Chapter 7 Resource Masters

Transcription

1 Chapter 7 Resource Masters

2 Consumable Workbooks Man of the worksheets contained in the Chapter Resource Masters booklets are available as consumable workbooks. Stud Guide and Intervention Workbook X Skills Practice Workbook Practice Workbook ANSWERS FR WRKBKS The answers for Chapter 7 of these workbooks can be found in the back of this Chapter Resource Masters booklet. Glencoe/McGraw-Hill Copright b The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced onl for classroom use; be provided to students, teacher, and families without charge; and be used solel in conjunction with Glencoe s Algebra. An other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGraw-Hill Companies 8787 rion Place Columbus, H ISBN: Algebra Chapter 7 Resource Masters

3 Contents Vocabular Builder vii Lesson 7- Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 7- Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 7-3 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 7-4 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 7-5 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 7-6 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 7-7 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 7-8 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson 7-9 Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Chapter 7 Assessment Chapter 7 Test, Form Chapter 7 Test, Form A Chapter 7 Test, Form B Chapter 7 Test, Form C Chapter 7 Test, Form D Chapter 7 Test, Form Chapter 7 pen-ended Assessment Chapter 7 Vocabular Test/Review Chapter 7 Quizzes & Chapter 7 Quizzes 3 & Chapter 7 Mid-Chapter Test Chapter 7 Cumulative Review Chapter 7 Standardized Test Practice Unit Test/Review (Ch. 5 7) First Semester Test (Ch. 7) Standardized Test Practice Student Recording Sheet A ANSWERS A A40 Glencoe/McGraw-Hill iii Glencoe Algebra

4 Teacher s Guide to Using the Chapter 7 Resource Masters The Fast File Chapter Resource sstem allows ou to convenientl file the resources ou use most often. The Chapter 7 Resource Masters includes the core materials needed for Chapter 7. These materials include worksheets, etensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Algebra TeacherWorks CD-RM. Vocabular Builder Pages vii viii include a student stud tool that presents up to twent of the ke vocabular terms from the chapter. Students are to record definitions and/or eamples for each term. You ma suggest that students highlight or star the terms with which the are not familiar. WHEN T USE Give these pages to students before beginning Lesson 7-. Encourage them to add these pages to their Algebra Stud Notebook. Remind them to add definitions and eamples as the complete each lesson. Stud Guide and Intervention Each lesson in Algebra addresses two objectives. There is one Stud Guide and Intervention master for each objective. WHEN T USE Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Skills Practice There is one master for each lesson. These provide computational practice at a basic level. WHEN T USE These masters can be used with students who have weaker mathematics backgrounds or need additional reinforcement. Practice There is one master for each lesson. These problems more closel follow the structure of the Practice and Appl section of the Student Edition eercises. These eercises are of average difficult. WHEN T USE These provide additional practice options or ma be used as homework for second da teaching of the lesson. Reading to Learn Mathematics ne master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. Additional questions ask students to interpret the contet of and relationships among terms in the lesson. Finall, students are asked to summarize what the have learned using various representation techniques. WHEN T USE This master can be used as a stud tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. Enrichment There is one etension master for each lesson. These activities ma etend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students perspectives on the mathematics the are learning. These are not written eclusivel for honors students, but are accessible for use with all levels of students. WHEN T USE These ma be used as etra credit, short-term projects, or as activities for das when class periods are shortened. Glencoe/McGraw-Hill iv Glencoe Algebra

5 Assessment ptions The assessment masters in the Chapter 7 Resource Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. Chapter Assessment CHAPTER TESTS Form contains multiple-choice questions and is intended for use with basic level students. Forms A and B contain multiple-choice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Forms C and D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with aes are provided for questions assessing graphing skills. Form 3 is an advanced level test with free-response questions. Grids without aes are provided for questions assessing graphing skills. All of the above tests include a freeresponse Bonus question. The pen-ended Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. A Vocabular Test, suitable for all students, includes a list of the vocabular words in the chapter and ten questions assessing students knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet. Intermediate Assessment Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. A Mid-Chapter Test provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions. Continuing Assessment The Cumulative Review provides students an opportunit to reinforce and retain skills as the proceed through their stud of Algebra. It can also be used as a test. This master includes free-response questions. The Standardized Test Practice offers continuing review of algebra concepts in various formats, which ma appear on the standardized tests that the ma encounter. This practice includes multiplechoice, grid-in, and quantitativecomparison questions. Bubble-in and grid-in answer sections are provided on the master. Answers Page A is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages This improves students familiarit with the answer formats the ma encounter in test taking. The answers for the lesson-b-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer kes are provided for the assessment masters in this booklet. Glencoe/McGraw-Hill v Glencoe Algebra

6 7 NAME DATE PERID Reading to Learn Mathematics Vocabular Builder This is an alphabetical list of the ke vocabular terms ou will learn in Chapter 7. As ou stud the chapter, complete each term s definition or description. Remember to add the page number where ou found the term. Add these pages to our Algebra Stud Notebook to review vocabular at the end of the chapter. Vocabular Term composition of functions Found on Page Definition/Description/Eample Vocabular Builder depressed polnomial end behavior Factor Theorem Fundamental Theorem of Algebra inverse function inverse relation leading coefficients location principle one-to-one (continued on the net page) Glencoe/McGraw-Hill vii Glencoe Algebra

7 NAME DATE PERID 7 Reading to Learn Mathematics Vocabular Builder (continued) Vocabular Term polnomial function Found on Page Definition/Description/Eample polnomial in one variable power function quadratic form Rational Zero Theorem relative maimum relative minimum remainder theorem square root function snthetic substitution sihn THEH tihk Glencoe/McGraw-Hill viii Glencoe Algebra

