9-. Plan Objetives To desribe polynomials To add and subtrat polynomials Examples Degree of a Monomial Classifying Polynomials Adding Polynomials Subtrating Polynomials 9- What You ll Learn To desribe polynomials To add and subtrat polynomials... And Why To ombine and simplify polynomials, as in Example Adding and Subtrating Polynomials Chek Skills You ll Need GO for Help Lesson - Simplify eah expression.. 6t + t 9t. g + g 9g. 7k - k 8k. b - 6 + 9b b 6. n - 7n n 6. 8x - x 7x New Voabulary monomial degree of a monomial polynomial standard form of a polynomial degree of a polynomial binomial trinomial Math Bakground A polynomial is made up of terms, whih are algebrai expressions ombined by addition or subtration. In a polynomial, there are no expressions that involve dividing by the variable or taking any root of a variable. More Math Bakground: p. 9C Lesson Planning and Resoures See p. 9E for a list of the resoures that support this lesson. Bell Ringer Pratie Chek Skills You ll Need For intervention, diret students to: The Distributive Property Lesson -: Example Extra Skills and Word Problem Pratie, Ch. Part Desribing Polynomials. Davis: m Brooks:.99s ± Martino:.99s ±.9g ±.89p Roky's Friends Bird Supplies bird seed ( lb) uttlebone ( t) spray millet ( lb) gravel paper ( pkg) perhes ( t) $.99 $.00 $.00 $.9 $.89 Ativity: Using Polynomials Business Suppose you work at a pet store. The spreadsheet below shows the details of several ustomers orders. A Customer Davis Brooks Casi Martino B Seed C Cuttlebone The following variables represent the number of eah item ordered. s = bags of birdseed m = bags of millet = pakages of uttlebone g = pakages of gravel paper p = pakages of perhes. Whih expression represents the ost of Casi s order? B A. 7.99(s + m) B..99s + m C. 7.99sm D Millet. Write expressions to represent eah of the other ustomers orders. See left.. Martino buys 0 bags of birdseed, pakages of gravel paper, and pakages of perhes. What is the total ost of his order? $.8 A monomial is an expression that is a number, a variable, or a produt of a number and one or more variables. Eah of the following is a monomial. y -x y E G. Paper F Perhes 9 Chapter 9 Polynomials and Fatoring 9 Speial Needs L Display the ativity. Have students write the expression for Brooks order..99s. Ask for student volunteers to point out how they piked up the data and built the expression. learning style: visual Below Level L Show students that is a monomial beause it an be rewritten as?. However, sine x =? x, it does not fit the definition of a monomial. learning style: verbal
Voabulary Tip The prefix mono means one. Quik Chek a. 9x ± 6x ± 7; fourth degree trinomial b. y ± y ; ubi trinomial. v ± 8; linear binomial Quik Chek The fration is a monomial, but the expression x is not a monomial beause there is a variable in the denominator. The degree of a monomial is the sum of the exponents of its variables. For a nonzero onstant, the degree is 0. Zero has no degree. Degree of a Monomial Find the degree of eah monomial. a. x Degree: x = x. The exponent is. b. 7x y Degree: The exponents are and. Their sum is.. - Degree: 0 The degree of a nonzero onstant is 0. Critial Thinking What is the degree of 9x 0? Explain. 