Chapter P Fundamental Concepts of Algebra
Section P.1
Check Point Exercises
1. 22 86(3)86(133) x +−=+− 2 86(10) 86(100) 8600 608 =+ =+ =+ =
2. a. Since2016is16yearsafter2000,substitute16 for x 2 2 3613193 (16)361(16)3193 8713 Txx=−++ =−++ = Theaveragecostoftuitionandfeesatpublic U.S.collegesfortheschoolyearendingin 2016was$8713.
b. Theformulaunderestimatestheactualanswer by$65.
3. Theelementscommonto{3,4,5,6,7}and {3,7,8,9}are3and7.
{3,4,5,6,7}{3,7,8,9}{3,7} ∩=
4. Theunionisthesetcontainingalltheelementsof eitherset.
{3,4,5,6,7}{3,7,8,9}{3,4,5,6,7,8,9} ∪=
5. 9,1.3,0,0.3,,9,10 2 π
a. Naturalnumbers:9because93 =
b. Wholenumbers:0,9
c. Integers:9,0,9
d. Rationalnumbers:9,1.3,0,0.3,9
e. Irrationalnumbers:,10 2 π
f. Realnumbers:9,1.3,0,0.3,,9,10 2 π
6. a. 12
Because21.4, ≈ thenumberinsidethe absolutevaluebarsisnegative.Theabsolute valueof x when x <0is–x. Thus, () 121221 −=−−=−
b. 3 π
Because3.14, π ≈ thenumberinsidethe absolutevaluebarsispositive.Theabsolute valueofapositivenumberisthenumberitself. Thus, 33.ππ−=−
c. x x
Because0,. xxx >= Thus,1 xx xx ==
7. 4(5)99 −−=−= Thedistancebetween–4and5is9.
8. 22 7(43)2(5) xxxx +++ 22 22 2 7(43)2(5) 2821102 3823 xxxx xxxx xx =+++ =+++ =+
9. 64[7(2)] x +−− 64[72)] 64[9] 6364 424 x x x x =+−+ =+− =+− =−
Concept and Vocabulary Check P.1
C1. expression
C2. b tothe nthpower;base;exponent
C3. formula;modeling;models
C4. intersection; AB ∩
C5. union; AB ∪
C6. natural
C7. whole
C8. integers
C9. rational
C10. irrational
C11. rational;irrational
C12. absolutevalue; x, x
C13. ba + ; ba
C14. ()abc ++ ;()abc
C15. abac +
C16. 0;inverse;0;identity
C17. inverse;1;identity
C18. simplified
C19. a
Exercise Set P.1
1. 75(10)75057 +=+=
2. () 86583038 +=+=
3. 6(3)818810 −=−=
4. () 83424420 −=−=
5. 2 83(8)642488 +=+=
6. () 2 656363066 +=+=
7. 2 76(7)3494237310 −+=−+=+=
8. () 2 8784645648412 −+=−+=+=
9. 33 45(97)45(2) 45(8)44044 +−=+ =+=+=
10. ()() () 336586652 658 64046 +−=+ =+ =+=
11. 2 83(82)643(6) 641846 −−=− =−=
12. ()() 2 84836445642044 −−=−=−=
13. 5(2)5(102) 2142(10)14 5(12) 6 52 10 x x ++ = = =⋅ =
14. 7(3)7(93)7(6)7321 2162(9)162 x x ===⋅=
15. 23 ;2,4 1 xy xy x + =−= + ()() 22344128 8 2111 −+ −+ ====− −+−−
16. 2 ;2and4 2 xy xy xyx + =−= () ()()() 224440 0 2422844 −+ −+ === −−−−+
17. 5(5032) 9 C =− 5(18)10 9 ==
50°Fisequivalentto10°C.
18. 555 (32)(8632)(54)30 999 CF=−=−==
86°Fisequivalentto30°C.
19. 22 46016460(2)16(2) 412016(4)412064 1246460 htt=+−=+− =+−=+− =−=
Twosecondsafteritiskicked,theball’sheightis 60feet.
2Copyright©2022PearsonEducation,Inc
20. 2 2 46016
460(3)16(3)
418016(9) 4180144 18414440 htt =+− =+− =+− =+− =−= Threesecondsafteritiskicked,theball’sheightis 40feet.
21. { } { } { }1,2,3,42,4,52,4 =
22. { } { } { }1,3,72,3,83 =
23. { } { } { } ,,,,,, settesset =
24. { } { } { } ,,,,,,,,, reallearreal =
25. { } { } { } 1,3,5,72,4,6,8,10 = Theemptysetisalsodenotedby ∅
26. { } { } { } 1,3,5,75,3,1−−−= or ∅
27. { } ,,, abcd ∅=∅
28. { } ,, wyz ∅=∅
29. { } { } { }1,2,3,42,4,51,2,3,4,5 =
30. { } { } { }1,3,7,82,3,81,2,3,7,8 =
31. { } { } {} 1,3,5,72,4,6,8,10 1,2,3,4,5,6,7,8,10 =
32. { } { } { }0,1,3,52,4,60,1,2,3,4,5,6 =
33. { } { } ,,,,,,,, aeiouaeiou ∅=
34. { } { } ,,,,,,,, emptyempty ∅=
35. a. 100
b. 0,100
c. 9,0,100
d. 4 9,,0,0.25,9.2,100 5
e. 3
f. 4 9,,0,0.25,3,9.2,100 5
36. a. 49
b. 0,49
c. 7,0,49
d. 7,0.6,0,49
e. 50
f. 7,0.6,0,49,50
37. a. 64
b. 0,64
c. 11,0,64
d. 5 11,,0,0.75,64 6
e. 5, π
f. 5 11,,0,0.75,5,,64 6 π
38. a. 4
b. 0,4
c. 5,0,4
d. 5,0.3,0,4
e. 2
f. 5,0.3,0,2,4
39. 0
40. Answerswillvary.Anexampleis 1 . 2
41. Answerswillvary.Anexampleis2.
42. Answerswillvary.Anexampleis2.
43. true;–13istotheleftof–2onthenumberline.
44. false;–6istotheleftof2onthenumberline.
45. true;4istotherightof–7onthenumberline.
46. true;–13istotheleftof–5onthenumberline.
47. true; −π=−π
48. true;–3istotherightof–13onthenumberline.
49. true;0istotherightof–6onthenumberline.
50. true;0istotherightof–13onthenumberline.
