38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils How to find specil products How to 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 free-flling object, s shown in Eercises 163 168 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, 4 3 3 2 8 1, 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, 3 4 2 3 1 3 4 2 0 3 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 3 2 5 3 2 is 3 3 2 2 5. 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 3 4 4 2 7 3 5, is of fourth degree nd its leding coefficient is 3. nd 9 5 Definition of Polynomil in Let 1, 2, 3,..., n be rel numbers nd let n be nonnegtive integer. A polynomil in is n epression of the form n n n 1 n 1... 2 2 1 0 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 2 1 5 is not polynomil becuse the eponent in The epression 3 3 1 2 is not polynomil becuse the eponent in 2 1 3 1 2 is negtive. is not n integer.
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. () 5 2 2 7 4 2 (b) 16 8 3 (c) 5 4 6 3 Leding Polynomil Stndrd Form Degree Coefficient () 5 2 2 7 4 2 2 7 5 2 2 4 7 2 (b) 16 8 3 8 3 16 3 8 (c) 5 4 6 3 4 6 3 5 4 1 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: 2 2 3 7 The prefi mono mens one, the prefi bi mens two, nd the prefi tri mens three. Emple 2 Evluting Polynomil Find the vlue of 3 5 2 6 3 when 4. When 4, the vlue of 3 5 2 6 3 is 3 5 2 6 3 4 3 5 4 2 6 4 3 64 80 24 3 5 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 2 3 2 5 nd 2 6. 2 3 2 5 2 6 Write originl polynomils. 2 3 2 2 5 6 Group like terms. 2 3 2 2 1
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 5 3 2 2 7, 3 2 4 7, 3 4 2 8. 5 3 2 2 7 3 2 4 7 3 4 2 8 4 3 9 2 5 6 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 3 2 2 4 from 3 3 5 2 3. 3 3 5 2 3 3 2 2 Write originl 4 polynomils. 3 3 5 2 3 3 2 2 4 Add the opposite. 3 3 3 5 2 2 2 3 4 Group like terms. 2 3 7 2 7 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 3 4 2 3 3 4 from 4 4 2 3 5 2 8. 2 2 3 2 2 2 2 2 3 2 2 2 4 4 2 3 5 2 8 3 4 2 3 3 4 Common error 4 4 2 3 5 2 8 3 4 2 3 3 4 4 5 2 4 12
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 2 2 5 3, multiply ech term of the polynomil by 3. 3 2 2 5 3 3 2 2 3 5 3 3 6 3 15 2 9 Emple 7 Finding Products with Monomil Multipliers Multiply the polynomil by the monomil. () 2 7 3 (b) 4 2 2 3 3 1 () 2 7 3 2 3 7 3 6 2 21 Properties of eponents (b) 4 2 2 3 3 1 4 2 2 3 4 2 3 4 2 1 8 5 12 3 4 2 Properties of eponents Outer First 3 2 2 7 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. 3 2 2 7 3 2 7 2 2 7 Product of First terms 3 2 3 7 2 2 2 7 6 2 21 4 14 Product of Outer terms 6 2 17 14 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. Emple 8 Multiplying Binomils () Use the to multiply 2 by 3. 2 3 3 2 3 2 3 2 6 2 6
42 Chpter P Prerequisites Emple 9 Multiplying Binomils (FOIL Method) Use the FOIL method to multiply the binomils. () 3 9 (b) 3 4 2 1 F O I L () 3 9 2 9 3 27 2 12 27 F O I L (b) 3 4 2 1 6 2 3 8 4 6 2 11 4 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) 4 2 3 1 2 5 4 2 2 5 3 2 5 1 2 5 8 3 20 2 6 2 15 2 5 8 3 26 2 13 5 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. 4 2 2 5 3 2 With verticl formt, line up like terms in the sme verticl columns, much s you lign digits in whole-number multipliction. 4 2 2 2 3 5 20 2 5 10 12 3 3 2 6 4 4 3 2 2 4 4 11 3 25 2 10 Stndrd form Stndrd form 5 4 2 2 3 4 2 2 2 4 2 2
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 4 3. 4 3 4 4 4 4 4 4 2 4 4 16 4 2 8 16 4 2 4 8 4 16 4 3 4 2 8 2 32 16 64 3 12 2 48 64 Write ech fctor. Associtive Property of Multipliction Find 4 4. Emple 13 An Are Model for Multiplying Polynomils Show tht 2 2 1 2 2 5 2. 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 1 1 + 2 2 2 1. Are width length 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 2 + 1 2 2 5 2. Are sum of rectngulr res Becuse ech method must produce the sme re, you cn conclude tht Figure P.11 2 2 1 2 2 5 2. 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 cncel, s follows. 3 3 2 3 3 9 2 9 Sum nd difference of two terms Product hs no middle term.
