Operations with Polynomials


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1 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply polynomils Use specil products to multiply polynomils Use opertions with polynomils in ppliction problems Why you should lern it: Opertions with polynomils enble you to model vrious spects of the physicl world, such s the position of freeflling object, s shown in Eercises on pge 50. Bsic Definitions An lgebric epression contining only terms of the form k, where is ny rel number nd k is nonnegtive integer, is clled polynomil in one vrible or simply polynomil. Here re some emples of polynomils in one vrible. 3 8, , In the term k, is clled the coefficient, nd k the degree, of the term. Note tht the degree of the term is 1, nd the degree of constnt term is 0. Becuse polynomil is n lgebric sum, the coefficients tke on the signs between the terms. For instnce, hs coefficients 1, 4, 0, nd 3. Polynomils re usully written in order of descending powers of the vrible. This is referred to s stndrd form. For emple, the stndrd form of is Stndrd form The degree of polynomil is defined s the degree of the term with the highest power, nd the coefficient of this term is clled the leding coefficient of the polynomil. For instnce, the polynomil , is of fourth degree nd its leding coefficient is 3. nd 9 5 Definition of Polynomil in Let 0, 1, 2, 3,..., n be rel numbers nd let n be nonnegtive integer. A polynomil in is n epression of the form n n n1 n where n 0. The polynomil is of degree n, nd the number n is clled the leding coefficient. The number is clled the constnt term. 0 The following re not polynomils, for the resons stted. The epression is not polynomil becuse the eponent in The epression is not polynomil becuse the eponent in integer. is negtive is not nonnegtive
2 Section P.4 Opertions with Polynomils 39 Emple 1 Identifying Leding Coefficients nd Degrees Write the polynomil in stndrd form nd identify the degree nd leding coefficient of the polynomil. () (b) (c) Leding Polynomil Stndrd Form Degree Coefficient () (b) (c) Now try Eercise 7. A polynomil with only one term is monomil. Polynomils with two unlike terms re binomils, nd those with three unlike terms re trinomils. Here re some emples. Monomil: 5 3 Binomil: 4 3 Trinomil: The prefi mono mens one, the prefi bi mens two, nd the prefi tri mens three. Emple 2 Evluting Polynomil Find the vlue of when 4. When 4, the vlue of is Now try Eercise 27. Substitute 4 for. Evlute terms. Simplify. Adding nd Subtrcting Polynomils To dd two polynomils, simply combine like terms. This cn be done in either horizontl or verticl formt, s shown in Emples 3 nd 4. Emple 3 Adding Polynomils Horizontlly Use horizontl formt to dd nd Write originl polynomils Group like terms Now try Eercise 31.
3 40 Chpter P Prerequisites To use verticl formt to dd polynomils, lign the terms of the polynomils by their degrees, s shown in the following emple. Emple 4 Using Verticl Formt to Add Polynomils Use verticl formt to dd , , Now try Eercise 33. nd To subtrct one polynomil from nother, dd the opposite. You cn do this by chnging the sign of ech term of the polynomil tht is being subtrcted nd then dding the resulting like terms. Emple 5 Subtrcting Polynomils Horizontlly Use horizontl formt to subtrct from Now try Eercise 39. Write originl polynomils. Add the opposite. Group like terms. Study Tip The common error illustrted to the right is forgetting to chnge two of the signs in the polynomil tht is being subtrcted. When subtrcting polynomils, remember to dd the opposite of every term of the subtrcted polynomil. Be especilly creful to get the correct signs when you re subtrcting one polynomil from nother. One of the most common mistkes in lgebr is to forget to chnge signs correctly when subtrcting one epression from nother. Here is n emple. Wrong sign Wrong sign Emple 6 Using Verticl Formt to Subtrct Polynomils Use verticl formt to subtrct from Now try Eercise 45. Common error
4 Multiplying Polynomils Section P.4 Opertions with Polynomils 41 The simplest type of polynomil multipliction involves monomil multiplier. The product is obtined by direct ppliction of the. For instnce, to multiply the monomil 3 by the polynomil , multiply ech term of the polynomil by Emple 7 Finding Products with Monomil Multipliers Multiply the polynomil by the monomil. () 2 73 (b) () Properties of eponents (b) Properties of eponents Now try Eercise 71. Outer First Inner Lst FOIL Digrm To multiply two binomils, you cn use both (left nd right) forms of the. For emple, if you tret the binomil 2 7 s single quntity, you cn multiply 3 2 by 2 7 s follows Product of First terms Product of Outer terms Product of Inner terms Product of Lst terms The four products in the boes bove suggest tht you cn put the product of two binomils in the FOIL form in just one step. This is clled the FOIL Method. Note tht the words first, outer, inner, nd lst refer to the positions of the terms in the originl product (see digrm t the left). Emple 8 Multiplying Binomils () Use the to multiply 2 by Now try Eercise 77.