8 7- NAME DATE PERID Stud Guide and Intervention Polnomial Functions Polnomial Functions A polnomial of degree n in one variable is an epression of the form Polnomial in a 0 n a n a n a n a n, ne Variable where the coefficients a 0, a, a,, a n represent real numbers, a 0 is not zero, and n represents a nonnegative integer. The degree of a polnomial in one variable is the greatest eponent of its variable. The leading coefficient is the coefficient of the term with the highest degree. A polnomial function of degree n can be described b an equation of the form Polnomial P() a 0 n a n a n a n a n, Function where the coefficients a 0, a, a,, a n represent real numbers, a 0 is not zero, and n represents a nonnegative integer. Lesson 7- Eample What are the degree and leading coefficient of ? Rewrite the epression so the powers of are in decreasing order This is a polnomial in one variable. The degree is 4, and the leading coefficient is. Eample Find f( 5) if f() f() riginal function f( 5) ( 5) 3 ( 5) 0( 5) 0 Replace with Evaluate. 5 Simplif. Eample 3 Find g(a ) if g() 3 4. g() 3 4 riginal function g(a ) (a ) 3(a ) 4 Replace with a. a 4 a 3a 3 4 Evaluate. a 4 a 6 Simplif. Eercises State the degree and leading coefficient of each polnomial in one variable. If it is not a polnomial in one variable, eplain wh. 8; ; ; not a polnomial in 5; 9 one variable; contains 6; two variables Find f() and f( 5) for each function. 7. f() 9 8. f() f() ; 6 3; ; 43 Glencoe/McGraw-Hill 375 Glencoe Algebra 6 3

9 7- NAME DATE PERID Stud Guide and Intervention (continued) Polnomial Functions Graphs of Polnomial Functions End Behavior of Polnomial Functions Real Zeros of a Polnomial Function If the degree is even and the leading coefficient is positive, then f() as f() as If the degree is even and the leading coefficient is negative, then f() as f() as If the degree is odd and the leading coefficient is positive, then f() as f() as If the degree is odd and the leading coefficient is negative, then f() as f() as The maimum number of zeros of a polnomial function is equal to the degree of the polnomial. A zero of a function is a point at which the graph intersects the -ais. n a graph, count the number of real zeros of the function b counting the number of times the graph crosses or touches the -ais. Eample Determine whether the graph represents an odd-degree polnomial or an even-degree polnomial. Then state the number of real zeros. f() As, f() and as, f(), so it is an odd-degree polnomial function. The graph intersects the -ais at point, so the function has real zero. Eercises Determine whether each graph represents an odd-degree polnomial or an evendegree polnomial. Then state the number of real zeros f() f() f() even; 6 even; double zero odd; 3 Glencoe/McGraw-Hill 376 Glencoe Algebra

10 7- NAME DATE PERID Skills Practice Polnomial Functions State the degree and leading coefficient of each polnomial in one variable. If it is not a polnomial in one variable, eplain wh.. a 8 ;. ( )(4 3) 3; ; ; 7 5. u 3 4u v v 4 6. r r No, this polnomial contains two variables, u and v. Find p( ) and p() for each function. No, this is not a polnomial because cannot be written in the form r n, r where n is a nonnegative integer. 7. p() 4 3 7; 8. p() 3 ; 0 r Lesson 7-9. p() 4 7; 0. p() ; 3. p() ; 38. p() 3 3; 3 If p() 4 3 and r() 3, find each value. 3. p(a) 4a 3 4. r(a) 6a 5. 3r(a) 3 9a 6. 4p(a) 6a 7. p(a ) 4a r( ) 7 3 For each graph, a. describe the end behavior, b. determine whether it represents an odd-degree or an even-degree polnomial function, and c. state the number of real zeroes f() f() f() f() as, f() as, f() as, f() as ; f() as ; f() as ; Glencoe/McGraw-Hill 377 Glencoe Algebra

11 7- State the degree and leading coefficient of each polnomial in one variable. If it is not a polnomial in one variable, eplain wh.. (3 )( 9) 4; 6. 5 a3 3 5 a 4 5 a 3; 5 m NAME DATE PERID Practice (Average) Polnomial Functions 3. 3m Not a polnomial; m cannot be written in the form No, this polnomial contains two m n for a nonnegative integer n. variables, and. Find p( ) and p(3) for each function. 5. p() p() p() ; 6 9; 39 0; p() p() p() ; 73 3; 93 6; 4 If p() 3 4 and r() 5, find each value.. p(8a). r(a ) 3. 5r(a) 9a 4 a 4 5a 40a 50a 5 4. r( ) 5. p( ) 6. 5[p( )] For each graph, a. describe the end behavior, b. determine whether it represents an odd-degree or an even-degree polnomial function, and c. state the number of real zeroes f() f() f() f() as, f() as, f() as, f() as ; f() as ; f() as ; even; even; odd; 5 0. WIND CHILL The function C(s) 0.03s s 7 estimates the wind chill temperature C(s) at 0 F for wind speeds s from 5 to 30 miles per hour. Estimate the wind chill temperature at 0 F if the wind speed is 0 miles per hour. about F Glencoe/McGraw-Hill 378 Glencoe Algebra