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. 2 5 2 2 5 2 5 4 2 10 10 25 4 2 20 25 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 v u v u 2 v 2 3 4 3 4 3 2 4 2 9 2 16 Squre of Binomil Emple u v 2 u 2 2uv v 2 4 9 2 4 2 2 4 9 9 2 162 72 81 u v 2 u 2 2uv v 2 6 2 2 2 6 6 2 2 12 36 2 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. () 3 2 3 2 (b) 2 7 2 (c) 2 b 2 () 3 2 3 2 3 2 2 2 Specil product 9 2 4 Simplify. (b) 2 7 2 2 2 2 2 7 7 2 Specil product 4 2 28 49 Simplify. (c) 2 b 2 2 2 2 2 b b 2 Specil product 2 4 4 2b 4b b 2 Simplify.
Applictions Section P.4 Opertions with Polynomils 45 There re mny pplictions tht require the evlution of polynomils. One commonly used second-degree 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 free-flling 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), nd if the initil velocity is negtive, the object ws projected downwrd. t = 0 t = 1 200 ft t = 2 t = 3 Figure P.13 Emple 15 Finding the Height of Free-Flling Object An object is thrown downwrd from the top of 200-foot 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 16 1 2 10 1 200 16 10 200 174 feet. When t 2, the height of the object is Height 16 2 2 10 2 200 64 20 200 116 feet. When t 3, the height of the object is Height 16 3 2 10 3 200 144 30 200 26 feet. 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 3.2368. Wht cn you conclude?
46 Chpter P Prerequisites Emple 16 Using Polynomil Models The numbers of pounds of poultry P nd of beef B consumed per person in the United Sttes from 1985 to 1998 cn be modeled by P 0.084t 2 3.43t 30.5, 5 t 18 Poultry (pounds per person) B 0.143t 2 3.95t 90.4, 5 t 18 Beef (pounds per person) where t 5 represents 1985. Find model tht represents the totl mount T of poultry nd beef consumed from 1985 to 1998. Estimte the totl mount T consumed in 1996. (Source: U.S. Deprtment of Agriculture) The sum of the two polynomil models would be P B 0.084t 2 3.43t 30.5 0.143t 2 3.95t 90.4 The model for the totl consumption of poultry nd beef is T P B 0.059t 2 0.52t 120.9. 0.059t 2 0.52t 120.9 Totl (pounds per person) Using this model, nd substituting t 16, you cn estimte the 1996 consumption to be T 0.059 16 2 0.52 16 120.9. 127.7 pounds per person. Emple 17 Geometry: Finding the Are of Shded Region Find n epression for the re of the shded portion of the figure. 2 + 5 + 3 3 + 1 First find the re of the lrge rectngle A 1 nd the re of the smll rectngle A 2. A 1 2 5 1 nd A 2 3 3 Then to find the re A of the shded portion, subtrct A 2 from A 1. A A 1 A 2 Write formul. 2 5 1 3 3 2 2 7 5 2 9 2 2 7 5 2 9 2 7 14 Substitute. Use FOIL Method nd specil product formul..