5 42 Chpter P Prerequisites Emple 9 Multiplying Binomils (FOIL Method) Use the FOIL method to multiply the binomils. () 3 9 (b) F O I L () F O I L (b) Now try Eercise 81. To multiply two polynomils tht hve three or more terms, you cn use the sme bsic principle tht you use when multiplying monomils nd binomils. Tht is, ech term of one polynomil must be multiplied by ech term of the other polynomil. This cn be done using either horizontl or verticl formt. Emple 10 Multiplying Polynomils (Horizontl Formt) Now try Eercise 97. When multiplying two polynomils, it is best to write ech in stndrd form before using either the horizontl or verticl formt. This is illustrted in the net emple. Emple 11 Multiplying Polynomils (Verticl Formt) Write the polynomils in stndrd form nd use verticl formt to multiply With verticl formt, line up like terms in the sme verticl columns, much s you lign digits in wholenumber multipliction Write in stndrd form Write in stndrd form Now try Eercise 101.
6 Section P.4 Opertions with Polynomils 43 EXPLORATION Use the FOIL Method to find the product of where is constnt. Wht do you notice bout the number of terms in your product? Wht degree re the terms in your product? Polynomils re often written with eponents. As shown in the net emple, the properties of lgebr re used to simplify these epressions. Emple 12 Multiplying Polynomils Epnd Now try Eercise 129. Write ech fctor. Associtive Property of Multipliction Multiply 4 4. Emple 13 An Are Model for Multiplying Polynomils Show tht An pproprite re model to demonstrte the multipliction of two binomils would be A lw, the re formul for rectngle. Think of rectngle whose sides re 2 nd 2 1. The re of this rectngle is Are widthlength Another wy to find the re is to dd the res of the rectngulr prts, s shown in Figure P.11. There re two squres whose sides re, five rectngles whose sides re nd 1, nd two squres whose sides re 1. The totl re of these nine rectngles is Are sum of rectngulr res Figure P Becuse ech method must produce the sme re, you cn conclude tht Now try Eercise 155. Specil Products Some binomil products hve specil forms tht occur frequently in lgebr. For instnce, the product 3 3 is clled the product of the sum nd difference of two terms. With such products, the two middle terms subtrct out, s follows Sum nd difference of two terms Product hs no middle term.
7 44 Chpter P Prerequisites Another common type of product is the squre of binomil. With this type of product, the middle term is lwys twice the product of the terms in the binomil Squre of binomil Outer nd inner terms re equl. Middle term is twice the product of the terms in the binomil. Specil Products Let u nd v be rel numbers, vribles, or lgebric epressions. Then the following formuls re true. Sum nd Difference of Sme Terms Emple u vu v u 2 v Squre of Binomil Emple u v 2 u 2 2uv v u v 2 u 2 2uv v b b The squre of binomil cn lso be demonstrted geometriclly. Consider squre, ech of whose sides re of length b. (See Figure P.12). The totl re includes one squre of re 2, two rectngles of re b ech, nd one squre of re b 2. So, the totl re is 2 2b b 2. + b Emple 14 Finding Specil Products b b + b Figure P.12 b 2 Multiply the polynomils. () (b) (c) 2 b 2 () Sum nd difference of sme terms Simplify. (b) Squre of binomil Simplify. (c) 2 b b b 2 Squre of binomil b 4b b 2 Simplify. Now try Eercise 107.