12 7- NAME DATE PERID Reading to Learn Mathematics Polnomial Functions Pre-Activit Where are polnomial functions found in nature? Read the introduction to Lesson 7- at the top of page 346 in our tetbook. In the honecomb cross section shown in our tetbook, there is heagon in the center, 6 heagons in the second ring, and heagons in the third ring. How man heagons will there be in the fourth, fifth, and sith rings? 8; 4; 30 Reading the Lesson There is heagon in a honecomb with ring. There are 7 heagons in a honecomb with rings. How man heagons are there in honecombs with 3 rings, 4 rings, 5 rings, and 6 rings? 9; 37; 6; 9. Give the degree and leading coefficient of each polnomial in one variable. degree leading coefficient a b c Lesson 7-. Match each description of a polnomial function from the list on the left with the corresponding end behavior from the list on the right. a. even degree, negative leading coefficient iii i. f() as ; f() as b. odd degree, positive leading coefficient iv ii. f() as ; f() as c. odd degree, negative leading coefficient ii iii. f() as ; f() as d. even degree, positive leading coefficient i iv. f() as ; f() as Helping You Remember 3. What is an eas wa to remember the difference between the end behavior of the graphs of even-degree and odd-degree polnomial functions? Sample answer: Both ends of the graph of an even-degree function eventuall keep going in the same direction. For odd-degree functions, the two ends eventuall head in opposite directions, one upward, the other downward. Glencoe/McGraw-Hill 379 Glencoe Algebra

13 7- NAME DATE PERID Enrichment Approimation b Means of Polnomials Man scientific eperiments produce pairs of numbers [, f()] that can be related b a formula. If the pairs form a function, ou can fit a polnomial to the pairs in eactl one wa. Consider the pairs given b the following table. 4 7 f () We will assume the polnomial is of degree three. Substitute the given values into this epression. f() A B( 0 ) C( 0 )( ) D( 0 )( )( ) You will get the sstem of equations shown below. You can solve this sstem and use the values for A, B, C, and D to find the desired polnomial. 6 A A B( ) A B 39 A B(4 ) C(4 )(4 ) A 3B 6C 54 A B(7 ) C(7 )(7 ) D(7 )(7 )(7 4) A 6B 30C 90D Solve.. Solve the sstem of equations for the values A, B, C, and D.. Find the polnomial that represents the four ordered pairs. Write our answer in the form a b c d Find the polnomial that gives the following values f () A scientist measured the volume f() of carbon dioide gas that can be absorbed b one cubic centimeter of charcoal at pressure. Find the values for A, B, C, and D f () Glencoe/McGraw-Hill 380 Glencoe Algebra

14 7- NAME DATE PERID Stud Guide and Intervention Graphing Polnomial Functions Graph Polnomial Functions Location Principle Suppose f() represents a polnomial function and a and b are two numbers such that f(a) 0 and f(b) 0. Then the function has at least one real zero between a and b. Determine the values of between which each real zero of the function f() is located. Then draw the graph. Make a table of values. Look at the values of f() to locate the zeros. Then use the points to sketch a graph of the function. f() 35 Eample f() The changes in sign indicate that there are zeros between and and between and. 9 Eercises Graph each function b making a table of values. Determine the values of at which or between which each real zero is located. Lesson 7-. f() 3. f() f() 4 f() 8 f() f() between 0 and ; between and 3; at at ; between and between and 4. f() f() f() f() f() f() at, between 0 and between 0 and ; between and 3 Glencoe/McGraw-Hill 38 Glencoe Algebra

15 NAME DATE PERID 7- Stud Guide and Intervention (continued) Graphing Polnomial Functions Maimum and Minimum Points A quadratic function has either a maimum or a minimum point on its graph. For higher degree polnomial functions, ou can find turning points, which represent relative maimum or relative minimum points. Graph f() Estimate the -coordinates at which the relative maima and minima occur. Make a table of values and graph the function. Eample f() indicates a relative maimum 4 6 f() A relative maimum occurs at 4 and a relative minimum occurs at zero between, indicates a relative minimum 4 9 Eercises Graph each function b making a table of values. Estimate the -coordinates at which the relative maima and minima occur.. f() 3 3. f() f() 3 3 f() f() f() ma. at 0, min. at ma. about, ma. about, min. about 0.5 min. about 4. f() f() 5 6. f() f() f() f() min. about ma. at 0, ma. about, min. about min. at 0 Glencoe/McGraw-Hill 38 Glencoe Algebra

16 7- NAME DATE PERID Skills Practice Graphing Polnomial Functions Complete each of the following. a. Graph each function b making a table of values. b. Determine consecutive values of between which each real zero is located. c. Estimate the -coordinates at which the relative maima and minima occur.. f() 3 3. f() f() f() 3 3 f() f() f() zeros between and 0, 0 and, zeros between and, 0 and, and and 3; rel. ma. at 0, and and ; rel. ma. at, rel. min. at rel. min. at f() zero between and 0; zero between and 0; rel. ma. at, rel. min. at, rel. ma. at 0 rel. min. at f() f() f() f() Lesson 7-5. f() 4 6. f() f() f() zeros between and, and zeros between and, and and ; rel. ma. at 0, 3, and, and and 3; rel. ma. at f() f() Glencoe/McGraw-Hill 383 Glencoe Algebra

17 7- Complete each of the following. a. Graph each function b making a table of values. b. Determine consecutive values of between which each real zero is located. c. Estimate the -coordinates at which the relative and relative minima occur.. f() f() NAME DATE PERID Practice (Average) Graphing Polnomial Functions f() f() f() f() zeros between 4 7 zeros between and 0, and, and, 0 and, and and 3; rel. ma. at, and 3 and 4; rel. ma. at, rel. min. at 0 rel. min. at 3. f() f() f() zeros between 3 and, and and ; rel. ma. at 0, rel. min. at and f() zeros between 3 and, and 0 and ; rel. min. at PRICES For Eercises 5 and 6, use the following information. The Consumer Price Inde (CPI) gives the relative price for a fied set of goods and services. The CPI from September, 000 to Jul, 00 is shown in the graph. Source: U. S. Bureau of Labor Statistics 5. Describe the turning points of the graph. rel ma. in Nov. and June; rel. min in Dec. 6. If the graph were modeled b a polnomial equation, what is the least degree the equation could have? Months Since September, LABR A town s jobless rate can be modeled b (, 3.3), (, 4.9), (3, 5.3), (4, 6.4), (5, 4.5), (6, 5.6), (7,.5), (8,.7). How man turning points would the graph of a polnomial function through these points have? Describe them. 4: rel. ma. and rel. min. Glencoe/McGraw-Hill 384 Glencoe Algebra Consumer Price Inde

