-. Plan Lesson Preview Check Skills You ll Need Adding and Subtracting Polnomials Lesson 9-: Eample Eercises 0 Etra Practice, p. 70 Multipling Binomials Lesson 9-: Eamples, Eercises 9 Etra Practice, p. 70 Lesson Resources Teaching Resources Practice, Reteaching, Enrichment Reaching All Students Practice Workbook - Spanish Practice Workbook - Basic Algebra Planning Guide - Presentation Assistant Plus! Transparencies Check Skills You ll Need - Additional Eamples - Student Edition Answers - Lesson Quiz - PH Presentation Pro CD - Computer Test Generator CD Technolog Resource Pro CD-ROM Computer Test Generator CD Prentice Hall Presentation Pro CD www.phschool.com Student Site Teacher Web Code: aek-00 Self-grading Lesson Quiz Teacher Center Lesson Planner Resources Plus Lesson Preview What You ll Learn To divide polnomials... And Wh To find the length of a rectangle, as in Eample Part - Dividing Polnomials Check Understanding Need Help? quotient divisorqdividend Chapter Rational Epressions and Functions Before the Lesson Diagnose prerequisite skills using: Check Skills You ll Need Dividing Polnomials Check Skills You ll Need (For help, go to Lessons 9- and 9-.). a ± 9a ± Write each polnomial in standard form.. ± ±. 9a - a +. - + -. - + 8t 8t Find each product.. See back of book.. ( + )( + ). (-n - )(n - ). Aa + (a B - 7) To divide a polnomial b a monomial, divide each term of the polnomial b the monomial divisor. Dividing a Polnomial b a Monomial Divide 8 + - b. 8 + - = A8 + - B Multipl b the reciprocal of. = 8 + Use the Distributive Propert. = + 0 Use the division rules for - eponents. = + - Simplif. a. Am - m + m B m b. A8t + t - t + tb t m ± m The process of dividing a polnomial b a binomial is similar to long division. For eample, consider dividing 77 b. q77 07 0 77 =. Divide: can go into 7 about times.. Multipl, then subtract from 7.. Bring down the 7. Divide: 07 N.. Multipl, and then subtract from 07.. The remainder is. You can summarize the process for long division as Interactive lesson includes instant self-check, tutorials, and activities. t ± t ± t Divide, multipl, subtract, bring down, and repeat as necessar. In the division above, the answer is written as a mied number: means +. remainder In dividing polnomials, write the answer as quotient + divisor. Ongoing 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
Need Help? When subtracting the polnomial -7 - from -7-0, think -7-0 (7 ) is -7-0 (7 ). - Check Understanding Check Understanding When the divisor and dividend are in standard form, divide the first term of the dividend b the first term of the divisor to find the first term of the quotient. Divide + - 0 b +. Dividing a Polnomial b a Binomial Step Begin the long division process. Align terms b their degrees. T So put above of the dividend. q 0 0 7 0 Step Repeat the process: divide, multipl, subtract, and bring down. 7 q 0 0 7 0 7 The answer is - 7 +, or - 7 -. a. Ab - b - B (b + ) b. Am - m - 7B (m + ) b m m When the dividend is in standard form and a power is missing, add a term of that power with 0 as its coefficient. For eample, rewrite b + b -, as b + 0b + b -. Reaching All Students Below Level Help students understand division of polnomials b connecting the process of long division with the process of dividing polnomials. Dividing Polnomials With a Zero Coefficient Geometr The width and area of a rectangle are shown in the figure. What is the length? Since A = Ow, divide the area b the width to find the length. b b b qb 0b b b b b b b b b b 0 Divide: Think. Multipl: ( ± ) ± 0. Then subtract. Bring down 0. Divide: 7 7. Multipl: 7( ± ) 7. Then subtract. The remainder is. Rewrite the dividend with 0b. W (b ) in. A (b b ) in. The length of the rectangle is Ab + b + B in. a. At + t + t - B (t - ) b. Ac - c + B (c + ) t ± t ± t ± c c ± c Lesson - Dividing Polnomials Advanced Learners Challenge students to predict the degree of the quotient when a polnomial of degree is divided b a polnomial of degree. Error Prevention See note on page.. Teach Math Background The process of dividing a polnomial b a binomial reviews the procedure of subtracting a term b adding its opposite. Teaching Notes Teaching Tip Point out to students that lining the terms up in vertical columns makes the division much easier to read. Error Prevention Students sometimes make errors when subtracting the binomials b subtracting onl the first term. Suggest to students that the put parentheses around the whole binomial to be subtracted. Additional Eamples Divide 8 + 9 - b. ± Divide + - b +. 8 ± The width and area of a rectangle are shown in the figure below. What is the length? ( ± ± ) in. ( 9) in. ( ) in. Divide -8 - + b - +. Closure Ask students to eplain how division of polnomials is similar to long division. The division of polnomials is similar to long division in that both processes involve dividing, multipling, and subtracting, then bringing down, and repeating as needed.