51. 300300 =
52. 203203−=
53. 1212−π=−π
54. 77−π=−π
55. 2552 −=−
56. 513135 −=−
57. 33 1 33 ==−
58. 77 1 77 ==−
59. 373744 −−−=−=−=
60. 51351388 −−−=−=−=
61. 2(5)33 xy+=+−=−=
62. 2(5)77 xy−=−−==
63. 25257 xy+=+−=+=
64. 25253 xy−=−−=−=−
65. 55 1 55 y y ===−
66. 2525 25251(1)0 xy xy +=+=+=+−=
67. Thedistanceis2171515 −=−=
68. Thedistanceis4151111 −=−= .
69. Thedistanceis2577 −−=−=
70. Thedistanceis681414 −−=−= .
71. Thedistanceis19(4)1941515 −−−=−+=−=
72. Thedistanceis26(3)2632323 −−−=−+=−=
73. Thedistanceis
3.6(1.4)3.61.42.22.2 −−−=−+=−=
74. Thedistanceis
5.4(1.2)5.41.24.24.2 −−−=−+=−=
75. 6+(–4)=(–4)+6; commutativepropertyofaddition
76. 11(74)117114; ⋅+=⋅+⋅ distributivepropertyofmultiplicationoveraddition
77. 6+(2+7)=(6+2)+7; associativepropertyofaddition
78. 6(23)6(32); ⋅⋅=⋅⋅ commutativepropertyofmultiplication
79. (2+3)+(4+5)=(4+5)+(2+3); commutativepropertyofaddition
80. 7(118)(118)7; ⋅⋅=⋅⋅ commutativepropertyofmultiplication
81. 2(–8+6)=–16+12; distributivepropertyofmultiplicationoveraddition
82. 8(311)24(88) −+=−+− ; distributivepropertyofmultiplicationoveraddition
83. () 1 31;3 3 xx x +=≠− + , inversepropertyofmultiplication
84. ()()440;xx++−+= inversepropertyofaddition
85. 5(34)453544 15204 1516 xx x x +−=⋅+⋅− =+− =+
4Copyright©2022PearsonEducation,Inc
86. 2(54)325243 1083 105 xx x x +−=⋅+⋅− =+− =+
87. 5(32)12535212 151012 2710 xxxx xx x −+=⋅−⋅+ =−+ =−
88. 2(51)14252114 10214 242 xxxx xx x −+=⋅−⋅+ =−+ =−
89. 7(35)2(43) 73752423 213586 2929 yy yy yy y −++ =⋅−⋅+⋅+⋅ =−++ =−
90. 4(26)3(510) 424635310 8241530 236 yy yy yy y −++ =⋅−⋅+⋅+⋅ =−++ =+
91. ()() 53272151072 812 yyyy y −−+=−−− =−
92. 4(53)(63)201263 1415 yyyy y −−+=−−− =−
93. () [ ] 7434574345 yy −−−=−−+ [ ] 7484 73216 1625 y y y =−− =−+ =−
94. [ ] [] 65[8(24)]65824 65122 66010 1054 yy y y y −−−=−−+ =−− =−+ =−
95. () () 22 22 22 22 22 22 184625 1846125 18467 18467 18647 186111211 xx xx xx xx xx xx +−−+ =+−−+ =+−− =+−+ =−++ =−+=+
Section P.1 AlgebraicExpressions,MathematicalModels,andRealNumbers
96. () () 22 22 22 22 22 2 2 145724 1457144 145710 145710 147510 14715 715 xx xx xx xx xx x x +−−+ =+−−+ =+−− =+−+ =−++ =−+ =+
97. –(–14x)=14x 98. ()1717yy−−=
99. –(2x –3y –6)=–2x +3y +6
100. () 51315131 xyxy −−−=−++
101. [] 1 3(3)(4)(4)0xyyxx ++−=+=
102. ()() 1 2770 2 yxxyy +−+=+=
103. 63 63 63 > Since63,63. >−>−
104. 2050 2050 2050 < Since2050,2050. <−<−
105. 3 0.6 5 0.60.6 0.60.6 0.60.6 = Since 3 0.60.6,0.6. 5 ==−
Copyright©2022PearsonEducation,Inc.5
106. 5 2.5 2 2.52.5 .52.5 2.52.5 2 = Since 5 2.52.5,2.5. 2 ==−
107. 3031415 4041514 303014 404015 15 14 01 01 < Since 3031415 01, 4041514 <−<⋅
108. 1718505 1817606 17 ⋅− 18 18 17 5050 6060 10 10 > Since1718505 10,> 1817606 >⋅−
109. 88 1 1313 813 1 138 11 11 ÷− = Since8811,=1. 1313 =÷−
110. 44 2 1717 417 2 174 21 21 −÷ > Since44 21,2>. 1717 >−÷
111. 22 8162436416443 64443 64163 483 45 −÷⋅−=−÷⋅− =−⋅− =−− =− =
112. 22 101005231001002523 100423 10083 923 89 −÷⋅−=−÷⋅− =−⋅− =−− =− = 113. 2 222 2 2 523529 [3(2)][9(2)] 109 [92] 109 11 1 121 = = + = = 114. 22 2 10234512 (1232)(126) 17 6 17 36 ÷+⋅+ = = =
115. 83[2(25)4(86)]83[2(3)4(2)] 83[68] 83[2] 86 14 −−−−−=−−−− =−− =−− =+ = 116. 83[2(57)5(42)]83[2(2)5(2)] 83[410] 83[6] 818 26 −−−−−=−−−− =−− =−− =+ = 117. 2(2)4(3)412 583 8 3 8 3 −−−−+ = = =−
6(4)5(3)2415
121. ()444xxxx −+=−−=−
122. ()8828xxxxx −−=−+=−
123. () 6530xx−=−
124. () 10440xx−=−
125. 523xxx −=
126. () 62628 xxxxx −−=+=
127. () 83683656 xxxxx −+=−−=−
128. () 8368318310 xxx −+=−−=−−
129. a. () () () 7 220 10 7 22020 10 7 200 10 140 Ha H =− =− = = Thelowerlimitoftheheartratefora20-yearoldwiththisexercisegoalis140beatsper minute.
b. () () () 4 220 5 4 22020 5 4 200 5 160 Ha H =− =− = =
Theupperlimitoftheheartratefora20-yearoldwiththisexercisegoalis160beatsper minute.
Thelowerlimitoftheheartratefora30-yearoldwiththisexercisegoalis95beatsper minute.
b. () () () 3 220 5 3 22030 5 3 190 5 114 Ha H =− =− = =
Theupperlimitoftheheartratefora30-yearoldwiththisexercisegoalis114beatsper minute.
131. a. Model1
97513,547
975(18)13,547
31,097
Model1estimatesthecosttohavebeen $31,097in2018,whichoverestimatesthe actualcostby$366.
Model2 2 2 3233115,479
32(18)331(18)15,479
31,805
Model2estimatesthecosttohavebeen $31,805in2018,whichoverestimatesthe actualcostby$1074.
b. Model2 2 2 3233115,479
32(30)331(30)15,479
54,209
Model2estimatesthecostwillbe$54,209in 2030.
132. a. Model1
23,297 Tx
97513,547
975(10)13,547
Model1estimatesthecosttohavebeen $23,297in2010,whichoverestimatesthe actualcostby$1028.
Model2 2 2 3233115,479
32(10)331(10)15,479
21,989
Model2estimatesthecosttohavebeen $21,989in2010,whichunderestimatesthe actualcostby$280.
b. Model2 2 2 3233115,479
Txx=++ =++ =
32(25)331(25)15,479
43,754
Model2estimatesthecostwillbe$43,754in 2025.
133. a. ()0.0150.01756000 0.0151050.0175 1050.0025 xx xx x +− =+− =−
b. () () () () () 0.0150.01756000 0.01544000.017560004400 0.01544000.01751600 94 xx +− =+− =+ = () 1050.0025 1050.00254400 94 x =− = Bothexpressionsdeterminethetotalinterestis $94
c. () 1050.0025 1050.00253000 97.5 x =− = Thetotalinterestis$97.50
134. a. ()0.060.550 0.06250.5 250.44 tt tt t +− =+− =−
b. () () () () 0.060.550 0.060.55020 0.06200.530 16.2 tt t +− =+− =+ = () 250.44 250.4420 16.2 t =− = Bothexpressionsdeterminethetotaldistanceis 16.2miles.
c. () 250.44 250.4425 14 t =− = Thetotaldistanceis14miles
135. – 143. Answerswillvary.
144. doesnotmakesense;Explanationswillvary. Sampleexplanation:Modelsdonotalways accuratelypredictfuturevalues.