18 7- NAME DATE PERID Reading to Learn Mathematics Graphing Polnomial Functions Pre-Activit How can graphs of polnomial functions show trends in data? Read the introduction to Lesson 7- at the top of page 353 in our tetbook. Three points on the graph shown in our tetbook are (0, 4), (70, 3.78), and (00, 9). Give the real-world meaning of the coordinates of these points. Sample answer: In 900, 4% of the U. S. population was foreign born. In 970, 3.78% of the population was foreign born. In 000, 9% of the population was foreign born. Reading the Lesson. Suppose that f() is a third-degree polnomial function and that c and d are real numbers, with d c. Indicate whether each statement is true or false. (Remember that true means alwas true.) a. If f(c) 0 and f(d) 0, there is eactl one real zero between c and d. false b. If f(c) f(d) 0, there are no real zeros between c and d. false c. If f(c) 0 and f(d) 0, there is at least one real zero between c and d. true. Match each graph with its description. a. third-degree polnomial with one relative maimum and one relative minimum; leading coefficient negative iii b. fourth-degree polnomial with two relative minima and one relative maimum i Lesson 7- c. third-degree polnomial with one relative maimum and one relative minimum; leading coefficient positive iv d. fourth-degree polnomial with two relative maima and one relative minimum ii i. ii. iii. iv. f() f() f() f() Helping You Remember 3. The origins of words can help ou to remember their meaning and to distinguish between similar words. Look up maimum and minimum in a dictionar and describe their origins (original language and meaning). Sample answer: Maimum comes from the Latin word maimus, meaning greatest. Minimum comes from the Latin word minimus, meaning least. Glencoe/McGraw-Hill 385 Glencoe Algebra

19 7- NAME DATE PERID Enrichment Golden Rectangles Use a straightedge, a compass, and the instructions below to construct a golden rectangle.. Construct square ABCD with sides of centimeters.. Construct the midpoint of A B. Call the midpoint M. 3. Using M as the center, set our compass opening at MC. Construct an arc with center M that intersects A B. Call the point of intersection P. 4. Construct a line through P that is perpendicular to A B. 5. Etend D C so that it intersects the perpendicular. Call the intersection point Q. APQD is a golden rectangle. Check this conclusion b finding the value of Q P. AP A figure consisting of similar golden rectangles is shown below. Use a compass and the instructions below to draw quarter-circle arcs that form a spiral like that found in the shell of a chambered nautilus. 6. Using A as a center, draw an arc that passes through B and C. C 7. Using D as a center, draw an arc that passes through C and E. 8. Using F as a center, draw an arc that passes through E and G. D J M L K F H E 9. Continue drawing arcs, using H, K, and M as the centers. B A G Glencoe/McGraw-Hill 386 Glencoe Algebra

20 7-3 NAME DATE PERID Stud Guide and Intervention Solving Equations Using Quadratic Techniques Quadratic Form Certain polnomial epressions in can be written in the quadratic form au bu c for an numbers a, b, and c, a 0, where u is an epression in. Eample Write each polnomial in quadratic form, if possible. a. 3a 6 9a 3 Let u a 3. 3a 6 9a 3 3(a 3 ) 9(a 3 ) b. 0b 49 b 4 Let u b. 0b 49 b 4 0( b ) 49( b ) 4 c. 4a 5 a 3 8 This epression cannot be written in quadratic form, since a 5 (a 3 ). Eercises Write each polnomial in quadratic form, if possible p 4 6p 8 ( ) 6( ) 8 4(p ) 6(p ) ( 4 ) ( 4 ) not possible ( ) 0( ) ( 4 ) 4 4 8( 3 ) ( 3 ) Lesson ( ) 9( ) 5 not possible ( 3 ) 3( 3 ) 0 63( 4 ) 5( 4 ) ( 5 ) 4( 5 ) ( 3 ) 7( 3 ) 3 0( 5 ) 7( 5 ) 7 Glencoe/McGraw-Hill 387 Glencoe Algebra

21 7-3 Solve Equations Using Quadratic Form If a polnomial epression can be written in quadratic form, then ou can use what ou know about solving quadratic equations to solve the related polnomial equation. Eample Solve riginal equation ( ) 40( ) 44 0 Write the epression on the left in quadratic form. ( 4)( 36) 0 Factor. 4 0 or 36 0 Zero Product Propert ( )( ) 0 or ( 6)( 6) 0 Factor. 0or 0 or 6 0or 6 0 Zero Product Propert or or 6or 6 Simplif. The solutions are and 6. Eample NAME DATE PERID Stud Guide and Intervention (continued) Solving Equations Using Quadratic Techniques Solve riginal equation ( ) 5 0 Write the epression on the left in quadratic form. ( 5)( 3) 0 Factor. 5 0 or 3 0 Zero Product Propert 5 or 3 Simplif. Since the principal square root of a number cannot be negative, 3 has no solution. 5 The solution is or Eercises Solve each equation , i 7, 3, i t 6 48t 0 5. m 6 6m ,, i, i 3, , 3 4,, , 9 9, 49 7, 8 Glencoe/McGraw-Hill 388 Glencoe Algebra

