Simplifying a Rational Expression. Factor the numerator and the denominator. = 1(x 2 6)(x 2 1) Divide out the common factor x 6. Simplify.

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1 - Plan Lesson Preview Check Skills You ll Need Factoring ± ± c Lesson -5: Eamples Eercises Etra Practice, p 70 Lesson Preview What You ll Learn BJECTIVE - To simplify rational epressions And Why To find the aking time for read, as in Eample ( ± )( ) ing Rational Epressions Check Skills You ll Need (For help, go to Skills Handook page 7 and Lesson -5 ) Write each fraction in simplest form Factor each quadratic epression ( ± )( ± ) ( 5)( ± ) See left ( ± ) ( ± )( ) ( )( ) Lesson Resources New Vocaulary rational epression Teaching Resources Practice, Reteaching, Enrichment Checkpoint Quiz BJECTIVE Part ing Rational Epressions Interactive lesson includes instant self-check, tutorials, and activities Practice Workook - Spanish Practice Workook - Reading and Math Literacy B Spanish Reading & Literacy B Spanish Checkpoint Quiz Basic Algera Planning Guide - Presentation Assistant Plus! Transparencies Check Skills You ll Need - Additional Eamples - Student Edition Answers - Lesson Quiz - PH Presentation Pro CD - Checkpoint Quiz Computer Test Generator CD Technology Resource Pro CD-RM Computer Test Generator CD Prentice Hall Presentation Pro CD wwwphschoolcom Student Site Teacher We Code: aek-5500 Reasoning & Puzzles pp, Self-grading Lesson Quiz Teacher Center Lesson Planner Resources Plus 5 Check Understanding 5 Chapter Rational Epressions and Functions Before the Lesson Diagnose prerequisite skills using: Check Skills You ll Need 5 7 Fractions like,, and are rational numers An epression with a variale in the denominator is a rational epression Here are some eamples of rational epressions f course, the value of the epression in the denominator cannot e zero, since division y zero is undefined For the rest of this chapter, assume that the values of the variales that make the denominator zero are ecluded from the domain Like rational numers, a rational epression is in simplest form if the numerator z 5 and denominator have no common factors ecept For eample, 0z is in simplest form since neither 0 nor z is a factor of z ing a Rational Epression ( ) Factor the numerator The denominator cannot e factored ( ) Divide out the common factor ± each epression a 5 c c c m d 0 t 5 c c m t 5 5 Recall that you learned to factor quadratic epressions in Lessons -5 and - You may need to factor a quadratic epression to simplify a rational epression ngoing Assessment and Intervention During the Lesson Monitor progress using: Check Understanding Additional Eamples Standardized Test Prep After the Lesson Assess knowledge using: Lesson Quiz Computer Test Generator CD Chapter Checkpoint (p 5)

2 Check Understanding ing a Rational Epression 7 ( ) ( )( ) Factor the numerator and the denominator 7 ( ) ( )( ) Divide out the common factor each epression a z c a d c c 0 z z a 5a c 5c c 5 z a c The numerator and denominator of are opposites To simplify the epression, you can factor - from - to get -(- + ), which you can rewrite as -( - ) Then simplify ( ) Teach Math Background When the numerator and denominator of a rational epression have a common factor, such as in Eample, the graph will have a hole, or discontinuity In this eample there is no y value when This hole may not e visile with some graphing programs in some windows BJECTIVE Teaching Notes Check Understanding Real-World Connection For a given volume of dough, the greater the surface area is, the shorter the aking time Recognizing pposite Factors ( ) ( )( ) Factor the numerator and the denominator 5( ) ( )( ) Factor from 5( ) ( )( ) Divide out the common factor 5 each epression a m c r d c m m r r c r You can use a rational epression to model some real-world situations Evaluating a Rational Epression Baking The aking time for read depends, in part, on its size and shape A good approimation for the aking time, in minutes, of a cylindrical loaf is 0? volume 0rh surface area, or r h, where the radius r and the length h of the aked loaf are in inches Find the aking time for a loaf that is 5 inches long and has a radius of inches Round your answer to the nearest minute 0rh 0()(5) r h 5 Sustitute for r and 5 for h 00 < 7 Round to the nearest whole numer The aking time is approimately 7 minutes Lesson - ing Rational Epressions 5 Math Tip Remind students that they must factor the numerator and the denominator in order to find the GCF Stress that factors, not terms, must e divided out Teaching Tip Students may e confused y using h for length Tell them to look at the length as the height of the cylinder Additional Eamples Find the aking time for a loaf of read that is inches long and has a radius of inches Round your answer to the nearest minute 5 minutes Closure Ask students to eplain how to simplify a rational epression If possile, factor the numerator and denominator Remove common factors and then simplify Below Level Students may e confused y opposite factors Suggest that they use the Distriutive Property to show ( - ) -( - ) Advanced Learners Ask students to justify the technique of dividing out a common factor A factor divided y itself equals, and does not need to e written as a factor Error Prevention See note on page 5 5