145. doesnotmakesense;Explanationswillvary. Sampleexplanation:Tousethemodel,substitute0 for x
146. makessense
147. doesnotmakesense;Explanationswillvary. Sampleexplanation:Thecommutativeproperty changesorderandtheassociativepropertychanges groupings.
148. false;Changestomakethestatementtruewillvary. Asamplechangeis:Somerationalnumbersarenot integers.
149. false;Changestomakethestatementtruewillvary. Asamplechangeis:Allwholenumbersareintegers.
150. true
151. false;Changestomakethestatementtruewillvary. Asamplechangeis:Someirrationalnumbersare negative.
152. false;Changestomakethestatementtruewillvary. Asamplechangeis:Theterm x hasacoefficient of1.
153. false;Changestomakethestatementtruewillvary. Asamplechangeis: 53(4)531237. xxx +−=+−=−
154. false;Changestomakethestatementtruewillvary. Asamplechangeis:2. xxx−−=−
155. true
156. 21.4 1.41.5
159. a. 437 ()() bbbbbbbbbb ⋅=⋅⋅⋅⋅⋅=
b. 5510 ()() bbbbbbbbbbbbb ⋅=⋅⋅⋅⋅⋅⋅⋅⋅=
c. addtheexponents
160. a. 7 4 3 bbbbbbbb b bbbb ⋅⋅⋅⋅⋅⋅ == ⋅⋅ b. 8 6 2 bbbbbbbbb b bbb ==
c. subtracttheexponents
161. 3 6.2106.21010106200 ×=×××= Itmovesthedecimalpoint3placestotheright.
Section P.2
Check Point Exercises
1. a. 32325 3333or243 + ⋅==
b. ()()34263246410410 xyxyxxyy =⋅⋅⋅⋅⋅ 3246 510 40 40 xy xy ++ =⋅ =
2. a. 6 3 3 (3)(3)27 (3) =−=−
b. 148148 14385113 3535 2727 99 33 xyxy xyxy xyxy =⋅⋅==
3. a. 2 2 11 5 525 ==
b. 3 3 111 (3)2727 (3) −===−
c. 2 2 2 2 114 1416 411 4 ==⋅==
d. 4 644 66 13 33 y xyy xx =⋅⋅=
4. a. () 2 3326 333or729 ==
b. () 2 77(2)14 14 1 yyy y ===
c. () 4 33(4)12 bbb ==
5. 3333 (4)(4)()64 xxx−=−=−
6. a. () 55 55 2 232 yyy −==
b. () 33 5 515 3 327 3 xxx ==
7. a. () () ()() 444 4 36361224 2216 xyxyxy ==
b. ()() () 253253 38 6363 18 xyxyxxyy xy −=−⋅⋅⋅⋅⋅ =−
c. 122122 164164 100100 2020 xyxy xyxy = 12162(4) 46 6 4 5 5 5 xy xy y x = = =
d. ()() () 222 42 4 55xx yy = ()() () 22 42 22 8 8 22 8 2 5 5 5 25 x y x y y x y x = = = =
8. a. 9 2.6102,600,000,000 −×=−
b. 6 3.017100.000003017 ×=
9. a. 95,210,000,0005.2110 =×
b. 0.000000068936.893108 −=−×
10. () 727 410104.11010 ×=×× () 27 9 4.11010 4.110
1313 88 5 13,500,000,000,0001.35101.3510 309,000,0003.093.091010 0.4410 44,000 × ==⋅ × =× = Eachcitizenwouldhavetopay$44,000in2010.This wouldbeanincreaseof$33,000,percitizen,between 2010and2020.
Concept and Vocabulary Check P.2
C1. mn b + ;add
C2. mn b ;subtract
C3. 1
C4. 1 n b
C5. false
C6. n b
C7. true
C8. anumbergreaterthanorequalto1andlessthan10; integer
C9. true
C10. false
Exercise Set P.2
1. 2 52(55)225250 ⋅=⋅⋅=⋅=
2. 2 62(66)236272 ⋅=⋅⋅=⋅=
3. 6 (2)(2)(2)(2)(2)(2)(2)64 −=−−−−−−=
4. 4 (2)(2)(2)(2)(2)16 −=−−−−=
5. 6 222222264 −=−⋅⋅⋅⋅⋅=−
6. 4 2222216 −=−⋅⋅⋅=−
7. 0 (3)1 −=
8. 0 (9)1 −=
9. 0 31−=−
10. 0 91−=−
11. 3 3 111 4 444464 ===
12. 6 6 111 2 222222264 ===
13. 23235 22222222232 + ⋅===⋅⋅⋅⋅=
14. 32325 333333333243 + ⋅===⋅⋅⋅⋅= 15. 23236 (2)2222222264 ===⋅⋅⋅⋅⋅=
16. 32326 (3)33333333729 ===⋅⋅⋅⋅⋅=
17. 8 844 4 2 22222216 2 ===⋅⋅⋅=
18. 8 844 4 3 33333381 3 ===⋅⋅⋅=
19. 3312 2 111 3333 3339 −−+− ⋅=====
20. 3312 2 111 2222 2224 −−+− ⋅=====
21. 3 374 74 2111 22 22222216 ===== ⋅⋅⋅
22. 4 473 73 3111 33 3333327 ===== ⋅⋅
23. 2 22 1 y xyy xx =⋅= 24. 3 33 1 x xyx yy =⋅= 25. 0555 1 xyyy =⋅= 26. 7077 1 xyxx ⋅=⋅=
27. 373710 xxxx + ⋅== 28. 11511516 xxxx + ⋅== 29. 5105105 xxxx −−+ ⋅== 30. 6126126 xxxx −−+ ⋅== 31. 373721()xxx == 32. 11511555()xxx == 33. 535315 15 1 ()xxx x === 34. 646424 24 1 ()xxx x === 35. 14 1477 7 x xx x == 36. 30 301020 10 x xx x == 37. 14 14(7)14721 7 x xxx x −−+ ===
38. 30 30(10)301040 10 x xxx x −−+ ===
39. 322322326 (8)8()864 xxxx === 40. 422422428 (6)(6)()636 xxxx ===
41. 33 33 4(4)64 xxx −==−
42. 33 33 6(6)216 yyy −==−
43. 25222252 2252 410 (3)(3)()() 9 9 xyxy xy xy −=−⋅ = =
44. 46334363 4363 1218 (3)(3)()() 27 27 xyxy xy xy −=−
45. 47474711 (3)(2)3266 xxxxxx + =⋅⋅==
46. 51251251217 (11)(9)1199999 xxxxxx + =⋅==
47. 364364 3614 95 (9)(2)(9)(2) 18 18 xyxyxxyy xy xy ++ −−=−− = =
48. 47114711 47111 1112 (5)(6)(5)(6) 30 30 xyxyxxyy xy xy ++ −−=−− = =
49. 2020 20416 44 88 44 22 xx xx xx ===
50. 2424 24618 66 2020 22 1010 xx xx xx ===
51. 134134 2323 13243 11 2525 55 5 5 abab abab ab ab = −⋅ =− =−
52. 146146 7373 14763 73 3535 77 5 5 abab abab ab ab = =− =− 53. 77 7147 14147 14142 22 77 bbbb bbb ==⋅==
35 375(9) 79 414 14 4 243 324 3 4 3 4 xy xy xy xy y x = = =
49 4129(3) 123 812 12 8 101 303 1 3 3 xy xy xy xy y x
60. 433 4 3 343 3 12 3 3 3 27 xy yx y x y x = = =