22 7-3 NAME DATE PERID Skills Practice Solving Equations Using Quadratic Techniques Write each epression in quadratic form, if possible ( ) ( ) not possible 3. 00a 6 a 3 00(a 3 ) a ( 4 ) 4( 4 ) ( ) 7( ) 6. 6b 5 3b 3 not possible 7. 5v 6 8v 3 9 5(v 3 ) 8(v 3 ) 9 8. a 9 5a 5 7a a[(a 4 ) 5(a 4 ) 7] Solve each equation. 9. a 3 9a 4a 0 0, 7, , 3. t 4 3t 3 40t 0 0, 5, 8. b 3 8b 6b 0 0, 4 3. m 4 4,, i, i 4. w 3 6w 0 0, 6, 6 5. m 4 8m 8 3, , 3, 3, 3i, 3i Lesson h 4 0h 9,, 3, 3 8. a 4 9a 0 0,, 5, v 4 v 35 0,, 3, 3 5, 5, 7, c 3 7c 3 0 0,,, 6, 6 64, 7 3. z 5 z 6 4, , 400 Glencoe/McGraw-Hill 389 Glencoe Algebra

23 7-3 NAME DATE PERID Practice (Average) Solving Equations Using Quadratic Techniques Write each epression in quadratic form, if possible.. 0b 4 3b d 6 5d 3 0(b ) 3(b ) not possible 8(d 3 ) 5(d 3 ) 4. 4s 8 4s b 5 8b 3 4(s 4 ) 4(s 4 ) 7 500( ) not possible 7. 3w 5 56w 3 8w 8. e 3 7e w[4(w ) 7(w ) ] ( e 3) 7( e 3) 0 ( 0 ) 9( 0 ) Solve each equation , 0, 9. s 5 4s 4 3s 3 0 8, 0, 4. m , 5, 5i, 5i 3. n 4 49n 0 0, 7, ,, 7, 7 5. t 4 t , 4, 5, r 6 9r 4 0 0, 3, ,, i 6, i 6 8. d 4 6d 48,, 3, 3 9. t , 7 7i 3, , , , 9 3. n 0 n w w 7 0 9, i 3 6. PHYSICS A proton in a magnetic field follows a path on a coordinate grid modeled b the function f() 4 5. What are the -coordinates of the points on the grid where the proton crosses the -ais? 5, 5 7. SURVEYING Vista count is setting aside a large parcel of land to preserve it as open space. The count has hired Meghan s surveing firm to surve the parcel, which is in the shape of a right triangle. The longer leg of the triangle measures 5 miles less than the square of the shorter leg, and the hpotenuse of the triangle measures 3 miles less than twice the square of the shorter leg. The length of each boundar is a whole number. Find the length of each boundar. 3 mi, 4 mi, 5 mi Glencoe/McGraw-Hill 390 Glencoe Algebra

24 7-3 NAME DATE PERID Reading to Learn Mathematics Solving Equations Using Quadratic Techniques Pre-Activit How can solving polnomial equations help ou to find dimensions? Read the introduction to Lesson 7-3 at the top of page 360 in our tetbook. Eplain how the formula given for the volume of the bo can be obtained from the dimensions shown in the figure. Sample answer: The volume of a rectangular bo is given b the formula V wh. Substitute 50 for, 3 for w, and for h to get V() (50 )(3 )() Reading the Lesson. Which of the following epressions can be written in quadratic form? b, c, d, f, g, h, i a b c. m 6 4m 3 4 d. 5 e. 5 3 f. r 4 6 r 8 g. p 4 8p h. r 3 r 6 3 i. 5 z z 3. Match each epression from the list on the left with its factorization from the list on the right. a vi i. ( 3 3)( 3 3) b v ii. ( 3)( 3) c. 6 9 i iii. ( 3) d. 9 ii iv. ( )( 4 ) Lesson 7-3 e. 6 iv v. ( 5) f. 6 9 iii vi. ( 5)( 8) Helping You Remember 3. What is an eas wa to tell whether a trinomial in one variable containing one constant term can be written in quadratic form? Sample answer: Look at the two terms that are not constants and compare the eponents on the variable. If one of the eponents is twice the other, the trinomial can be written in quadratic form. Glencoe/McGraw-Hill 39 Glencoe Algebra

25 7-3 NAME DATE PERID Enrichment dd and Even Polnomial Functions Functions whose graphs are smmetric with respect to the origin are called odd functions. If f( ) f() for all in the domain of f(), then f () is odd. f() 3 f() 4 4 Functions whose graphs are smmetric with respect to the -ais are called even functions. If f ( ) f() for all in the domain of f(), then f () is even. 6 4 f() f() Eample f() 3 3 f( ) ( ) 3 3( ) 3 3 ( 3 3) f () Therefore, f () is odd. Determine whether f() 3 3 is odd, even, or neither. Replace with. Simplif. Factor out. Substutute. The graph at the right verifies that f () is odd. The graph of the function is smmetric with respect to the origin. f() 4 f() Determine whether each function is odd, even, or neither b graphing or b appling the rules for odd and even functions.. f () 4. f () f () 7 4. f () 3 5. f () f () f () f () f () f () Complete the following definitions: A polnomial function is odd if and onl if all the terms are of degrees. A polnomial function is even if and onl if all the terms are of degrees. Glencoe/McGraw-Hill 39 Glencoe Algebra