3 Practice Assignment Guide jective A B Core C Etension 7 Standardized Test Prep 5 Mied Review 5 Error Prevention Eercises When only one factor is left in the denominator, some students ring that factor into the numerator Have students put a aove and elow factors that were divided into the numerator and denominator to help them to see the answer is over the remaining factor Connection To History Eercises In March, in Johannesurg, South Africa, a team aked the world s largest pan loaf The loaf weighed l 0 oz and had dimensions of ft 0 in y ft in y ft 7 in Sketch the rectangular loaf and lael it with the dimensions Enrichment - Reteaching - Practice - Name Class Date Practice - ing Rational Epressions Check Understanding EXERCISES Practice and Prolem Solving A Practice y Eample Eample (page 5) Eample (page 5) Eample (page 5) Eample (page 5) a Find the aking time for a loaf that is inches long and has a radius of inches Round your answer to the nearest minute 5 min 0? volume 0pr Critical Thinking The ratio for a cylinder is h surface area this pr prh epression to show that it is the same as the epression evaluated in Eample 0rh r h each epression a a m 5 7 m 5 p 5 p 7 m m m m 7 0 w 7w w a a a m 7m m w w 7 5a 5 5 m m m c c c 5 ± 5 m c c c 5 m m m 5 n 7 t m n 5 t m t t m 0 v 5 w m v 5 5 v w w w Baking Use the epression 0rh r h to estimate the aking time in minutes for each type of read Round your answer to the nearest minute aguette: r 5 in, h in pita: r 5 in, h 05 in min min iscuit: r in, h 075 in min For more practice, see Etra Practice each epression 5a h n 5a h n 5 7 t t a a t a a Pearson Education, Inc All rights reserved y y 0 y y y 7 y 0 5 y 0 y perimeter of rectangle 7 Write and simplify the ratio The perimeter of area of rectangle the rectangle is 0w and the area of the rectangle is w The ratio? volume of cone determines the height of a cone Find the area of ase height when the volume is r + r and the area of the ase is r? area of triangle The ratio determines the length of the ase of a height of triangle triangle Find the length of the ase when the area is n + n and the height is n + volume of rectangular solid 0 The ratio determines the height of a area of rectangular ase rectangular solid Find the height when the volume is 5s + 0s and the area is 5s B Apply Your Skills each epression 5 r r 5 r 7z z 7z 7 r 0r 5 r 5 z z z a a z (z ) z 0 a a a z z a m a a 5 m m a m 7 5 a a m m 5t t t 5t s s s s s 0c c c 5c c 5t t c(c 5) 5c pen-ended Write an epression that has and - ecluded from the domain Answers may vary Sample: ( )( ) 5 Chapter Rational Epressions and Functions Algera Chapter Lesson - Practice 5