61. () 334223 103104 53 6 15 18 153 5 3 27 abb aba b a b a = = =
62. () 3314882 1721714 103 30 9 303 10 3 3 27 abb aba b a b a = = =
63. 520 34 3 1 12 ab ab =
64. 530 35 4 1 12 ab ab =
65. 2 3.810380 ×=
66. 2 9.210920 ×=
67. 4 6100.0006 ×=
68. 5 7100.00007 ×=
69. 6 7.16107,160,000 −×=−
70. 6 8.17108,170,000 −×=−
71. 1 7.9100.79 ×=
72. 1 6.8100.68 ×=
73. 3 4.15100.00415 −×=−
74. 3 3.14100.00314 −×=−
75. 10 6.000011060,000,100,000 −×=−
76. 10 7.000011070,000,100,000 −×=−
77. 432,0003.210 =×
78. 464,0006.410 =×
79. 17 638,000,000,000,000,000 6.3810=×
80. 17579,000,000,000,000,0005.7910 =×
81. 357165.71610 −=−×
82. 338293.82910 −=−×
83. 0.00272.7103 =×
84. 0.00838.3103 =×
85. 0.000000005045.04109 −=−×
86. 0.000000004054.05109 −=−×
87. ()() () ()4343 437 3102.11032.11010 6.3106.310 + ××=×× =×=×
88. ()()437 2104.1108.210 ××=×
89. ()() () () 15111511 15(11) 4 1.6104101.641010 6.410 6.410 +− ××=×× =× =×
90. ()() () () 15111511 15(11) 4 1.4103101.431010 4.210 4.210 +− ××=×× =× =×
91. ()() () () 8484 8(4) 12 11 6.1102106.121010 12.210 12.210 1.2210 −+− ××=×× =× =× =×
92. ()()8412 11 5.11031015.310 1.5310 ××=× =×
93. ()() () () 84 84 84 12 1313 4.3106.210 4.36.21010 26.6610 26.6610 2.666102.6710 + ×× =×× =× =× =×≈×
94. ()() 84 8412 1313 8.2104.610 37.721037.7210 3.772103.7710 + ×× =×=× =×≈×
95. 88 55 853 8.4108.410 4 41010 2.1102.110 × =× × =×=×
96. 8 853 5 6.910 2.3102.310 310 × =×=× ×
97. 44 22 4(2) 65 3.6103.610 9 91010 0.410 0.410410 × =× × =× =×=×
98. () 4 4(2)6 2 165 1.210 0.6100.610 210 61010610 × =×=× × =××=×
99. 22 66 268 4.8104.810 2.4 2.41010 210210 × =× × =×=×
100. 2 268 6 7.510 310310 2.510 × =×=× ×
101. 22 66 2(6) 43 2.4102.410 4.8 4.81010 0.510 0.510510 × =× × =× =×=×
102. 2 2(6) 6 43 1.510 0.510 510 0.510510 × =× × =×=×
103. 11 4 11 4 11(4) 15
480,000,000,0004.810 0.000121.210 4.810 1.210 410 410 × = × =× =× =×
11 3 11(3) 14 282,000,000,0002.8210 0.001411.4110 210 210 × = × =× =× 105. ()() 43 4 43 3 4 0.000720.003 0.00024 7.210310 2.410 7.231010910 2.410 × ×× = × ×⋅ =×=×
106. ()() ()() () 43 33 1 16 6 7 660000.0016.610110 0.0030.002310210 6.610 1.110 610 1.110 ×× × = × ×× × ==× × =×
107. () () () 2363 21363 33 66001 xyxy xyxy xyxy = ===
108. () () () 2224 2363 7 43 2687 8 xyxy xyxy y xyxy x = ===
109. () () 5 363655 486 48686 2222 2 216 xyzxxyzx yy xyz xzxz =⋅⋅ ===
110.
115. a. 115.1910 × b. 74.810 × c. 1111 77 4 5.19105.1910 4.8 4.81010 1.081310 10,813 × =× × ≈× ≈ $10,813yearlyperperson
d. 10,813901 12 ≈ $901monthlyperperson
116. a. 103.310 × b. 72.5710 × c. 1010 77 3 3.3103.310 2.57 2.571010 1.28410 1284 × =× × ≈× ≈ $1284yearlyperperson d. 1284 107 12 = $107monthlyperperson 117. Medicaid 1111 77 4 1.98101.9810 5.34 5.341010 0.370810 3708 × =× × ≈× ≈ Medicare 1111 77 4 2.94102.9410 4.23 4.231010 0.695010 6950 × =× × ≈× ≈
Medicareprovidesthegreateryearlyperperson benefitby,$6950$3708,or$3242.
118. a. 132.5510 × b. 112.5410 × c. 1313 1111 2 2.55102.5510 2.54 2.541010 1.0010 100 × =× × ≈× ≈ approximately100years
119. a. 132.5510 × b. 4610 × c. 1313 44 9 2.55102.5510 6 61010 0.42510 425,000,000 × =× × =× = 425,000,000Americans
120. a. 119.8410 × b. 73.210 ×
c. 1111 77 4 9.84109.8410 3.2 3.21010 3.07510 30,750 × =× × =× = 30,750years
121. – 128. Answerswillvary.
129. doesnotmakesense;Explanationswillvary. Sampleexplanation:392712 36()36not36. xxx =
130. makessense
131. doesnotmakesense;Explanationswillvary. Sampleexplanation:12 4.610 × representsover4 trillion.Theentireworldpopulationismeasuredin billions(910).
132. makessense
133. false;Changestomakethestatementtruewillvary. Asamplechangeis:23 44. >
134. true
135. false;Changestomakethestatementtruewillvary. Asamplechangeis:441 (2)2because16. 16 −≠≠
136. false;Changestomakethestatementtruewillvary. Asamplechangeis:2255 5522. ⋅=⋅
137. false;Changestomakethestatementtruewillvary. Asamplechangeis:534.75347. ≠
138. false;Changestomakethestatementtruewillvary. Asamplechangeis: 30 30(5)35 5 810 410410. 210 × =×=× ×
139. false;Changestomakethestatementtruewillvary. Asamplechangeis: 53 (710)(210)700,000.002. ×+×=
140. true
141. Thedoctorhasgathered: 12112132222444 2 +=+=+=
So, 31 1 44 −= isremaining.
142. ,, ACD bMNbMbN === ACDbbb ACD = =+
143. 70bts min 60min ⋅ hr 24hrs ⋅ day 365days ⋅ yr 80yrs ⋅ 9 9
70602436580beats
2943360000beats
2.9433610beats
2.9410beats =⋅⋅⋅⋅ = =× ≈× Theheartbeatsapproximately9 2.9410 × timesovera lifetimeof80years.