26 7-4 NAME DATE PERID Stud Guide and Intervention The Remainder and Factor Theorems Snthetic Substitution Remainder Theorem The remainder, when ou divide the polnomial f( ) b ( a), is the constant f(a). f() q() ( a) f(a), where q() is a polnomial with degree one less than the degree of f(). Eample Eample If f() , find f( ). Method Snthetic Substitution B the Remainder Theorem, f( ) should be the remainder when ou divide the polnomial b The remainder is 8, so f( ) 8. Method Direct Substitution Replace with. f() f( ) 3( ) 4 ( ) 3 5( ) ( ) or 8 So f( ) 8. If f() 5 3, find f(3). Again, b the Remainder Theorem, f(3) should be the remainder when ou divide the polnomial b The remainder is 40, so f(3) 40. Eercises Use snthetic substitution to find f( 5) and f for each function.. f() ;. f() ; 3 3. f() ; 4. f() 4 899; Use snthetic substitution to find f(4) and f( 3) for each function. 5. f() f() ; 7 78; f() f() ; 69 30; f() f() ; 98 67; 77. f() f() ; 8 805; 46 9 Lesson 7-4 Glencoe/McGraw-Hill 393 Glencoe Algebra

27 7-4 NAME DATE PERID Stud Guide and Intervention (continued) The Remainder and Factor Theorems Factors of Polnomials The Factor Theorem can help ou find all the factors of a polnomial. Factor Theorem The binomial a is a factor of the polnomial f() if and onl if f(a) 0. Eample Show that 5 is a factor of Then find the remaining factors of the polnomial. B the Factor Theorem, the binomial 5 is a factor of the polnomial if 5 is a zero of the polnomial function. To check this, use snthetic substitution Since the remainder is 0, 5 is a factor of the polnomial. The polnomial can be factored as ( 5)( 3 ). The depressed polnomial 3 can be factored as ( )( ). So ( 5)( )( ). Eercises Given a polnomial and one of its factors, find the remaining factors of the polnomial. Some factors ma not be binomials ; ; 3 ( 4)( ) ( 5)( ) ; ; 4 ( 3)( 5) ( )( 9) ; ; 4 ( 3)( ) (3 )( 5) ; ; (4 )(3 ) (7 3)( 3) ; ; 4 ( 5) ( 3 7) ; ; (3 5 4) ( )( 3)( 3) Glencoe/McGraw-Hill 394 Glencoe Algebra

28 7-4 NAME DATE PERID Skills Practice The Remainder and Factor Theorems Use snthetic substitution to find f() and f( ) for each function.. f() 6 5, 0. f() 3, 3 3. f(), 4. f() 3 5, 6 5. f() 3 3 3, 3 6. f() , 0 7. f() 3 3 4, 7 8. f() , 9. f() 4 9 5, 6 0. f() , 4. f() f() , 0 3, 6 Given a polnomial and one of its factors, find the remaining factors of the polnomial. Some factors ma not be binomials ; ;,, ; ; 3,, ; ; 4 3, 6 4, ; ;, 4 5, 3 Lesson ;. 3 5 ;,, ; ; 3, Glencoe/McGraw-Hill 395 Glencoe Algebra

29 7-4 NAME DATE PERID Practice (Average) The Remainder and Factor Theorems Use snthetic substitution to find f( 3) and f(4) for each function.. f() 3 6, 7. f() , 6 3. f() 5 4 0, 8 4. f() 3 3 7, f() 3 5 4, 0 6. f() , 4 7. f() 3 8 3, 3 8. f() , f() , f() , 80. f() , 7. f() , f() , f() , f() f() , , 588 Given a polnomial and one of its factors, find the remaining factors of the polnomial. Some factors ma not be binomials ; ;, 4 3, ; ; 3 3, 3, ; ; 4 3, 4 3, ; ; 3 3, 3, ; ; 3 3, 3, ; ;,,,,, 5 9. PPULATIN The projected population in thousands for a cit over the net several ears can be estimated b the function P() , where is the number of ears since 000. Use snthetic substitution to estimate the population for , VLUME The volume of water in a rectangular swimming pool can be modeled b the polnomial If the depth of the pool is given b the polnomial, what polnomials epress the length and width of the pool? 3 and Glencoe/McGraw-Hill 396 Glencoe Algebra

30 7-4 NAME DATE PERID Reading to Learn Mathematics The Remainder and Factor Theorems Pre-Activit How can ou use the Remainder Theorem to evaluate polnomials? Read the introduction to Lesson 7-4 at the top of page 365 in our tetbook. Show how ou would use the model in the introduction to estimate the number of international travelers (in millions) to the United States in the ear 000. (Show how ou would substitute numbers, but do not actuall calculate the result.) Sample answer: 0.0(4) 3 0.6(4) 6(4) 5.9 Reading the Lesson. Consider the following snthetic division a. Using the division smbol, write the division problem that is represented b this snthetic division. (Do not include the answer.) ( ) ( ) b. Identif each of the following for this division. dividend divisor quotient remainder c. If f() , what is f()? 3 3. Consider the following snthetic division a. This division shows that 3 is a factor of 3 7. b. The division shows that 3 is a zero of the polnomial function f() 3 7. c. The division shows that the point ( 3, 0) is on the graph of the polnomial function f() 3 7. Lesson 7-4 Helping You Remember 3. Think of a mnemonic for remembering the sentence, Dividend equals quotient times divisor plus remainder. Sample answer: Definitel ever quiet teacher deserves proper rewards. Glencoe/McGraw-Hill 397 Glencoe Algebra

31 NAME DATE PERID 7-4 Enrichment Using Maimum Values Man times maimum solutions are needed for different situations. For instance, what is the area of the largest rectangular field that can be enclosed with 000 feet of fencing? Let and denote the length and width of the field, respectivel. Perimeter: Area: A (000 ) 000 This problem is equivalent to finding the highest point on the graph of A() 000 shown on the right. A Complete the square for 000. A ( ) 500 ( 500) 500 Because the term ( 500) is either negative or 0, the greatest value of A is 500. The maimum area enclosed is 500 or 50,000 square feet. 000 Solve each problem.. Find the area of the largest rectangular garden that can be enclosed b 300 feet of fence.. A farmer will make a rectangular pen with 00 feet of fence using part of his barn for one side of the pen. What is the largest area he can enclose? 3. An area along a straight stone wall is to be fenced. There are 600 meters of fencing available. What is the greatest rectangular area that can be enclosed? Glencoe/McGraw-Hill 398 Glencoe Algebra

