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Para cada uno de los siguientes polinomios indica qué números racionales son candidatos a ser raíces suyas y, después, determina cuáles lo son:

a) 3x^2 + 4x + 5

b) 2x^3 + 9x^2 + 12x + 2

","status","pending_classification","hasRelatedVideo","relatedVideos",[133],{"_134":135,"_153":154,"_51":169,"_203":204,"_221":222},"content",{"_43":136,"_55":137,"_138":-5,"_139":140,"_141":142,"_143":-5,"_45":144,"_147":148,"_149":148,"_150":-5,"_151":152},"468e3e62-35e0-47be-a15f-68dd43bb32bb","Óptica Geométrica - Parte 1","description","content_type_id","1051eee7-c62e-4b2c-9900-77a29fcec080","resource_id","fa6aaf42-4cc4-4d76-8915-7a67081f7e9a","user_id",{"_145":146},"external_id","4963","created_at","2025-08-20T12:27:41.608438+00:00","updated_at","content_group_id","slug","optica-geometrica-parte-1","prometheusData",{"_155":156,"_162":163,"_164":165,"_166":167,"_28":168},"provider",{"_43":157,"_158":159,"_160":161},"06f120be-2194-4548-8336-7ba44ffbae14","key","grupo_q_proteus","kind","proteus","reference","d80487a0-0f98-42c8-99bb-551165ba3068","state","completed","thumbnail_url","https://proteus.grupoq.io/videos/d80487a0-0f98-42c8-99bb-551165ba3068/thumbnailsthumb.0000000.jpg","https://proteus.grupoq.io/videos/d80487a0-0f98-42c8-99bb-551165ba3068/levels.m3u8",[170],{"_43":171,"_55":172,"_57":173,"_174":175,"_176":-5,"_147":177,"_149":177,"_45":178,"_151":173,"_184":185},"9772e958-72c6-4499-ad47-877cf94d670d","Exatas","exatas","tag_group_id","c1ce2cf2-b217-4cc7-8182-81c56bd3522d","parent_id","2025-04-29T01:56:45.284172+00:00",{"_179":180,"_181":88,"_182":183},"childrenCount",18,"content_count","question_count",135489,"children",[186],{"_43":187,"_55":188,"_57":189,"_174":175,"_176":171,"_147":190,"_149":190,"_45":191,"_151":194,"_184":195},"ca6fee0e-79c6-4f1c-b500-ee081a071949","Mecânica","mecanica","2025-04-29T02:19:12.828432+00:00",{"_179":192,"_181":88,"_182":193},10,19948,"mecanica-2",[196],{"_43":197,"_55":79,"_57":80,"_174":175,"_176":187,"_147":198,"_149":198,"_45":199,"_151":201,"_184":202},"ac55bdba-154d-42e7-bc25-72dde822663d","2025-04-29T03:44:29.139762+00:00",{"_179":88,"_181":88,"_182":200},3376,"resistencia-dos-materiais-4",[],"video",{"_43":142,"_13":137,"_138":-5,"_205":206,"_207":208,"_28":-5,"_209":-5,"_155":208,"_210":211,"_166":-5,"_212":-5,"_213":-5,"_214":-5,"_215":-5,"_45":216,"_217":88,"_218":-5,"_128":219,"_147":220,"_149":220},"duration",20,"source_type","prometheus","iframe_url","provider_video_id","1cd3d633-d690-4f4b-9995-9b0e44994e35","format","quality","file_path","file_size",{"_145":146},"views_count","last_played_at","draft","2025-08-19T11:56:20.758718+00:00","tag",{"_43":171,"_55":172,"_57":173,"_174":175,"_176":-5,"_147":177,"_149":177,"_45":223,"_151":173,"_184":224},{"_179":180,"_181":88,"_182":183},[225],{"_43":187,"_55":188,"_57":189,"_174":175,"_176":171,"_147":190,"_149":190,"_45":226,"_151":194,"_184":227},{"_179":192,"_181":88,"_182":193},[228],{"_43":197,"_55":79,"_57":80,"_174":175,"_176":187,"_147":198,"_149":198,"_45":229,"_151":201,"_184":230},{"_179":88,"_181":88,"_182":200},[],"similarQuestions",[233,263,283],{"_40":234,"_42":41,"_43":235,"_101":236,"_125":251,"_126":252,"_51":253},"112831da-be9c-458e-94cb-e8292d0dba12","3c0b9441-b878-4896-8bac-fa4ca61e3f13",[237,240,243,246,249],{"_43":238,"_105":106,"_107":239},"265bd71a-6749-41f5-b291-67efd646ca64","y'' + 2y' - 3y = 0",{"_43":241,"_105":111,"_107":242},"6d3c3a1d-8e80-4ff3-b97a-d5b5bca91940","6y'' - y' - y = 0",{"_43":244,"_105":115,"_107":245},"4168ac67-64fd-46ed-a69c-a01dbfcde7ea","y'' - 2y' + y = 0",{"_43":247,"_105":119,"_107":248},"cc7bc8df-6995-4264-ac5a-32ea9b4ed7db","9y'' + 6y' + y = 0",{"_43":234,"_105":123,"_107":250},"y'' - 2y' + 6y = 0","Somente em y''−2y'+6y=0 o polinômio característico r²−2r+6=0 tem Δ<0, gerando raízes complexas conjugadas (1±i√5) e, portanto, solução em termos de exponencial modulando seno e cosseno. Nos demais itens (a–d) o discriminante é ≥0, produzindo raízes reais (distintas ou repetidas).","