144. Answerswillvary.
145. a. 164428 ⋅=⋅=
b. 164648 ⋅==
c. 164164 ⋅=⋅
146. a. 30017.32 ≈
b. 10317.32 ≈
c. 300103 =
147. a. 211031 xxx += b. 212102312 +=
Section P.3
Check Point Exercises
1. a. 819 = b. 93−=−
c. 11 255 =
d. 366410010 +==
e. 36646814 +=+=
2. a. 7525325353 =⋅==
⋅=⋅
5. a. 52712 + 59343 53323 15323 (152)3 173 =⋅+⋅ =⋅+ =+ =+ = b. 61848xx 692442 632422 18282 (188)2 102 xx xx xx x x =⋅−⋅
xxxx
b. 2 2 2 510510
3. a. 25255 164 16 ==
b. 33150150 22 xx xx =
x x = =⋅ =
4. a. 813913(89)3 1713 +=+ = b. 172017 1172017 (120)17 1917 xx xx x x =− =− =−
6. a. Ifwemultiplynumeratoranddenominatorby 3,thedenominatorbecomes 3393. ⋅== Therefore,multiplyby1, choosing 3 3 for1. 5535353 33393 =⋅==
b. The smallest numberthatwillproduceaperfect squareinthedenominatorof6 12 is3 because123⋅= 366. = Somultiplyby 1,choosing 3 3 for1. 6636363 3 12123366 =⋅===
7. Multiplyby 45 45 8845 454545 8(45) 224(5) 8(45) 165 8(45)3285 or 1111 =⋅ ++− = = =
8. a. 33333 40858525 =⋅=⋅=
b. 555555 886432222 ⋅==⋅=
c. 3 3 3 1251255 273 27 ==
9. 33 38143 33 33 33 3 3 327343 33343 9343 (94)3 53 =⋅− =⋅− =− =− =
10. a. 1 2 25255 ==
b. 1 33 882 ==
c. 1 44 81813 −=−=−
d. () 1 33 882 −=−=−
e. 1 3 13 3 111 27 273 27 ===
11. a. () 4 3344 2727(3)81 ===
b. () 3 2233 44(2)8 ===
c. () 2 5 222 5 5 1111 32 323224 ====
12. a. ()() 4/38/325xx () 4/38/3 4/3(8/3) 12/3 4 25 10 10 10 xx x x x + =⋅⋅ = = = b. 33 22 442020 55 xx xx = () () () 3 2 83 22 5 2 44 4 4 x x x = = = 13. 31 662 3 xxxx === Concept and Vocabulary Check P.3
C1. principal
C2. 28
C3. a
C4. ab
C5. a b
C6. 183
C7. 5;63
C8. 73
C9. 102 + C10. index;radicand
C11. 5(2)
C12. a; a
C13. na
C14. 2;8 Exercise Set P.3 1. 2 3666 == 2. 2 2555 == 3. 2 3666 −=−=− 4. 2 2555 −=−=−
5. 36,Thesquarerootofanegativenumberisnot real.
6. 25,Thesquarerootofanegativenumberisnot real.
7. 251693 −==
8. 1442516913 +==
9. 2516541 −=−=
10. 1442512517 +=+=
11. 2 (13)16913 −==
12. 2 (17)28917 −==
13. 5025225252 =⋅== 14. 27939333 =⋅== 15. 22 2 2 4595 95 95 35 xx x x x =⋅ = = = 16. 22 2 2 125255 255 255 55 xx x x x =⋅ = = =
17. 2626 xxxx ⋅=⋅ 2 2 12 43 23 x x x = =⋅ = 18. 2 108108 80 xxxx x ⋅=⋅ = 2 165 45 x x =⋅ = 19. 32 xxxxx =⋅= 20. 32 yyyyy =⋅= 21. 22 2626 xxxx ⋅=⋅ 3 2 12 43 23 x xx xx = =⋅ = 22. 226363 xxxx ⋅=⋅ 2 183 92 32 x xx xx = =⋅ = 23. 111 819 81 == 24. 111 497 49 == 25. 49497 164 16 == 26. 12112111 93 9 == 27. 33 48482164 33 xx xx xx === 28. 33 7272293 88 xx xx xx === 29. 44 3 2 150150 33 50 252 52 xx xx x xx xx = = =⋅ =
30. 44 3 2 2424 33 8 42 22 xx xx x xx xx = = =⋅ =
31. () 33 11 31 4 4 200200 1010 20 20 45 2 25 xx xx x x x x = = = =⋅ =
32. () 33 31 11 442 500500 50 1010 5025252 xx x xx xxx == ==⋅=
33. 7363(76)3133 +=+=
34. 85115(811)5195 +=+=
35. 617817(68)17217 xxxx −=−=−
36. 413613(46)13213 xxxx −=−=−
37. 8324232 2232 (23)2 52 +=⋅+ =+ =+ =
38. 20654565 2565 (26)5 85 +=⋅+ =+ =+ =
39. 50825242 5222 (52)2 32 xxxx xx x x −=⋅−⋅ =− =− =
40. 63289747 3727 (32)7 7 xxxx xx x x −=⋅−⋅ =− =− =
41. 3185503925252 332552 92252 (925)2 342 +=⋅+⋅ =⋅+⋅ =+ =+ =
42. 4122754432253 423253 83103 (810)3 23 −=⋅−⋅ =⋅−⋅ =− =− =−
43. 383237275 −+− 3421623362253 3224236253 624218253 20253 =⋅−⋅+⋅−⋅ =⋅−+⋅− =−+− =−
44. 35422496463 −−+ 396246166497 33622646437 964646127 6127 =⋅−⋅−⋅+⋅ =⋅−⋅−+⋅ =−−+ =+
45. 1177 7777 =⋅=
46. 221021010 101010105 =⋅==
47. 22510 5555 =⋅=
48. 77321 3333 =⋅=
20Copyright©2022PearsonEducation,Inc
49. 22 1313311 311311311 13(311) 3(11) 13(311) 911 13(311) 2 =⋅ ++− = = =
50. 22 3337 373737 3(37) 3(7) 3(37) 97 3(37) 2 =⋅ ++− = = =
51. 22 7752 525252 7(52) (5)2 7(52) 54 7(52) + =⋅ −−+ + = + = =+
52. 22 5531 313131 5(31) (3)1 5(31) 31 5(31) 2 + =⋅ −−+ + = + = + =
53. 22 6653 535353 6(53) (5)(3) 6(53) 53 6(53) 2 3(53) =⋅ ++− = = = =− 54. 22 111173 737373 11(73) (7)(3) 11(73) 73 11(73) 4 + =⋅ −−+ + = + = + = 55. 33312555 == 56. 333822 == 57. 33 3 8(2)2 −=−=− 58. 33 3 125(5)5 −=−=−
59. 416isnotarealnumber.
60. 481isnotarealnumber.
61. 44(3)33 −=−= 62. 44(2)22 −=−= 63. 55(3)3 −=− 64. 55(2)2 −=− 65. 55 5 111 322 2 −=−=−
66. 6 6 66 111 642 2 ==
67. 33333 32848424 =⋅==⋅
68. 3150cannotbesimplifiedfurther.
69. 33433 xxxxx =⋅=⋅
70. 3335322 xxxxx ==
71. 3333333 965427227232 ⋅==⋅==
72. 33333 124488626 ⋅==⋅=
73. 566 555 5 6464 322 22 xx xx xx ===
74. 455 444 4 162162 813 22 xx xx xx ===
75. 555 423272 +=
76. 555 632383 +=
77. 3333 51654582272 +=⋅+⋅ 33 33 3 52232 10232 132 =⋅+ =+ =
78. 3333 3248183273 +=⋅+⋅ 33 33 3 32333 6333 93 =⋅+ =+ =
79. 33354128 xyyx
333 33 3 272642 3242 2 xyyx yxyx yx =⋅−⋅ =− =−
80. 3332481 xyyx
333 33 3 83273 2333 3 xyyx yxyx yx =⋅−⋅ =− =−
81. 3 2822 +=+
82. 3 315 + willnotsimplify.