0; the degree of a nonzero onstant is 0. A polynomial is a monomial or the sum or differene of two or more monomials. x + x - 7x + degree S 0 The polynomial shown above is in standard form. Standard form of a polynomial means that the degrees of its monomial terms derease from left to right. The degree of a polynomial in one variable is the same as the degree of the monomial with the greatest exponent. The degree of x + x - 7x + is. After you simplify a polynomial by ombining like terms, you an name the polynomial based on its degree or the number of monomials it ontains. Polynomial 7x x x x 9x x Degree 0 Name Using Degree linear quadrati ubi fourth degree onstant Classifying Polynomials Number of Terms Write eah polynomial in standard form. Then name eah polynomial based on its degree and the number of its terms. a. - x b. x - + x + x -x + Plae terms in order. x + x + x - Plae terms in order. linear binomial 8x + x - Combine like terms. fourth degree trinomial Write eah polynomial in standard form. Then name eah polynomial based on its degree and the number of its terms. See left. a. 6x + 7-9x b. y - - y. 8 + 7v - v Name Using Number of Terms binomial trinomial monomial binomial monomial. Teah Guided Instrution Ativity Diversity Ask knowledgeable students to desribe the purpose of the bird supplies. Math Tip Explain to students that it is reasonable that the degree of - is 0 beause - an be written as -x 0, for x 0. Tatile Learners Have eah student write a polynomial similar to one shown in the table above Example. Redistribute the papers. Diret students to move, using instrutions suh as: All linear binomials go to the front of the room. All ubi monomials go to the windows. Use different ombinations of polynomial names until all students have moved to an area of the room. Have students hek with eah other to make sure they are in the orret loations. Additional Examples Find the degree of eah monomial. a. 8 0 b. xy. 6 Write eah polynomial in standard form. Then name eah polynomial based on its degree and the number of its terms. a. - + 7x 7x ; linear binomial b. x - - x + 7x x ± 7x ; fifth degree trinomial Lesson 9- Adding and Subtrating Polynomials 9 Advaned Learners L Have students simplify x x - 7 x x. x 6 x learning style: verbal English Language Learners ELL Ask: What numbers do the first syllables of eah of the terms monomial, binomial, and trinomial make you think of?,, What does poly mean on the front of a word? many Stress that this number refers to the number of terms in an expression. learning style: verbal 9
Alternative Method Use tiles to demonstrate adding polynomials. Have students model eah polynomial. Group like tiles together. Remove zero pairs. Then write an expression for the remaining tiles. Remind students that red tiles represent negative amounts. Error Prevention Remind students to distribute the negative sign to all terms in the seond parentheses. You may want to enourage students to mark the terms, suh as by irling the whole parentheses, as a reminder. Additional Examples Simplify (6x + x + 7) + (x - 6x - ). 8x x ± Simplify (x + x - 6) - (x + x - ). x ± x x Part Adding and Subtrating Polynomials For: Polynomial Addition Ativity Use: Interative Textbook, 9- Quik Chek You an add polynomials by adding like terms. Adding Polynomials Simplify Ax + 6x + 7 B + Ax - 9x +. B Method Add vertially. Line up like terms. Then add the oeffiients. x + 6x + 7 x 9x 6x - x + 8 Method Add horizontally. Group like terms. Then add the oeffiients. Ax + 6x + 7 B + Ax - 9x + B = Ax + x B + (6x - 9x) + (7 + ) = 6x - x + 8 Simplify eah sum. 0m ± 9 a. Am + B + A8m + B b. At - 6 B + At + B t ±. A9w + 8w B + A7w + B d. Ap + 6p + 0p B + A9p + p + pb 6w ± 8w ± p ± 7p ± p In Chapter, you learned that subtration means to add the opposite. So when you subtrat a polynomial, hange eah of the terms to its opposite. Then add the oeffiients. Resoures Daily Notetaking Guide 9- L Daily Notetaking Guide 9- Adapted Instrution L Closure Say the name of a polynomial using its degree and number of terms. Instrut students to write a polynomial to math the name. Repeat for different