Section P.4 Opertions with Polynomils 47 P.4 Eercises In Eercises 1 12, write the polynomil in stndrd form, nd find its degree nd leding coefficient. 1. 10 4 2. 3 2 8 3. 5 3y 4 4. 3 3 2 2 3 5. 8z 16z 2 6. 35t 16t 2 7. 8. 9. 6t 4t 5 t 2 3 10 3 2 15 5 7 5 3 5 2 10. 16 z 2 8z 4z 3 11. 12. 4 In Eercises 13 18, determine whether the polynomil is monomil, binomil, or trinomil. 13. 12 5y 2 14. t 3 15. 3 2 2 4 16. 2u 7 9u 3 17. 1.3 2 18. 2 4 4z 2 In Eercises 19 26, give n emple of polynomil in one vrible stisfying the conditions. (Note: There is more thn one correct nswer.) 19. A monomil of degree 3 20. A trinomil of degree 3 21. A trinomil of degree 4 nd leding coefficient 2 22. A binomil of degree 2 nd leding coefficient 8 23. A monomil of degree 1 nd leding coefficient 7 24. A binomil of degree 5 nd leding coefficient 3 25. A monomil of degree 0 26. A monomil of degree 2 nd leding coefficient 9 In Eercises 27 30, find the vlues of the polynomil t the given vlues of the vrible. 27. 3 12 () 2 (b) 0 28. 1 4 4 2 2 () 2 (b) 2 29. 5 4 4 3 16 16 () 1 (b) 2 30. 3t 4 4t 3 () t 1 (b) t 2 3 In Eercises 31 34, perform the ddition using horizontl formt. 31. 2 2 3 5 2 6 32. 3 3 2 8 3 5 33. 2 3 8 2 2 4 3 2 34. 5y 6 4y 2 6y 3 In Eercises 35 38, perform the ddition using verticl formt. 35. 5 2 3 4 3 2 4 36. 4 3 2 2 8 4 2 6 37. 2b 3 b 2 2b 7 b 2 38. v 2 v 3 4v 1 2v 2 3v In Eercises 39 42, perform the subtrction using horizontl formt. 39. 3 2 2 1 2 2 1 40. 5y 4 2 3y 4 2 41. 10 3 15 6 3 11 42. y 2 3y 4 y 4 y 2 In Eercises 43 46, perform the subtrction using verticl formt. 43. 2 3 2 44. 3z 2 z z 3 2z 2 z 45. 2 3 15 25 2 3 13 12 46. 0.2t 4 5t 2 t 4 0.3t 2 1.4 In Eercises 47 64, perform the opertions. 47. 3 2 8 7 5 2 48. 20s 12s 2 32 15s 2 6s 49. 4 2 5 6 2 2 4 5 50. 13 3 9 2 4 5 5 3 7 3 51. 10 2 11 7 3 12 2 15 52. 15y 4 18y 18 11y 4 8y 8 53. 5s 6s 30s 8 54. 3 2 2 3 9 2 55. 8 3 4 2 3 3 4 2 5 5 56. 5y 2 2y y 2 y 3y 2 6y 2 57. 5 2 3 1 3 3 12 2 4 2 3 2 2 1
48 Chpter P Prerequisites 58. 2 y 2 3y 9 3 4y 4 62. 63. 64. 5 y 2 2y 3 59. 2 t 2 12 5 t 2 5 6 t 2 5 60. 10 v 2 8 v 1 3 v 9 61. 2z 2 z 11 3 z 2 4z 5 2 2z 2 5z 10 7 3t 4 2t 2 t 5t 4 9t 2 4t 3 8t 2 5t 2 5 3 13 4 3 9 2 3 3 3 2 2 6 5 5 t 3 2t 2 t 8 3t 3 t 2 4t 2 4 2t 2 3t 1 t 3 1 In Eercises 65 68, use clcultor to perform the opertions. 65. 8.04 2 9.37 2 5.62 2 66. 11.98y 3 4.63y 3 6.79y 3 67. 4.098 2 6.349 11.246 2 9.342 68. 27.433k 2 19.018k 14.61k 2 3.814k In Eercises 69 96, perform the multipliction nd simplify. 69. 2 2 8 70. 6n 3n 2 71. 2y 5 y 72. 5z 2z 7 73. 4 3 2 2 3 5 74. 3y 2 3y 2 7y 3 75. 2 2 5 3 2 7 3 76. 3 2 11 3 77. 7 4 78. y 2 y 3 79. 5 3 80. 2 y 4 y 81. 2t 1 t 8 82. 3z 5 2z 7 83. 3 4 5 2 4 7 84. 8b 5 1 3b 5 2 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 2 y 3 2y 2 y 3 2y 5 2 3y 2 y 3 2b 5 4 3 3b 5 4y 1 3 12y 9 5t 3 4 2t 16 3 5 5 2 4t 3t t 2 1 5 2 3 2 3 4 2 1 9 3 95. 96. 2t 1 t 1 3 2t 5 5 8y 3 2y 1 y 7 In Eercises 97 100, perform the multipliction using horizontl formt. 97. 3 3 2 2 98. t 3 t 2 5t 1 99. u 5 2u 2 3u 4 100. 1 2 4 6 In Eercises 101 104, perform the multipliction using verticl formt. 101. 7 2 14 9 3 102. 4 4 6 2 9 2 2 3 103. 2 2 1 2 1 104. 2s 2 5s 6 3s 4 In Eercises 105 138, perform the multipliction. 105. 4 4 106. y 7 y 7 107. 6c 6c 108. 8n m 8n m 109. 2t 9 2t 9 110. 5z 1 5z 1 111. 2 1 4 2 1 4 112. 2 3 7 2 3 7 113. 114. 115. 0.2t 0.5 0.2t 0.5 4 0.1b 4 0.1b 3 4 3 4 116. 5 3 5 3 117. 5 2 118. 9 2 119. 5 2 2 120. 3 8 2 121. 2 3b 2 122. 4 5y 2 123. 2 4 3 2 124. y 7 4z 2 125. 126. 127. 128. 129. 2 4 2 2 4 2 2 3 2 2 2 3 t 2 5t 1 2t 2 5 2z 2 3z 7 3z 4 5 3 130. y 2 3 131. 2 3 3 132. 3y 4 3 133. 134. 2 9 5 2 3 t 2 2t 7 2t 2 8t 3 135. 2 y 2 136. 4 y 2 137. 2z y 1 2 138. u v 3 2 In Eercises 139 142, perform the opertions nd simplify. 139. 3 3 2 8 2 140. k 8 k 8 k 2 k 3 141. t 3 2 t 3 2 142. 6 2 6 2