32 7-5 NAME DATE PERID Stud Guide and Intervention Roots and Zeros Tpes of Roots The following statements are equivalent for an polnomial function f(). c is a zero of the polnomial function f(). ( c) is a factor of the polnomial f(). c is a root or solution of the polnomial equation f() 0. If c is real, then (c, 0) is an intercept of the graph of f(). Fundamental Theorem of Algebra Corollar to the Fundamental Theorem of Algebras Descartes Rule of Signs Ever polnomial equation with degree greater than zero has at least one root in the set of comple numbers. A polnomial equation of the form P() 0 of degree n with comple coefficients has eactl n roots in the set of comple numbers. If P() is a polnomial with real coefficients whose terms are arranged in descending powers of the variable, the number of positive real zeros of P() is the same as the number of changes in sign of the coefficients of the terms, or is less than this b an even number, and the number of negative real zeros of P() is the same as the number of changes in sign of the coefficients of the terms of P( ), or is less than this number b an even number. Eample Eample Solve the equation and state the number and tpe of roots ( ) 0 Use the Zero Product Propert. 3 0or 0 0or i The equation has one real root, 0, i and two imaginar roots,. Eercises State the number of positive real zeros, negative real zeros, and imaginar zeros for p() Since p() has degree 4, it has 4 zeros. Use Descartes Rule of Signs to determine the number and tpe of real zeros. Since there are three sign changes, there are 3 or positive real zeros. Find p( ) and count the number of changes in sign for its coefficients. p( ) 4( ) 4 3( ) 3 ( ) ( ) Since there is one sign change, there is eactl negative real zero. Solve each equation and state the number and tpe of roots , 7; real 0, 5; 3 real 5i 3 0, ; real, imaginar State the number of positive real zeros, negative real zeros, and imaginar zeros for each function. 4. f() ; or 0; 0 or 5. f() or 0; 0; or 4 Lesson 7-5 Glencoe/McGraw-Hill 399 Glencoe Algebra

33 7-5 Find Zeros NAME DATE PERID Stud Guide and Intervention (continued) Roots and Zeros Comple Conjugate Theorem Suppose a and b are real numbers with b 0. If a bi is a zero of a polnomial function with real coefficients, then a bi is also a zero of the function. Eample Find all of the zeros of f() Since f() has degree 4, the function has 4 zeros. f() f( ) Since there are 3 sign changes for the coefficients of f(), the function has 3 or positive real zeros. Since there is sign change for the coefficients of f( ), the function has negative real zero. Use snthetic substitution to test some possible zeros So 3 is a zero of the polnomial function. Now tr snthetic substitution again to find a zero of the depressed polnomial So 5 is another zero. Use the Quadratic Formula on the depressed polnomial 4 to find the other zeros, i 3. The function has two real zeros at 3 and 5 and two imaginar zeros at i 3. Eercises Find all of the zeros of each function.. f() 3 9 9, 3i. f() , i 3. p(a) a 3 0a 34a 40 4, 3 i 4. p() , i 5. f() f() , 3i, 4, 5i Glencoe/McGraw-Hill 400 Glencoe Algebra

34 7-5 NAME DATE PERID Skills Practice Roots and Zeros Solve each equation. State the number and tpe of roots ; real 6i; imaginar , 0, 0, i, i; 3 real, imaginar 5i, 5i, 5i, 5i; 4 imaginar ; real 0, 3, 3, 3i, 3i; 3 real, imaginar State the possible number of positive real zeros, negative real zeros, and imaginar zeros of each function. 7. g() h() or 0; ; or 0 or 0; ; or 0 9. f() p() or ; 0; or 0 3 or ; 0; or 0. q() f() ; ; or 0; or 0; 4 or or 0 Find all the zeros of each function. 3. h() g() ,,, i, i 5. h() q() ,, 3,, 4 7. g() f() 4 80,,, 4 4, 4, 5, 5 Write a polnomial function of least degree with integral coefficients that has the given zeros. 9. 3, 5, 0. 3i f() f() 9. 5 i., 3, 3 f() 0 6 f() i, 5i 4.,, i 6 f() f() Lesson 7-5 Glencoe/McGraw-Hill 40 Glencoe Algebra

35 7-5 Solve each equation. State the number and tpe of roots ; real 3,,, ; 4 real , 3, 3, 3i, 3i; 3 real, imaginar, 3, 3 ; 3 real , 3i; real, imaginar,, i, i; real, imaginar State the possible number of positive real zeros, negative real zeros, and imaginar zeros of each function. 7. f() p() 4 3 or 0; ; or 0 3 or ; ; or 0 9. q() h() or 0; or 0; 4,, or 0 or 0; or 0; 4,, or 0 Find all the zeros of each function.. h() p() , 3, 4 NAME DATE PERID Practice (Average) Roots and Zeros, i 6, i 6 3. h() q() , i, i i, i, 7i, 7i 5. g() f() ,,, 4,, 3 i, 3 i Write a polnomial function of least degree with integral coefficients that has the given zeros. 7. 5, 3i 8., 3 i f() f() , 4, 3i 0., 5, i f() f() CRAFTS Stephan has a set of plans to build a wooden bo. He wants to reduce the volume of the bo to 05 cubic inches. He would like to reduce the length of each dimension in the plan b the same amount. The plans call for the bo to be 0 inches b 8 inches b 6 inches. Write and solve a polnomial equation to find out how much Stephen should take from each dimension. (0 )(8 )(6 ) 05; 3 in. Glencoe/McGraw-Hill 40 Glencoe Algebra