types of polynomials. Let students hoose two of the polynomials to both add and subtrat. a. 8v ± v v b. 8d 0d d Quik Chek Subtrating Polynomials Simplify Ax + x - xb - Ax - 8x + B. Method Subtrat vertially. x + x - x Line up like terms. Ax 8x B x + x - x Then add the opposite of eah term in the polynomial x 8x being subtrated. x + x - x- Method Subtrat horizontally. Ax + x - xb - Ax - 8x + B = x + x - x - x + 8x - = Ax - x B + Ax + 8x B - x - Group like terms. = x + x - x - Simplify. Write the opposite of eah term in the polynomial being subtrated. Simplify eah differene. a b. See left. a. Av + 6v - vb - A9v - 7v + vb b. A0d - 9d - db - Ad + d B. Ax + x + B - A6x + x + 8 B x ± x 7 96 Chapter 9 Polynomials and Fatoring 96 pages 97 99 Exerises. x ± x; quadrati binomial 6. x ± 9; linear binomial 7. ± ; quadrati trinomial 8. z ± z ; quadrati trinomial 9. y 8 7y ± y; eighth degree trinomial
EXERCISES Pratie and Problem Solving A GO Pratie by Example for Help Example (page 9) Example (page 9) 9. quadrati trinomial 0. linear binomial. ubi trinomial Example (page 96) Example (page 96) Find the degree of eah monomial.. x. 7. -6 0. 6y w 8 0. 8ab 6. 6 0 7. -9x 8. 0 Name eah expression based on its degree and number of terms. 9. See left. 9. x - x + 0. z +. 7a + a -. x +. -. w + not a polynomial onstant monomial quadrati binomial Write eah polynomial in standard form. Then name eah polynomial based on its degree and number of terms. 0. See margin p. 96.. x - x 6. x + 9 7. - + 8. 9z - z + z - 9. y - 7y + y 8 0. -0 + q - 8q + q Simplify eah sum. 7. See margin.. m + 9 8m ±. k - 8 0k ±. w + w - m 6 7k 7w w 8 8w w ±. A8x + B + Ax + 6B 0x ± 7. Ag + g B + A9g + 7gB 6. Aa + a + B + Aa - 8a + 0B 7. A7y - y + y B + A8y + y B Simplify eah differene. 8. See margin. 8. 6-9. b + 6 0. 7h + h - 8 ( 9) (b ) Ah h 0B. A7n + n B - A0n + n B. Ax + xb - A9x + xb. A6w - w + B - Aw + w - 9B. A-x + x B - Ax + 8x - xb B w w ± 0 x x 7x ± x Apply Your Skills Simplify. Write eah answer in standard form. 8. See left.. 8y ± y. A7y - y + y B + A8y + y + yb 6. Ax - x - B - A8x + - 8x B 6. 6x ± x 7. A-7z + z - B - A-6z + z + B 8. A7a - a + a B + A8a - a - B 7. 7z ± 6z ± z 8. 7a ± a a Geometry Find an expression for the perimeter of eah figure. GPS 9. 8 6 9 0 0. 9x 7 9x For more exerises, see Extra Skill and Word Problem Pratie. x 7x 6 8x. Error Analysis Kwan s work is shown below. What mistake did he make? See margin. (x x ) (x x ) x x x x x x x x x 7x. Pratie Assignment Guide A B -0,, A B -, -0 C Challenge - Test Prep 6-6 Mixed Review 6-86 Homework Quik Chek To hek students understanding of key skills and onepts, go over Exerises,, 6, 9,. Error Prevention! Exerises 0 Many students forget that the sign in front of a term must move with the term. Have students irle eah term, inluding the sign in front of the term. Everything that is inside eah irle must move together. Error Prevention! Exerise Some students may use zero for the oeffiient of w. Remind them that w means w. Exerises 9, 0 Tell students these are multi-step problems in whih they must add the known sides before subtrating from the perimeter. GPS Enrihment Guided Problem Solving Reteahing Adapted Pratie Pratie Name Class Date Pratie 9- Adding and Subtrating Polynomials Write eah polynomial in standard form. Then name eah polynomial based on its degree and number of terms.. y - y + - y. x + x - 6. x +. m - 7m + m. - x + x 6. 7x + x 7. n - n 8. 6 + 7x 9. a + a - a + 0. + x. 7-8a + 6a. x + - x. + x - x. x - x. y - 7 - y 6. x - 6x - 7. v - v + v 8. 