8 Applictions Section P.4 Opertions with Polynomils 45 There re mny pplictions tht require the evlution of polynomils. One commonly used seconddegree polynomil is clled position polynomil. This polynomil hs the form 16t 2 v 0 t s 0 Position polynomil where t is the time, mesured in seconds. The vlue of this polynomil gives the height (in feet) of freeflling object bove the ground, ssuming no ir resistnce. The coefficient of t, v 0, is clled the initil velocity of the object, nd the constnt term, s 0, is clled the initil height of the object. If the initil velocity is positive, the object ws projected upwrd (t t 0), if the initil velocity is negtive, the object ws projected downwrd, nd if the initil velocity is zero, the object ws dropped. t = 0 t = ft t = 2 t = 3 Figure P.13 Emple 15 Finding the Height of FreeFlling Object An object is thrown downwrd from the top of 200foot building. The initil velocity is 10 feet per second. Use the position polynomil 16t 2 10t 200 to find the height of the object when t 1, t 2, nd t 3 (see Figure P.13). When t 1, the height of the object is Height feet. When t 2, the height of the object is Height feet. When t 3, the height of the object is Height feet. Now try Eercise 167. In Emple 15, the initil velocity is 10 feet per second. The vlue is negtive becuse the object ws thrown downwrd. If it hd been thrown upwrd, the initil velocity would hve been positive. If it hd been dropped, the initil velocity would hve been zero. Use your clcultor to determine the height of the object in Emple 15 when t Wht cn you conclude?
9 46 Chpter P Prerequisites Emple 16 Using Polynomil Models The numbers of vehicles (in thousnds) fueled by compressed nturl gs G nd by electricity E in the United Sttes from 1995 to 2003 cn be modeled by G 0.079t t 3.2, 5 t 13 Vehicles fueled by nturl gs E 1.090t t 51.6, 5 t 13 Vehicles fueled by electricity where t represents the yer, with t 5 corresponding to Find model tht represents the totl numbers T of vehicles fueled by compressed nturl gs nd by electricity from 1995 to Then estimte the totl number T of vehicles fueled by compressed nturl gs nd by electricity in (Source: Science Applictions Interntionl Corportion nd Energy Informtion Administrtion) The sum of the two polynomil models is s follows. G E 0.079t t t t 51.6 So, the polynomil tht models the totl numbers of vehicles fueled by compressed nturl gs nd by electricity is T G E 1.169t t t t 54.8 Using this model, nd substituting t 12, you cn estimte the totl number of vehicles fueled by compressed nturl gs nd by electricity in 2002 to be T thousnd vehicles. Now try Eercise Figure P.14 A 1. Emple 17 Geometry: Finding the Are of Shded Region Find n epression for the re of the shded portion in Figure P.14. First find the re of the lrge rectngle A 1 nd the re of the smll rectngle A 2. A nd A Then to find the re A of the shded portion, subtrct A 2 from A A 1 A Write formul. Substitute Use FOIL Method nd 9 specil product formul Now try Eercise 149.
10 Section P.4 Opertions with Polynomils 47 P.4 Eercises VOCABULARY CHECK: Fill in the blnks. 1. The epression n n n1 n n 0 is clled. 2. The of polynomil is the degree of the term with the highest power, nd the coefficient of this term is the of the polynomil. 3. A polynomil with one term is clled, while polynomil with two unlike terms is clled, nd polynomil with three unlike terms is clled. 4. The letters in FOIL stnd for the following. F O I L 5. The product u vu v u 2 v 2 is clled the nd of terms. 6. The product u v 2 u 2 2uv v 2 is clled the of. 7. The epression 16t 2 v 0 t s 0 is clled the, nd v 0 is the initil nd s 0 is the initil. In Eercises 1 12, write the polynomil in stndrd form, nd find its degree nd leding coefficient y z 16z t 16t t 4t 5 t z 2 8z 4z In Eercises 13 18, determine whether the polynomil is monomil, binomil, or trinomil y t u 7 9u z 2 In Eercises 19 26, give n emple of polynomil in one vrible stisfying the conditions. (There re mny correct nswers.) 19. A monomil of degree A trinomil of degree A trinomil of degree 4 nd leding coefficient A binomil of degree 2 nd leding coefficient A monomil of degree 1 nd leding coefficient A binomil of degree 5 nd leding coefficient A monomil of degree A monomil of degree 2 nd leding coefficient 9 In Eercises 27 30, evlute the polynomil for ech specified vlue of the vrible () 2 (b) () 2 (b) () 1 (b) 30. 3t 4 4t 3 () t 1 (b) t 2 3 In Eercises 31 34, perform the ddition using horizontl formt y 6 4y 2 6y 3 9 2y 11y 2 In Eercises 35 38, perform the ddition using verticl formt b 3 b 2 2b 7 b v 2 v 3 4v 1 2v 2 3v In Eercises 39 42, perform the subtrction using horizontl formt y 4 2 3y y 2 3y 4 y 4 y 2 In Eercises 43 46, perform the subtrction using verticl formt z 2 z z 3 2z 2 z t 4 5t 2 t 4 0.3t