83. 1/2 36366 ==
84. 1/2 12112111 ==
85. 1/33 882 ==
86. 1/33 27273 ==
87. () 2 2/32 3 125125525 ===
88. () 2/32 3 884 ==
89. 4/5 4/54 111 32 32216 ===
90. 5/2 5/255 1111 16 16(16)41024 ====
91. ()()1/31/41/31/4 1/31/4 7/12 7272 14 14 xxxx x x + =⋅⋅ =⋅ =
92. 2/33/42/33/4 2/33/4 17/12 (3)(4)34 12 12 xxxx x x + =⋅⋅ =⋅ =
93. 1/21/2 1/41/4 1/21/4 1/4 2020 55 4 4 xx xx x x = =⋅ =
94. 3/43/4 3/41/35/12 1/31/3 7272 88 99 xx xx xx ==⋅=
95. ()3 2/32/332 xxx ==
96. 4/554/554()xxx ==
97. 461/21/241/261/223(25)255xyxyxy ==
98. 961/31/39/36/332(125)1255xyxyxy == 99. 13 43 431 412 11 1212 3 27 27 y y y yy == 82 1232727yy==
100. () () 441/541/5 3/103/10 22yy yy = 4/5 4/53/101/2 3/10 16 1616 y yy y ===
101. 422/41/2 5555 ===
102. 422/41/2 7777 ===
103. 366/32 xxx ==
104. 43 1212/4 xxx ==
105. 66/2344/22 xxx ==
106. 99/3366/32 xxx ==
107. 6321 63293 9933 xyxyxyxy ===
108. 4812 12482 1212333 xyxyxyxy ===
109. 333 4 16625225273 +=+==
113. ()() ( ) () ()()()() 5/41/366 5/43/41/3 3/4 2/41/362/461/36 3 32 2 xy xy x xyxy x xy y = == ==
114. ()() ( ) () ()()()() 1/27/444 1/27/45/4 5/4 1/22/441/242/44 2 22 2 xy xy y xyxy y xy x = == ==
110. 333 3 3 33 16991000216 133106 1616 448 2 +++ =+++ =+ =+= = 111. ()() () ()()() ()()()() () () 1/2 241/2 1/21/21/2241/2 21/241/21/2 1/2 121/21121/2 2 23/2 3/2 49 49 1 49 11 77 1 77 xyxy xyxy xyxy xyxyxy x xy y −+ −+ = = =⋅= == 112. ()() ()()()()()()()() () 1/36635/61/3 1/361/331/35/661/36 21522512 3 31 8 8 22 2 2 xyxy xyxy xyxyxy x xy y +− −−−+ = == ==
115. a. () () () () 13 5 251325 525 25135 55 18 xx P x + = + = + = = 18%of25-year-oldsmustpaymoretaxes.
b. () 2 13 5 13 5 13 5 13 5 xx P x xxxx xx xxx x xx + = + =⋅ + = + =
c. () 13 5 132525 5 13525 5 18 xx P + = + = + = = Yeswegetthesameamount,18%. No,thisdoesnotprovewehavecorrectly rationalizedthedenominator.Explanationswill vary.
116. a. For2020:534.1 51034.1 Ex=+ =+ For2050:534.1 54034.1 521034.1 101034.1 Ex=+ =+ =⋅+ =+ Difference: (101034.1)(51034.1) 101034.151034.1 101051034.134.1 510 +−+ =+−− =−+− = Thedifferenceis510.
b. 51015.8 ≈ Thisunderestimatesthedifferenceprojectedby thegraphof65.847.318.5 −= by2.7.This representsadifferenceof2.7millionpeople.
117. 2512(51) 515151 2(51) 4 51 2 1.62 ++ ⋅= −+ + = + =
About1.62to1.
f RR R R R = = = = Ifyouaregonefor44weeks,then100weekswill havepassedforyourfriend.
119. Perimeter: 22 21252220 2255445 255425 10585 185feet
Alw = =⋅ =⋅ = =⋅ =
Plw =+ =⋅+⋅ =⋅⋅+⋅ =⋅+⋅ =+ = Area: 125220 212520 22500 250 100squarefeet
120. Perimeter: 22 2420280 8452165 825245 16585 245feet
132. doesnotmakesense;Explanationswillvary. Sampleexplanation:Findingthe nthrootfirstoften givessmallernumbersonthemiddlestep.
133. false;Changestomakethestatementtruewillvary. Asamplechangeis: 11221 7777. ⋅==
134. false;Changestomakethestatementtruewillvary.
Asamplechangeis: () () 1 3 13 3 111 8 882 ===
135. false;Changestomakethestatementtruewillvary. Thecuberootof–8istherealnumber–2.
136. false;Changestomakethestatementtruewillvary.
Asamplechangeis: 205 84 =
137. () () 535322 +−= 25322 33 −= =
Alw = =⋅ =⋅ = =⋅ =
Plw =+ =⋅+⋅ =⋅+⋅ =⋅+⋅ =+ = Area: 42080 42080 41600 440 160squarefeet
121. – 128. Answerswillvary.
129. makessense
130. makessense
131. doesnotmakesense;Explanationswillvary.
Sampleexplanation:220475 + simplifiesto 45203 + andthustheradicaltermsarenot common.
138. 147255xx = 139. 7 132 32 ++ + 732 132 3232 2172 132 92 2172 132 7 13232 16 4 =++⋅ +− =++ =++ =++− = =
140. a. 11 2333 > CalculatorCheck:1.73211.4422 > b. 718718 +>+ CalculatorCheck:6.88845 >
141. a.
b. Son’sportion:
Section P.4
Check Point Exercises
1. a. 3232 (174115)(163315) xxxxxx −+−−+−+− 3322 32 (1716)(43)(113)(515) 820 xxxxxx xxx =−++−+−++−− =−+−−
b. 2232 (13971)(7259) xxxxxx −−+−−+−+ 3232 3322 32 (13971)(7259) (137)(92)(75)(19) 201128 xxxxxx xxxxxx xxx =−−++−+− =++−−+−++− =−−−
2. 2 (52)(354) xxx−−+ 22 22 322 32 5(354)2(354) 535554232524 1525206108 1531308 xxxxx xxxxxxx xxxxx xxx =−+−−+ =⋅−⋅+⋅−⋅+⋅−⋅ =−+−+− =−+−
3. 2 2 (75)(43)747(3)(5)4(5)(3) 28212015 284115 xxxxxx xxx xx −−=⋅+−+−+−− =−−+ =−+
4. a. Usethespecial-productformulashown. 22 22 2 ()() (78)(78)(7)(8) 4964 ABABAB xxx x +−=− +−=− =−
b. Usethespecial-productformulashown. 22 3333 322 6 ()() (25)(25)(25)(25) (2)(5) 425 ABABAB yyyy y y +−=− −+=+− =− =−
5. a. Usethespecial-productformulashown. 222 222 2 ()2 (10)2()(10)10 20100 ABAABB xxx xx +=++ +=++ =++
b. Usethespecial-productformulashown. 222 222 2 ()2 (54)(5)2(5)(4)4 254016 ABAABB xxx xx +=++ +=++ =++
6. a. Usethespecial-productformulashown. 222 222 2 ()2 (9)2()(9)9 1881
ABAABB xxx xx −=−+ −=−+ =−+
b. Usethespecial-productformulashown. 222 222 2 ()2 (73)(7)2(7)(3)3 49429
ABAABB xxx xx −=−+ −=−+ =−+
7. 3223323 3223323 3322233 223 (45)(6) (45)(6) ()(46)(5)() 252 xxyxyyxxyy xxyxyyxxyy xxxyxyxyyy xyxyy −+−−−+ =−+−+−+− =−+−+++−− =+−
8. a. 22 22 (76)(3)(7)(3)(7)()(6)(3)(6)() 217186 21256 xyxyxxxyyxyy xxyxyy xxyy −−=+−+−+−− =−−+ =−+
b. 222 22 (24)(2)2(2)(4)(4) 41616 xyxxyy xxyy +=++ =++
Concept and Vocabulary Check P.4
C1. whole
C2. standard
C3. monomial
C4. binomial
C5. trinomial
C6. n
C7. like;
C8. distributive;32486 xx−+ ; 73 x
C9. 5 x ;3;like
C10. 32 x ;5 x ;21x ;35
C11. 22AB ;minus
C12. 22 2 AABB ++ ;squared;productoftheterms;squared
C13. 22 2 AABB −+ ;minus;productoftheterms;plus
C14. nm +
Exercise Set P.4
1. yes; 22 235325 xxxx +−=+−
2. no;Theterm1 3 x doesnothaveawholenumberexponent.