8d + d Simplify. Write eah answer in standard form. 9. (x - x) - (x + x + ) 0. (x - x + ) + (x - x + ) L L L L L. (y - y + ) - (y + y + ). (x + x ) - (x + 7x - ) Lesson 9- Adding and Subtrating Polynomials 97. (a + a + 7a) + (a + a - a). (8y - y + 7) - (6y + y - ). (x - 6) + (x + x - ) 6. (n - 7) - (n + n - ) 7. (n + n + ) - (n + n - ) 8. (y - 7y + ) - (y + - y ) 9. (x + 9x - 7) + (x - 6x - ) 0. ( - x - x ) + (x + x - ). (x + x - x ) - (x + x + x). (d + 8 - d) - (d + d - d + ). (x + 7x ) + (x - x ). (6 + - ) - ( + 8) 0. q ± q 8q 0; fourth degree polynomial with terms. 0g ± g 6. 6a 7a ± 7. 8y ± 7y ± y 8. 9. b ± 0. h ± 6h 8. 7n ± n. x ± x. Kwan did not take the opposite of eah term in the polynomial being subtrated.. (y - y - 7) + (y - 6y + 7) 6. ( - 8 + ) - (7 + - 8) 7. (x + x + 9) + (x + x + 6) 8. (x - x + ) - (x + 8x) 9. (7x - x + ) + (7x + ) 0. (x + 7x - 9x ) + ( - x - x). (y + y - ) + (y - 6y ). (x - 8x - ) - (x + 8x - 8). (x - x + 9) - (x - x + 7). (x - 6x + ) - (x + x + ). (x - x - 7) + (9x + + x) 6. (a + a - ) + (a + a + ) 7. (x + - x) - ( + x - x) 8. (n - n - ) + (n - n + ) 9. (x + x) - (x + 6 - x) 0. (7s + s + ) + (s + - s ). (6x - x + 9) - (x + x - ). (x - x + ) + (x - x + 9). (y + y - ) - (y + y + ). ( + x + x) - (x + x + x ). (x + x + ) + (x - x + 7) 6. (7-8x ) + (x - x + ) 7. (x + ) - (x - ) + (x + ) 8. (x + ) - (x - ) + (x - x) Pearson Eduation, In. All rights reserved. 97
. Assess & Reteah Lesson Quiz Write eah expression in standard form. Then name eah polynomial by its degree and number of terms.. - + x - x x ± x ; quadrati trinomial. b - b + 6 b + b ± 6; ubi trinomial. (x + x - ) + (-x + + x ) x ; fourth degree monomial. (-r + r - ) - (r + 6r - ) 9r ; linear binomial Alternative Assessment Diret one student to state the name of a polynomial. Instrut another student to give an example of the named polynomial. Write the polynomial on the board. Let another student name a polynomial. Ask another student to give an example of the polynomial while you write it next to the previous polynomial. On their own piees of paper, have eah student add the two polynomials, write the new polynomial in standard form, and write its name. Repeat for subtration. pages 97 99 Exerises a. monogram: a design omposed of one or more letters, typially the initials of a name; used as an identifying mark binoular: relating to, used by, or involving both eyes at the same time triuspid: having three usps, usually said of a molar tooth polyglot: a person with a speaking, reading, or writing knowledge of several languages GO nline Homework Help Real-World Visit: PHShool.om Web Code: ate-090 C Challenge a. y x ; y 0.x ± 8. or d. The lines interset at x. Connetion There were about million students enrolled in ollege in 980,.8 million in 990, and million in 000.. a. Writing Write the definition of eah word. Use a ditionary if neessary. monogram binoular triuspid polyglot a b. See margin. b. Open-Ended Find other words that begin with mono, bi, tri, or poly.. Do these prefixes have meanings similar to those in mathematis? yes Simplify. Write eah answer in standard form. 8. See margin.. (x + x) + (x - x ). (6g - 7g 8 ) - (g + g + g ). (h - h 9 ) - (-8h + h 0 ) 6. (-t - 9t + 6) + (t + t ) 7. (8b - 6b 7 + b 8 ) + (b 7 - b 9 ) 8. ( + k - 6k ) - (k - k ) Geometry Find an expression for eah missing length. 9. Perimeter = x + 8 0. Perimeter = a - 7 x x ± 8 98 Chapter 9 Polynomials and Fatoring b. Answers may vary. Sample: monopoly, biathlon, tripod, polyhrome x x 6x 8 6a 8 9a 6 8a 9. Critial Thinking Is it possible to write a binomial with degree 0? Explain. No; both terms of a binomial annot be onstants.. a. Write the equations for line P and line Q. y P Q Use slope-interept form. a, d. See left. 6 b. Use the expressions on the right side of eah equation to write a funtion for the vertial distane D(x) between points on lines P and Q with the same x-value. D(x).x. For what value of x does D(x) equal zero? O d. Critial Thinking How does the x-value in part () relate to the graph? 