11 48 Chpter P Prerequisites In Eercises 47 68, perform the indicted opertion(s) s 12s s 2 6s y 4 18y 18 11y 4 8y s 6s 30s y 2 2y y 2 y 3y 2 6y y 2 3y 9 34y 4 5y 2 2y t t 2 5 6t v 2 8v 1 3v z 2 z 11 3z 2 4z 5 22z 2 5z 10 73t 4 2t 2 t 5t 4 9t 2 4t 38t 2 5t t 3 2t 2 t 8 3t 3 t 2 4t 2 42t 2 3t 1 t y y y k k 14.61k k In Eercises 69 96, perform the multipliction nd simplify n3n y5 y 72. 5z2z y 2 3y 2 7y y 2y y4 y 81. 2t 1t z 52z b 5 13b y3 2y y3 2y y 2 y b b y 1 312y 9 5t 3 42t t3tt t 1t 1 32t y 3 2y 1y 7 In Eercises , perform the multipliction using horizontl formt t 3t 2 5t u 52u 2 3u In Eercises , perform the multipliction using verticl formt s 2 5s 63s 4 In Eercises , perform the multipliction y 7y c 6c n m8n m t 92t z 15z t t b4 0.1b b y y 7 4z t 2 5t 12t 2 5 2z 2 3z 73z y y t 2 2t 72t 2 8t y y z y u v 3 2
12 Section P.4 Opertions with Polynomils 49 In Eercises , perform the indicted opertions nd simplify k 8k 8 k 2 k t 3 2 t Geometry In Eercises , write n epression for the perimeter or circumference of the figure y Geometric Modeling In Eercises , () perform the multipliction lgebriclly nd (b) use geometric re model to verify your solution to prt () yy t 3t z 5z 1 Geometric Modeling In Eercises 157 nd 158, use the re model to write two different epressions for the totl re. Then equte the two epressions nd nme the lgebric property tht is illustrted b + b 2y y y Geometry In Eercises , write n epression for the re of the shded portion of the figure y t 7t t 2 6t 159. Geometry The length of rectngle is times its width w. Write epressions for () the perimeter nd (b) the re of the rectngle Geometry The bse of tringle is 3 nd its height is 5. Write n epression for the re A of the tringle Compound Interest After 2 yers, n investment of $1000 compounded nnully t n interest rte of r will yield n mount r 2. Find this product Compound Interest After 2 yers, n investment of $1000 compounded nnully t n interest rte of 3.5% will yield n mount Find this product
13 50 Chpter P Prerequisites FreeFlling Object In Eercises , use the position polynomil to determine whether the freeflling object ws dropped, thrown upwrd, or thrown downwrd. Then determine the height of the object t time t t t 2 50t t 2 24t t 2 32t FreeFlling Object An object is thrown upwrd from the top of 200foot building (see figure). The initil velocity is 40 feet per second. Use the position polynomil 16t 2 40t 200 to find the height of the object when t 1, t 2, nd t ft 250 ft Per cpit consumption (in gllons) (b) During the given time period, the per cpit consumption of beverge milks ws decresing nd the per cpit consumption of bottled wter ws incresing (see figure). Ws the combined per cpit consumption of both beverge milks nd bottled wter incresing or decresing over the given time period? y Figure for 169 Beverge milks Bottled wter Yer (5 1995) t Synthesis Figure for 167 Figure for FreeFlling Object An object is thrown downwrd from the top of 250foot building (see figure). The initil velocity is 25 feet per second. Use the position polynomil 16t 2 25t 250 to find the height of the object when t 1, t 2, nd t Beverge Consumption The per cpit consumption of ll beverge milks M nd bottled wter W in the United Sttes from 1995 to 2003 cn be pproimted by the following two polynomil models. M t t Beverge milks W t t 9.40 Bottled wter In these models, the per cpit consumption is given in gllons nd t5 t 13 represents the yer, with t 5 corresponding to (Source: USDA/Economic Reserch Service) () Find polynomil model tht represents the per cpit consumption of both beverge milks nd bottled wter during the given time period. Use this model to find the per cpit consumption of beverge milks nd bottled wter in 1999 nd Writing Eplin why y 2 is not equl to 2 y Think About It Is every trinomil seconddegree polynomil? Eplin Think About It Cn two thirddegree polynomils be dded to produce seconddegree polynomil? If so, give n emple Perform the multiplictions. () 1 1 (b) (c) From the pttern formed by these products, cn you predict the result of ? 174. Writing Eplin why is not polynomil.
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