3. no;Theformofapolynomialinvolvesadditionandsubtraction,notdivision.
4. yes; 23443255xxxxxx −+−=−+−
5. 32 x hasdegree2
5x hasdegree1 4hasdegree0 2 354 xx−+ hasdegree2.
6. 43 x hasdegree3
72 x hasdegree2 –11hasdegree0 32 4711 xx −+− hasdegree3.
7. 2 x hasdegree2
43 x hasdegree3
9x hasdegree1 124 x hasdegree4 63hasdegree0 234491263xxxx−+−+ hasdegree4.
8. 2 x hasdegree2
83 x hasdegree3
154 x hasdegree4 91hasdegree0 23481591xxx−++ hasdegree4.
9. 32323322 32 (6589)(172413)(617)(52)(84)(913) 117124 xxxxxxxxxxxx xxx −+−+++−−=−++++−−+− =+−− Thedegreeis3.
10. 32323322 32 (761113)(1911717)(719)(611)(117)(1317) 12544 xxxxxxxxxxxx xxx −+−++−+−=−++−+−++− =−−− Thedegreeis3.
11. 32323232 3322 32 (17543)(59811)(17543)(59811) (175)(59)(48)(311) 1241214 xxxxxxxxxxxx xxxxxx xxx −+−−−−+=−+−+−++− =−+−++++−− =++− Thedegreeis3.
12. 43434343 4433 43 (18278)(9657)(18278)(9657) (189)(26)(75)(87) 9421 xxxxxxxxxxxx xxxxxx xxx −−+−−−+=−−++−++− =−+−++−++− =+−+
Thedegreeis4.
13. 222222 222 2 (578)(237)(43)(578)(237)(43) (52)(734)(873) 662 xxxxxxxxxxxx xxxxxx xx −−+−+−−−=−−+−++−++ =+−+−−++−++ =−+
Thedegreeis2.
14. 22322232 3222 32 (875)(34)(653)(875)(34)(653) 6(835)(74)(53) 610118 xxxxxxxxxxxx xxxxxx xxx +−−−−−−+=+−+−+++− =+−++++−− =++−
Thedegreeis3.
15. 222 322 3 (1)(1)()11()111 1 1 xxxxxxxxxx xxxxx x +−+=−⋅+⋅+−⋅+⋅ =−++−+ =+
16. () ()222 5525()(5)(25)5()5(5)5(25) xxxxxxxxxx +−+=−++−+ 322 3 525525125) 125 xxxxx x =−++−+ =+
17. 222 322 32 (23)(35)(2)()(2)(3)(2)(5)(3)()(3)(3)(3)(5) 26103915 291915 xxxxxxxxxx xxxxx xxx −−+=+−++−+−−+− =−+−+− =−+−
18. 222 322 32 (21)(43)(2)()(2)(4)(2)(3)(1)()(1)(4)(1)(3) 28643 29103 xxxxxxxxxx xxxxx xxx −−+=+−++−+−−+− =−+−+− =−+−
19. 22 (7)(3)37211021 xxxxxxx ++=+++=++
20. 22 (8)(5)58401340 xxxxxxx ++=+++=++
21. 22 (5)(3)3515215 xxxxxxx −+=+−−=−−
22. 22 (1)(2)222 xxxxxxx −+=+−−=+−
23. 22 (35)(21)(3)(2)3(1)5(2)5631056135 xxxxxxxxxxx ++=+++=+++=++
24. 22 (74)(31)(7)(3)7(1)4(3)4(1)21712421194 xxxxxxxxxxx ++=+++=+++=++
25. 22 (23)(53)(2)(5)(2)(3)(3)(5)(3)(3)1061591099 xxxxxxxxxxx −+=++−+−=+−−=−−
30Copyright©2022PearsonEducation,Inc
26. 22 (25)(72)(2)(7)(2)(2)(5)(7)(5)(2)1443510143110 xxxxxxxxxxx −+=++−+−=+−−=−−
27. 22222242242 (54)(37)(5)(3)(5)(7)(4)(3)(4)(7)15351228154728 xxxxxxxxxxx −−=+−+−+−−=−−+=−+
28. 22222242242 (72)(35)(7)(3)(7)(5)(2)(3)(2)(5)2135610214110 xxxxxxxxxxx −−=+−+−+−−=−−+=−+
29. ()()()()() ()() () ()()
323232 835885335 xxxxxx +−=+−++− 532 840315 xxx =−+−
30. ()()()()() ()() () ()()
323232532 752772552714510 xxxxxxxxx +−=+−++−=−+−
31. 222 (3)(3)39 xxxx +−=−=−
32. 222 (5)(5)525 xxxx +−=−=−
33. 222 (32)(32)(3)294 xxxx +−=−=−
34. 222 (25)(25)(2)5425 xxxx +−=−=−
35. 222(57)(57)5(7)2549 xxxx −+=−=−
36. 222(43)(43)4(3)169 xxxx −+=−=−
37. 2222242 (45)(45)(4)(5)1625 xxxxxxxx +−=−=−
38. 2222242 (34)(34)(3)(4)916 xxxxxxxx +−=−=−
39. ()() () () 22 555101111yyyy −+=−=−
40. ()() () () 22 555102224yyyy −+=−=−
41. 2222 (2)22244 xxxxx +=+⋅⋅+=++
42. 2222 (5)2551025 xxxxx +=+⋅⋅+=++
43. 2222 (23)(2)2(2)(3)34129 xxxxx +=++=++
44. 2222 (32)(3)2(3)(2)29124 xxxxx +=++=++
45. 2222 (3)23369 xxxxx −=−⋅⋅+=−+
46. 2222 (4)244816 xxxxx −=−⋅⋅+=−+
47. 22222242 (41)(4)2(4)(1)11681 xxxxx −=−+=−+
48. 22222242 (53)(5)2(5)(3)325309 xxxxx −=−+=−+
49. 22222 (72)72(7)(2)(2)4928442849 xxxxxxx −=−+=−+=−+
50. 2222 (95)92(9)(5)(5)819025 xxxxx −=−+=−+ or 2 259081 xx−+
51. 3322332 (1)31311331 xxxxxxx +=+⋅⋅+⋅+=+++
52. 3322332 (2)323226128 xxxxxxx +=+⋅⋅+⋅⋅+=+++
53. 3322332 (23)(2)3(2)33(2)338365427 xxxxxxx +=+⋅⋅+⋅+=+++
54. 3322332 (34)(3)3(3)43(3)442710814464 xxxxxxx +=+⋅+⋅+=+++
55. 3332332 (3)3333392727 xxxxxxx −=−⋅⋅+⋅⋅−=−+−
56. 3322332 (1)31311331 xxxxxxx −=−⋅+⋅−=−+−
57. 3322332 (34)(3)3(3)43(3)442710814464 xxxxxxx −=−⋅+⋅−=−+−
58. 3322332 (23)(2)3(2)33(2)338365427 xxxxxxx −=−⋅+⋅−=−+−
59. 2222 2 2 (53)(2)(52)(3) (52)(31) 74isofdegree3. xyxyxyxyxyxyxyxy xyxy xyxy −+−=++−− =++−− =−
60. 2222 2 2 (2)(47)(24)(7) (24)(17) 28isofdegree3. xyxyxyxyxyxyxyxy xyxy xyxy −+++=−+++ =−+++ =+