6 x Simplify eah expression.. (ab + ba ) + (a b - ab - ab). (9pq 6 - p q) - (-pq 6 + p q ) a b ab p q p q ± pq 6. Graduation You an model the number of men and women in the United States who enrolled in ollege within a year of graduating from high shool with the linear equations shown below. Let t equal the year of enrollment, with t = 0 orresponding to 990. Let m(t) equal the number of men in thousands, and let w(t) equal the number of women in thousands. m(t) =.t + 6.8 men enrolled in ollege w(t) =.6t + 8. women enrolled in ollege a. Add the expressions on the right side of eah equation to model the total number of reent high shool graduates p(t) who enrolled in ollege between 990 and 998. p(t) 7t ±. b. Use the equation you reated in part (a) to find the number of high shool graduates who enrolled in ollege in 99.,67,00. Critial Thinking If you had subtrated the expressions on the right side of eah equation above, what information would the resulting expression model? the differene between the number of men and the number of women enrolled in a ollege. x ± x ± x. 7g 8 g g ± g. h 0 h 9 ± 8h ± h 6. t ± t ± 6 7. b 9 ± b 8 b 7 ± 8b 8. k ± k k ± 98
Standardized Test Prep Test Prep Multiple Choie 6. Whih expression represents the sum of an odd integer n and the next three odd integers? D A. n + 6 B. n + C. n + 6 D. n + 7. Simplify (8x + ) + (7x + 0). G F. x - 7 G. x + H. x + J. 6x + 0 8. Whih sum is equivalent to (k + k + )? A A. (k + k - ) + (7k + k + 6) B. (k - ) + (k + k + ) C. (8k + k) + (k + ) D. (k + ) + (6k - ) 9. How many terms does the following sum have when it is written in standard form? (x 9 - x + 0x - ) + (x 9 + x + x) G F. 7 G. H. J. 60. Simplify (x + x + x) - (x - x + 6). C A. x + 0 B. x - x C. x + 9x - 6 D. x + x - x + 6 Test Prep Resoures For additional pratie with a variety of test item formats: Standardized Test Prep, p. Test-Taking Strategies, p. 0 Test-Taking Strategies with Transparenies Exerise 6 Explain to students that sine they are finding the sum of four terms, eah ontaining n, the result must inlude n. Tell students to always be areful when working with integer problems. If the problem has onseutive odd or onseutive even integers, eah term inreases by, not. Short Response 6. [] (x ) + (x + ) = 6x + x + = 8x [] no work shown 6. Write and simplify a polynomial expression that represents the perimeter of the retangle. Show your work. See left. 6. Simplify (9x - x + ) - (x + ). Show your work. See margin. x x Mixed Review GO for Help Lesson 8-8 Identify the growth fator in eah funtion. 6. y = 6? x 6. y = 0.6?. x. 6. y =? x 66. y = 0.? x Identify eah funtion as exponential growth or exponential deay. 67 70. See margin. 67. y = 0? x 68. y =.8? 0. x 69. y = 0.? 7 x 70. y = 0.? 0. x 7. a b 0 7. t 7 76. ( ) 77. h 0 78. 0q 7 Lesson 8- Lesson 6-8 Simplify eah expression. 7 78. See left. 7. 7 8? 7 0 7 8 7. 6? -7 7. (x )(9x ) 6x 7. (ab)(a b) 7. (-t )(6t 9 ) 76. (-) 6? (-) 77. (6h )(-h 8 ) 78. (q )(q ) Write an equation for eah translation of y»x». 79 8. See margin. 79. units up 80. right 6 units 8. units down 8. 7 units up 8. left 0 units 8. left 0. units 8.. units up 86.. units down y»x 0» y»x 0.» y»x». y»x». lesson quiz, PHShool.om, Web Code: ata-090 Lesson 9- Adding and Subtrating Polynomials 99 6. [] (9x x ± ) (x ± ) 9x x ± x (9x ) ± ( x x ) ± ( ) 9x x [] one inorret term OR no work shown 67. exponential growth 68. exponential deay 69. exponential growth 70. exponential deay 79. y»x» 80. y»x 6» 8. y»x» 8. y»x» 7 99