator. In 200, the per capita income in Alabama was about $3,707. Monitoring Progress Find the quotient Help in English and Spanish at BigIdeasMath.com ( ) Section 6.3 Multiplying and Dividing Rational Epressions 327

6 6.3 Eercises Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check. WRITING Describe how to multiply and divide two rational epressions. 2. WHICH ONE DOESN T BELONG? Which rational epression does not belong with the other three? Eplain your reasoning Monitoring Progress and Modeling with Mathematics In Eercises 3 0, simplify the epression, if possible. (See Eample.) ( ) 2. ERROR ANALYSIS Describe and correct the error in simplifying the rational epression = In Eercises 20, find the product. (See Eamples 2, 3, and.). y 3 2 y y y 3 y 2 y 6 3 y ERROR ANALYSIS Describe and correct the error in finding the product ( + 5)( 5) = ( + 5)( 5)( 3) = (3 )( + 5) = 5, 3, ( ) 3 3 ( + 5) 9 ( 3)( + 6) 3 ( 9)( + 8) (2 + 28) USING STRUCTURE Which rational epression is in simplified form? A C B D COMPARING METHODS Find the product below by multiplying the numerators and denominators, then simplifying. Then find the product by simplifying each epression, then multiplying. Which method do you prefer? Eplain. 2 y 2 2y Chapter 6 Rational Functions

7 25. WRITING Compare the function (3 7)( + 6) f() = to the function g() = + 6. (3 7) 26. MODELING WITH MATHEMATICS Write a model in terms of for the total area of the base of the building In Eercises 27 3, find the quotient. (See Eamples 5 and 6.) y y 8 y yz 3 z 3 6y 2 2 z ( ) ( ) In Eercises 35 and 36, use the following information. Manufacturers often package products in a way that uses the least amount of material. One measure of the effi ciency of a package is the ratio of its surface area S to its volume V. The smaller the ratio, the more effi cient the packaging. 36. A popcorn company is designing a new tin with the same square base and twice the height of the old tin. a. Write an epression for the efficiency ratio S V of each tin. b. Did the company make a good decision by creating the new tin? Eplain. 37. MODELING WITH MATHEMATICS The total amount I (in millions of dollars) of healthcare ependitures and the residential population P (in millions) in the United States can be modeled by 7,000t +,36,000 I = and t P = 2.96t where t is the number of years since Find a model M for the annual healthcare ependitures per resident. Estimate the annual healthcare ependitures per resident in 200. (See Eample 7.) 38. MODELING WITH MATHEMATICS The total amount I (in millions of dollars) of school ependitures from prekindergarten to a college level and the enrollment P (in millions) in prekindergarten through college in the United States can be modeled by 7,93t + 709,569 I = and P = t t where t is the number of years since 200. Find a model M for the annual education ependitures per student. Estimate the annual education ependitures per student in s s h s s 2h 35. You are eamining three cylindrical containers. a. Write an epression for the efficiency ratio S V of a cylinder. b. Find the efficiency ratio for each cylindrical can listed in the table. Rank the three cans according to efficiency. Soup Coffee Paint Height, h 0.2 cm 5.9 cm 9. cm Radius, r 3. cm 7.8 cm 8. cm 39. USING EQUATIONS Refer to the population model P in Eercise 37. a. Interpret the meaning of the coefficient of t. b. Interpret the meaning of the constant term. Section 6.3 Multiplying and Dividing Rational Epressions 329

8 0. HOW DO YOU SEE IT? Use the graphs of f and g to determine the ecluded values of the functions h() = ( fg)() and k() = ( g) f (). Eplain your reasoning. y f 6 y g. CRITICAL THINKING Find the epression that makes the following statement true. Assume 2 and = USING STRUCTURE In Eercises 5 and 6, perform the indicated operations (6 + 9) 3 2. DRAWING CONCLUSIONS Complete the table for the function y = + 2. Then use the trace feature of 6 a graphing calculator to eplain the behavior of the function at = MAKING AN ARGUMENT You and your friend are asked to state the domain of the epression below Your friend claims the domain is all real numbers ecept 5. You claim the domain is all real numbers ecept 9 and 5. Who is correct? Eplain. 3. MATHEMATICAL CONNECTIONS Find the ratio of the perimeter to the area of the triangle shown. 8 6 y 5 Maintaining Mathematical Proficiency Solve the equation. Check your solution. (Skills Review Handbook) ( 3 + 8) REASONING Animals that live in temperatures several degrees colder than their bodies must avoid losing heat to survive. Animals can better conserve body heat as their surface area to volume ratios decrease. Find the surface area to volume ratio of each penguin shown by using cylinders to approimate their shapes. Which penguin is better equipped to live in a colder environment? Eplain your reasoning. Galapagos Penguin 53 cm radius = 6 cm King Penguin 9 cm radius = cm Not drawn to scale 8. THOUGHT PROVOKING Is it possible to write two radical functions whose product when graphed is a parabola and whose quotient when graphed is a hyperbola? Justify your answer. 9. REASONING Find two rational functions f and g that have the stated product and quotient. g) (fg)() = 2, ( f () = ( )2 ( + 2) 2 Reviewing what you learned in previous grades and lessons + = = = = 3 5 Write the prime factorization of the number. If the number is prime, then write prime. (Skills Review Handbook) Chapter 6 Rational Functions