. Practice Assignment Guide Objective A C B Core 7 Etension 8 Standardized Test Prep 7 Mied Review 8 7 Enrichment - Reteaching - Practice - Pearson Education, Inc. All rights reserved. Name Class Date Practice -. 0.. ( ). (0 ). (-8 + - + ) 8. ( - 0) 7. ( - + 9) ( - 7) 8. ( - + ) ( - ) 9. ( - ) ( + ) 0. ( + - ) ( - ). ( + 0 + ) ( + ). ( - - ) ( - ). ( + + 0) ( + ). ( - 8-9) ( - ). ( - - ) ( - ). ( + + + ) ( + ) 7. ( - + ) ( - ) 8. ( + + ) ( - ) 9. ( + - ) ( + ) 0. ( + - 0) ( + ). (8 + + ) ( - ). ( + - ) ( - ). ( + - - ) ( - ). (0 + + 0 ) ( + ). ( - + ) ( - ). (- - + - 7) ( - 7) 7. The volume of a rectangular prism is + 8 - -. The height of the prism is +, and the width of the prism is +. Find the length of the prism. 8. The width of a rectangle is +, and the area is + - - cm. What is the length of the rectangle? Dividing Polnomials Algebra Chapter Lesson - Practice Need Help? To review standard form, see Lesson 9-. Check Understanding Practice and Problem Solving To use the process for long division, write an divisor or dividend in standard form before ou begin to divide. Reordering Terms and Dividing Polnomials Divide - + + 9 b +. Rewrite - + + 9 as 9 - + and + as +. Then divide. q9 9 The answer is - +. a. A0 - + 8 B ( + ) b. A9 - a - ab (a - ) ± a a Ke Concepts Summar Dividing a Polnomial b a Polnomial EXERCISES Step Step Step Step Step Arrange the terms of the dividend and divisor in standard form. Divide the first term of the dividend b the first term of the divisor. This is the first term of the quotient. Multipl the first term of the quotient b the divisor and place the product under the dividend. Subtract this product from the dividend. Bring down the net term. Repeat Steps as necessar until the degree of the remainder is less than the degree of the divisor. For more practice, see Etra Practice. A Practice b Eample Eample (page ) Eample (page ) Eample (page ) 8. See margin.. A - + B. A 8-8 B. A9c + c - c B c. An - 8n + n B n. A8q - qb q. A-7t + t - 8t + t B 7t 7. A - + B ( - ) 8. At + t - B (t - ) 9. An - n + B (n - ) 0. A - + B ( + ). A - 0 + B ( - ). A- q - q + B (q + ). At - 00B (t + 0). Aw + w - B (w - ). Ab - 0b + B (b - ). Ac - c - B (c - ) 7. At - t - B (t + ) 8. An - n - 0B (n + ) pages Eercises. ±.. c ± c Chapter Rational Epressions and Functions. n 8n ±. q. t ± t t ± 7. 8. t ± 9 ± t 9. n 8 0. ±.. q 0 ± q. t 0. w 0 ± w ± w. b b ± b. c c 7. t t 8. n n n 8
B m Appl Your Skills C Eample (page ) a. Answers ma var. Sample: (c ± c c ); (c ± ) b. (c ± c c ) (c ± ) c ± c c. vertical asmptote: horizontal asmptote: d. Answers ma var. Sample: d d ± d d ± d e. d d ± d d ± d Challenge 9. Geometr The width of a rectangle is (r - ) cm and the area is Ar - r - B cm. What is the length? (r ± r ± ) cm 0. Geometr The base of a triangle is (c + ) ft and the area is Ac + ft. What is the height? Hint: The formula for the area of a triangle is A = B Q bh. R (c 8c ± ) ft 0. See margin.. A9 + b + b B (b + ). Aa - + ab ( + a). A9w + + 0w B (7 + w). At + t - 9B ( + t). A- + - - B ( - ). A - q + q - q B (q - ) 7. A + - B 8. Ac + c - c + 8B c 9. A8b + b B (b - ) 0. A + - 7B ( - ). Aa + a - B (a + ). At - 0t + B (t + ). Ak - 0.9k -.kb k. A-7s + s + B (s + ). A-z - z + z + B (z + ). Am + m + 70B (m + ) 7. Ac - B ( - c) 8. A - r - 0r + r B Ar - B 9. At - t + t - B At + B 0. Az + z - B (z + ). a. Open-Ended Write a binomial and a trinomial using the same variable. b. Divide the trinomial b the binomial. a b. See left.. Use the function rule =. a. Rewrite the function rule as a quotient plus a remainder. b. Make a table of values and graph the function. See margin. c. What are the vertical and horizontal asmptotes? See left.. Writing Suppose ou divide a polnomial b a binomial. Eplain how ou know if the binomial is a factor of the polnomial. See margin.. Geometr The volume of the rectangular prism shown at the left is m + 8m + 9m +. Find the area of the base of the prism. m ± m ±. a. Find Ad - d + B (d + ). d ± d b. Find Ad - d + d - B (d + ). d d ± d c. Find Ad - d + d - d + B (d + ).d d ± d ± d d. Patterns Predict the result of dividing d - d + d - d + d - b d +. e. Verif our prediction b dividing the polnomials. d e. See left.. Critical Thinking Find the value of k if + is a factor of - - k. d 7. a. Solve d = rt for t. t r b. Use our answer from part (a) to find an epression for the time it takes to travel a distance of t - t + t + miles at a rate of t + miles per hour. t 7t ± 8. Aa b - a b + 0a b B ab a b ab ± ab 9. A + 7 - B ( - ) ± 0. A90r + 8r + r + r + r B (9r + ) 0r ± r ± r. Ab + b - b + b + 8b - B Ab + b - B b b ±. Assess Lesson Quiz -. ( 8 - + ) ±. ( - - ) ( + ). ( + + ) ( + ) ± ±. (9 + ) ( + ) ± 9 + Alternative Assessment Have students use algebra tiles to model the following problem: ( + + ) ( + ). Then have students model this problem: ( + + 9) ( + ). Have students eplain what is different about the two problems. Students should realize that the first problem can be divided evenl with no remainder. The second problem cannot be done with algebra tiles; therefore, the problem has a remainder.. z ± z ± z. m m ± 99 m 7. c 0c 8. r ± r 7 t 9. t ± 0. z z t ± 0z 88 0 ± z b. Answers ma var. Sample: 0 7 9. b ± ± b. q ± q ± ± q. a a 7. ± 9,0. 0w 8 ± w 7 8. c 8 ± c ± c 9. t t 9. b 0 ± b ± 0 ± b. ± ± 0. ± ± 9 ± 8 Lesson - Dividing Polnomials. 8a. t t ± t 88 7 ± t. k 0.k 0.. s 8 ± 9 s O. The binomial is a factor of the polnomial if there is no remainder from the division.
Standardized Test Prep Resources For additional practice with a variet of test item formats: Standardized Test Prep, pp. 97 99 Test-Taking Strategies, p. 9 Test-Taking Strategies with Transparencies Eercise Remind students that the remainder is a polnomial whose degree is less than the divisor. Thus, answer choices H and I can be eliminated. pages Eercises 7. [] a. ± q 0 b. Tables ma var. Sample: 8 vertical asmptote: horizontal asmptote: ; 0 8 [] appropriate methods, but with one computational error [] no asmptotes OR no graph [] one part onl O d.. 0 f() O 0 0 Standardized Standardized Test Prep Test Prep Multiple Choice Take It to the NET Online lesson quiz at www.phschool.com Web Code: aea-0 Short Response Etended Response Mied Review Chapter Rational Epressions and Functions. In Lesson -, ou saw horizontal and vertical asmptotes of rational functions. Some rational functions have asmptotes that are neither horizontal nor vertical. Consider the function f() =. a. Divide + - b +. ± 0 0 b. Rewrite the function using the quotient. f() ± c. As «increases, the remainder will become smaller, approaching zero, though never reaching it. So an asmptote is defined b the quotient without the remainder. Write an equation of that asmptote. d. Graph the original equation and the asmptote. See left.. Which of the following epressions equals ( - - ) ( + )? C A. - - 7 + B. - - 7-8 C. - - D. - 7 + 8. What is the remainder when - is divided b -? G F. - G. H. I.. Which of the following must be true for A + + B ( + )? B I. The remainder is negative. II. The dividend is in standard form. III. The quotient is larger than the divisor for positive values of. A. I onl B. II onl C. I and II D. II and III. The volume of a rectangular prism is + + -. The height of the prism is -, and the length of the prism is +. Find the width of the prism. Show our work. See back of book. 0 7. a. Write the function = as a quotient plus a remainder. b. Make a table and find the vertical and horizontal asmptotes. Graph the function. a b. See margin. (t )(t )(t ) Lesson - Multipl or divide. (t )(t )(t) 8. n 7n 8? n n ± 9. t 0t? t t n n n t t 8t ( )( ) ( 7)( 8) 0. c c c. 9 0 c 8 c 7c c c 7 See left. Lesson - Lesson 0- Find the distance between each pair of points. If necessar, round to the nearest tenth.. (, ), (, ). (-, ), (, -) 9.. (7, ), (-8, ). (7, ), (-9, -) 7.9. (-, ), (-, -7).0 7. (0, -), (-, 0). Find the value of each epression. If the value is irrational, round to the nearest hundredth..7. 8.!8.9.8 9.!89 7 70.!.9 7.!000 7.!0.