61. 2222 2 2 (4811)(252)(42)(85)(112) (42)(85)13 21313isofdegree3. xyxyxyxyxyxyxyxy xyxy xyxy +++−++=−++++ =−+++ =++
62. 4222422242422222 4222 4222 (753)(186)(718)(56)(3) (718)(56)(31) 11112isofdegree6. xyxyxyxyxyxyxyxyxyxyxyxy xyxyxy xyxyxy −++−−−=−+−−+− =−+−−+− =−−+
63. 323232 3322 32 32 (75)(64)(75) (6)(7)(54) (16)(71)(54) 589isofdegree3. xxyyxxyyxxyy xxxyxyyy xxyy xxyy +−−−+=+− =−+++−− =−+++−− =−+−
64. 43434343 4433 43 43 (75)(634)(75)(634) (6)(73)(54) (16)(73)(54) 549isofdegree4. xxyyxxyyxxyyxxyy xxxyxyyy xxyy xxyy −−−−+=−−+−+− =−+−++−− =−+−++−− =−−−
65. 423423423423 424233 423 423 (353)(2346)(353)(2346) (32)(53)(34)6 (32)(53)(34)6 86isofdegree6. xyxyyxyxyyxxyxyyxyxyyx xyxyxyxyyyx xyxyyx xyxyyx +−−−−+=+−+−++− =−+++−+− =−+++−+− =++−
66. 423423423423 424233 423 423 (567)(3568)(567)(3568) (53)(65)(76)8 (53)(65)(76)8 2118isofdegree6. xyxyyxyxyyxxyxyyxyxyyx xyxyxyxyyyx xyxyyx xyxyyx +−−−−+=+−+−++− =−+++−+− =−+++−+− =+−−
67. 2 22 (5)(73)(7)(3)(5)(7)(5)(3) 733515 73815 xyxyxxxyyxyy xxyxyy xxyy 2 ++=+++ =+++ =++
68. 22 22 (9)(67)(6)(7)(9)(6)(9)(7) 675463 66163 xyxyxxxyyxyy xxyxyy xxyy ++=+++ =+++ =++
69. 22 22 (3)(27)(2)(7)(3)(2)(3)(7) 27621 221 xyxyxxxyyxyy xxyxyy xxyy −+=++−+− =+−− =+−
70. 22 22 (3)(25)(3)(2)(3)(5)()(2)()(5) 61525 6135 xyxyxxxyyxyy xxyxyy xxyy −+=++−+− =+−− =+−
71. 22 22 (31)(52)(3)(5)(3)(2)(1)(5)(1)(2) 15652 152 xyxyxyxyxyxy xyxyxy xyxy −+=++−+− =+−− =+−
72. 222222 4222 422 (71)(23)(7)(2)(7)(3)(1)2(1)(3) 142123 14193 xyxyxyxyxyxy xyxyxy xyxy +−=+−++− =−+− =−−
73. 22222 (75)(7)2(7)(5)(5)497025 xyxxyyxxyy +=++=++
74. 22222 (97)(9)2(9)(7)(7)8112649 xyxxyyxxyy +=++=++
75. 2222222224422 (3)()2()(3)369 xyxyxyxyxy −=−+=−+
76. 2222222224422 (5)()2()(5)51025 xyxyxyxyxy −=−+=−+
77. 222222
322223 33 ()()()()()()()()()()() xyxxyyxxxxyxyyxyxyyy xxyxyxyxyy xy −++=+++−+−+− =++−−− =−
78. 222222
322223 33 ()()()()()()()() xyxxyyxxxxyxyyxyxyyy xxyxyxyxyy xy +−+=+−+++−+ =−++−+ =+
79. 2222(35)(35)(3)(5)925 xyxyxyxy +−=−=−
80. 2222(73)(73)(7)(3)499 xyxyxyxy +−=−=−
81. ()()() () 22 222242 71071071049100 xyyxyyxyyxyy −+=−=−
82. ()()() () 22 222242 343434916 xyyxyyxyyxyy −+=−=−
83. ()()()()()()()()()() ()() 222222 2222 2222 34343234432344 9241692416 9241692416 48 xyxyxxyyxxyy xxyyxxyy xxyyxxyy xy +−−=++−−+ =++−−+ =++−+− =
84. ()()()()()()()()()() ()() 222222 2222 2222 52525252252522 2520425204 2520425204 40 xyxyxxyyxxyy xxyyxxyy xxyyxxyy xy +−−=++−−+ =++−−+ =++−+− =
85. ()()()() ()() 22 22 22 2 57324561 15102114244305 15311424345 15311424345 939 xxxx xxxxxx xxxx xxxx xx =−−+−−−+ =−+−−+ =−+−+− =−++
86. ()()()() ()() ()() 22 22 22 2 3529721 62710457722 61745792 61745792 847 xxxx xxxxxx xxxx xxxx xx +−−−− =−+−−−−+ =−−−−+ =−−−+− =−−−
87. ()() () () () ()() () () 2 222 22 222 4 2525425 25425 425425 425 16625 xxx xx xx x x +−+ =−+ =−+ =− =−
88. ()() () () () ()() () () 2 222 22 222 4 3434916 34916 916916 916 81256 xxx xx xx x x +−+ =−+ =−+ =− =−
89. () () () () ()()()() 5 53 3 2 22 2 27 27 27 27 22277 42849 x x x x xx xx =− =− =−+ =−+
90. () () () () ()()()() 6 64 4 2 22 2 53 53 53 53 52533 25309 x x x x xx xx =− =− =−+
91.
b. 32 32 0.32.86.730
0.3(5)2.8(5)6.7(5)30 31 Sxxx S S =−++ =−++ =
Themodelgivesascoreof31forthegroupinthe45-54agerangewhichisthesameasthescoredisplayedbythe bargraph.
92. a.
0.3(6)2.8(6)6.7(6)30 34.2 Sxxx S S =−++ =−++ =
Themodelgivesascoreof34.2forthegroupinthe55-64agerange.Thisoverestimatesthescoreshownonthebar graphby0.2.
93. ()() () 82102803642 xxxxxx −−=−+ 23 32 80364 43680 xxx xxx =−+ =−+
94. ()() () 8252402642 xxxxxx −−=−+ 23 32 40264 42640 xxx xxx =−+ =−+
95. ()()()() 9351xxxx ++++ ()22 22 122765 122765 622 xxxx xxxx x =++−++ =++−−− =+
96. ()()()() 4321xxxx ++++ ()22 22 71232 71232 410 xxxx xxxx x =++−++ =++−−− =+
97. – 102. Answerswillvary.
103. makessense
104. doesnotmakesense;Explanationswillvary.Sampleexplanation:FOILisusedtomultiplytwobinomials.
105. makessense
106. makessense,althoughanswersmayvary
107. false;Changestomakethestatementtruewillvary.Asamplechangeis:336(32)(32)94 xxx+−=−