4 Real-World Connection A cylinder with a height of 500 ft and a volume of 0 ft has aout % of the surface area of a square prism with the same height and volume The student canceled terms instead of factors C m n m 0n a a v 7w v w Challenge 5 a Construction To keep heating costs down for a structure, architects want the ratio of surface area to volume as small as possile Find an epression for the ratio of the surface area to volume for each shape h h r i square prism h ii cylinder rh Find the ratio for each figure when ft, h ft, and r ft ; Error Analysis Eplain what error the student made in simplifying the rational epression at the right See left 7 Writing Eplain why is not the same as - is not in the domain of Proaility If a point is selected at random from a figure and is equally likely to e any point in the figure, then the proaility that the point is in a shaded part of the area of shaded part figure is area of whole figure Find the proaility that a point will e in the shaded part of each figure 5w 5w 5w y 0 (y ) y 5w h each epression See left y m n a 5a v w m mn 0n a a v vw w Reasoning Determine whether each statement is sometimes, always, or never true for real numers a and 5 sometimes a a sometimes 7 a a 5 a never r t h t t (t ) a 5 Assess Lesson Quiz - each epression 5 0 Alternative Assessment Write rational epressions on individual inde cards Make enough for half of the class Write the simplifications of the rational epressions on other inde cards for the other half of the class Have students find matching cards Shuffle the cards and repeat Standardized Test Prep Resources For additional practice with a variety of test item formats: Standardized Test Prep, pp 7 Test-Taking Strategies, p Test-Taking Strategies with Transparencies Eercises 5, 5 Encourage students to simplify the epressions as much as possile efore comparing them Standardized Standardized Test Prep Test Prep Multiple Choice Which epression simplifies to -? C A B r C n D p r n p Lesson - ing Rational Epressions 55 55

5 Chapter Checkpoint To check understanding of Lessons - to -: Checkpoint Quiz (p 5) Teaching Resources Checkpoint Quiz (also in Prentice Hall Assessment System) Reading and Math Literacy B Spanish versions availale pages Eercises 5 [] The student put the in the numerator rather than in the denominator [] no eplanation R no correctly simplified epression 5 vertical asymptote: 0 y ; h() 55 vertical asymptote: y 0; f() vertical asymptote: 0 y ; g() 5 ( 5) Quantitative Comparison Take It to the NET nline lesson quiz at wwwphschoolcom We Code: aea-0 Short Response Mied Review Lesson - 5 Chapter Rational Epressions and Functions D C y y I y y y F G H y I 50 Which epression is in simplest form? B A t B n C c 7 D r t n 7 c r Compare the quantity in Column A with the quantity in Column B Choose the est answer A The quantity in Column A is greater B The quantity in Column B is greater C The two quantities are equal D The relationship cannot e determined from the information given Column A Column B 5 5( 5) -0? 5,, y y 5 5 A student simplified 0 to What error did the student make? Show how to simplify the epression correctly See margin 5 5 See margin Identify the asymptotes of each function Then graph the function 5 h() + 55 f() 5 g() - Lesson - each radical epression!0 " 57!0?!0 5 "a 7 c 5 " 0 m!0 Ä5m 5m 0" a c " Lesson 0- rder each group of quadratic functions from widest to narrowest graph y, y, y, y, y - y, y, y 5 y y, y 05, y - y -, y, y - y, y, y 05, y, y y, y, y y 5 Checkpoint Quiz Lessons Checkpoint - through Quiz - Lessons - through - Instant self-check quiz online and on CD-RM Each pair of points is on the graph of an inverse variation Find the missing value (, ) and (, ) (, -) and (, y) (05, 7) and (0, y) Identify the asymptotes of each function Then graph the function See margin y 5 5 f() y + each epression 7 c ( ) c k c c k k k 0 pen-ended Write an epression that is not defined when the variale equals 5 Answers may vary Sample: page 5 Checkpoint Quiz 0, y 0; y 5, y 0; f() 0, y ; y