36 NAME DATE PERID 7-5 Reading to Learn Mathematics Roots and Zeros Pre-Activit How can the roots of an equation be used in pharmacolog? Read the introduction to Lesson 7-5 at the top of page 37 in our tetbook. Using the model given in the introduction, write a polnomial equation with 0 on one side that can be solved to find the time or times at which there is 00 milligrams of medication in a patient s bloodstream. 0.5t 4 3.5t 3 00t 350t 00 0 Reading the Lesson. Indicate whether each statement is true or false. a. Ever polnomial equation of degree greater than one has at least one root in the set of real numbers. false b. If c is a root of the polnomial equation f() 0, then ( c) is a factor of the polnomial f(). true c. If ( c) is a factor of the polnomial f(), then c is a zero of the polnomial function f. false d. A polnomial function f of degree n has eactl (n ) comple zeros. false. Let f() a. What are the possible numbers of positive real zeros of f? 5, 3, or b. Write f( ) in simplified form (with no parentheses) What are the possible numbers of negative real zeros of f? c. Complete the following chart to show the possible combinations of positive real zeros, negative real zeros, and imaginar zeros of the polnomial function f. Number of Number of Number of Total Number Positive Real Zeros Negative Real Zeros Imaginar Zeros of Zeros Helping You Remember 3. It is easier to remember mathematical concepts and results if ou relate them to each other. How can the Comple Conjugates Theorem help ou remember the part of Descartes Rule of Signs that sas, or is less than this number b an even number. Sample answer: For a polnomial function in which the polnomial has real coefficients, imaginar zeros come in conjugate pairs. Therefore, there must be an even number of imaginar zeros. For each pair of imaginar zeros, the number of positive or negative zeros decreases b Lesson 7-5 Glencoe/McGraw-Hill 403 Glencoe Algebra

37 7-5 NAME DATE PERID Enrichment The Bisection Method for Approimating Real Zeros The bisection method can be used to approimate zeros of polnomial functions like f () Since f () 4 and f () 3, there is at least one real zero between and. The midpoint of this interval is.5. Since f(.5).875, the zero is between.5 and. The midpoint of this interval is Since f(.75) is about 0.7, the zero is between.5 and.75. The midpoint of this interval is and f(.65) is about The zero is between.65 and.75. The midpoint of this interval is Since f (.6875) is about 0.4, the zero is between.6875 and.75. Therefore, the zero is.7 to the nearest tenth. The diagram below summarizes the results obtained b the bisection method. sign of f(): + + value : Using the bisection method, approimate to the nearest tenth the zero between the two integral values of for each function.. f () 3 4, f (0), f (). f () 4 5, f (), f () 3. f() 5 3, f () 3, f () 4 4. f () 4 3 7, f ( ), f ( ) 5 5. f () , f (4) 4, f (5) 6 Glencoe/McGraw-Hill 404 Glencoe Algebra

38 7-6 NAME DATE PERID Stud Guide and Intervention Rational Zero Theorem Identif Rational Zeros Rational Zero Let f() a 0 n a n a n a n a n represent a polnomial function Theorem with integral coefficients. If p q is a rational number in simplest form and is a zero of f(), then p is a factor of a n and q is a factor of a 0. Corollar (Integral If the coefficients of a polnomial are integers such that a 0 and a n 0, an rational Zero Theorem) zeros of the function must be factors of a n. Lesson 7-6 Eample List all of the possible rational zeros of each function. a. f() If p q is a rational root, then p is a factor of 0 and q is a factor of 3. The possible values for p are,, 5, and 0. The possible values for q are and 3. So all of the possible rational zeros are p q,, 5, 0, 3, 3, 5 3, and. 30 b. q() Since the coefficient of 3 is, the possible rational zeros must be the factors of the constant term 36. So the possible rational zeros are,, 3, 4, 6, 9,, 8, and 36. Eercises List all of the possible rational zeros of each function.. f() g() ,, 4, 8,, 4, 5, 0, 0 3. h() p() , 7, 49 5, 5,, 5. q() r() 4 5 8,, 3, 4, 6,,,, 3, 6, 9, 8, 4,, ,,,,, 7. f() g() ,, 3, 4, 5, 6, 8, 0,,, 3, 5, 5,, 5, 0, 4, 30, 40, 60, 0 9. h() p() , 3, 7,,,,,,,,, 7,,, 7 Glencoe/McGraw-Hill 405 Glencoe Algebra

39 7-6 NAME DATE PERID Stud Guide and Intervention (continued) Rational Zero Theorem Find Rational Zeros Eample Find all of the rational zeros of f() From the corollar to the Fundamental Theorem of Algebra, we know that there are eactl 3 comple roots. According to Descartes Rule of Signs there are or 0 positive real roots and negative real root. The possible rational zeros are,, 3, 4, 6,, 3 4 6,,,,,.Make a table and test some possible rational zeros p q Since f() 0, ou know that is a zero. The depressed polnomial is 5 7, which can be factored as (5 3)( 4). 3 B the Zero Product Propert, this epression equals 0 when or The rational zeros of this function are,, and 4. 5 Eample Find all of the zeros of f() There are 4 comple roots, with positive real root and 3 or negative real roots. The possible rational zeros are, 3,,,,,, and Make a table and test some possible values. p q is a root. Since f 0, we know that The zeros of this function are, 3, and i. 4 Eercises The depressed polnomial is Tr snthetic substitution again. An remaining rational roots must be negative. p q is another rational root. The depressed polnomial is 8 8 0, which has roots i. Find all of the rational zeros of each function.. f() , 4, 7. f() Find all of the zeros of each function. 3. f() f() , 5, i 3,, 3i Glencoe/McGraw-Hill 406 Glencoe Algebra