Section P.4 Opertions with Polynomils 49 Geometry In Eercises 143 146, find n epression for the perimeter or circumference of the figure. 143. 144. 2 2 + 3 + 3 145. 3y 146. 2y y + 4 2 4 4y 5 Geometry In Eercises 147 152, find n epression for the re of the shded portion of the figure. 147. 148. 3 + 1 2 5 y + 2 152. Geometric Modeling In Eercises 153 156, () perform the multipliction lgebriclly nd (b) use geometric re model to verify your solution to prt (). 153. 3 154. 2y y 1 155. t 3 t 2 156. 2z 5 z 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. 157. b 0.8 2 1.6 + b 149. 3 + 10 158. + 3 + 4 150. 4 + 2 1 1 4 + 2 + + 151. 6t 7t + 4 5t 2 6t 159. Geometry The length of rectngle is 1 1 2 times its width w. Find epressions for () the perimeter nd (b) the re of the rectngle. 160. Geometry The bse of tringle is 3 nd its height is 5. Find n epression for the re A of the tringle. 161. Personl Finnce After 2 yers, n investment of $1000 compounded nnully t n interest rte of r will yield n mount 1000 1 r 2. Find this product.
50 Chpter P Prerequisites 162. Personl Finnce After 2 yers, n investment of $1000 compounded nnully t n interest rte of 9.5% will yield n mount 1000 1 0.095 2. Find this product. Free-Flling Object In Eercises 163 166, use the position polynomil to determine whether the free-flling object ws dropped, thrown upwrd, or thrown downwrd. Also determine the height of the object t time t 0. 163. 16t 2 100 164. 16t 2 50t 165. 16t 2 24t 50 166. 16t 2 32t 300 167. Free-Flling Object An object is thrown upwrd from the top of 200-foot 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 3. (b) During the given period, the per cpit consumption of whole milk ws decresing nd the per cpit consumption of lowft milk ws incresing (see figure). Ws the combined per cpit consumption of milk incresing or decresing? Milk (in gllons per cpit) 11 10 9 8 7 6 5 4 3 2 1 y 0 1 2 3 4 5 6 7 8 Yer (0 1990) Whole milk Lowft milk t Synthesis 200 ft 250 ft Figure for 167 Figure for 168 168. Free-Flling Object An object is thrown downwrd from the top of 250-foot 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 3. 169. The per cpit consumption (verge consumption per person) of whole milk W nd lowft milk L in the United Sttes between 1990 nd 1998 cn be pproimted by these two polynomil models. W 0.024t 2 0.46t 10.2 0 t 8 Whole milk L 0.016t 2 0.35t 4.9 0 t 8 Low-ft milk In these models, W nd L represent the verge consumption per person in gllons nd t represents the yer, with t 0 corresponding to 1990. (Source: U.S. Deprtment of Agriculture) () Find polynomil model tht represents the per cpit consumption of milk (of both types) during this time period. Use this model to find the per cpit consumption of milk in 1993 nd in 1997. 170. Eplin why y 2 is not equl to 2 y 2. 171. Is every trinomil second-degree polynomil? Eplin. 172. Cn two third-degree polynomils be dded to produce second-degree polynomil? If so, give n emple. 173. Perform the multiplictions. () 1 1 (b) 1 2 1 (c) 1 3 2 1 From the pttern formed by these products, cn you predict the result of 1 4 3 2 1? 174. Eplin why 2 3 is not polynomil.