Resolva as seguintes EDOs determinando a solução geral e, quando houver condições iniciais do PVI, a solução particular.

(a) y'' + 2y' - 3y = 0

(b) 6y'' - y' - y = 0

(d) 9y'' + 6y' + y = 0

(e) y'' - 2y' + 6y = 0

(f) 4y'' + 9y = 0

(g) 6y'' - 5y' + y = 0; \\, y(0) = 4; \\, y'(0) = 0

(h) y'' + 4y' + 4y = 0; \\, y(-1) = 2; \\, y'(-1) = 1

(i) y'' + y = 0; \\, y\bigg(\\frac{\\pi}{3}\\bigg) = 2; \\, y'\\bigg(\\frac{\\pi}{3}\\bigg) = -4

(j) y'' - 2y' - 3y = 3e^{2t}

(k) y'' + 9y = t^2 e^{3t} + 6

(l) 2y'' + 3y' + y = t^2 + 3 \\sin(t)

(m) y'' + y' - 2y = 2t; \\, y(0) = 0; \\, y'(0) = 1

(n) y'' - 2y' + y = te^t + 4; \\, y(0) = 1; \\, y'(0) = 1

(o) y'' + 4y = 3 \\sin(2t); \\, y(0) = 2; \\, y'(0) = -1

",[254],{"_43":255,"_55":256,"_57":257,"_174":175,"_176":258,"_147":259,"_149":259,"_45":260,"_151":257},"6516bbbd-f2a3-4710-8382-702240b17836","Mecânica na Engenharia de Petróleo","mecanica-na-engenharia-de-petroleo","b787e438-0ab3-454c-ab43-5d3d14c2720d","2025-04-29T03:44:35.328997+00:00",{"_179":261,"_181":88,"_182":262},4,12302,{"_40":264,"_42":41,"_43":265,"_101":266,"_125":278,"_126":279,"_51":280},"6831a9b9-c9a4-4a76-bd7a-2ce98bfea875","93a32dd4-5fc6-4ccb-a76f-4724bf113742",[267,270,273,275],{"_43":268,"_105":106,"_107":269},"13e3f476-a0a5-421f-b47e-0969e8e9a7e0","

Verdadeiro, verdadeiro, falso

",{"_43":271,"_105":111,"_107":272},"c9a20855-d84b-4860-8b35-9784818282cf","

Falso, verdadeiro, verdadeiro

",{"_43":264,"_105":115,"_107":274},"

Verdadeiro, falso, verdadeiro

",{"_43":276,"_105":119,"_107":277},"3c98539b-1af9-4d44-aecd-c266754a2075","

Falso, falso, falso

","85,5 % decomposto → 14,5 % restante.\nN/N0 = 0,145 ⇒ t = ln(1/0,145)/(ln2/5730) ≈ 1,6×10⁴ anos.\nItens prováveis: (1) resta 14,5 % do C-14 (V); (2) idade ≈ 1,2×10⁴ anos ou 3 meias-vidas (F); (3) t½ do C-14 = 5730 anos (V). Sequência: V-F-V, correspondendo à alternativa c.","

Qual é a idade da madeira na qual 85,5\\% do C-14 se decompos?

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a) ????3 de ????4????5
b) ????2 de ????2????????2
c) ????1 de 2????????1
d) ????1 de 2????2????",[304],{"_43":255,"_55":256,"_57":257,"_174":175,"_176":258,"_147":259,"_149":259,"_45":305,"_151":257},{"_179":261,"_181":88,"_182":262},[307,309,318,324,328,330,332,340,345,353,355,360,366],{"_43":255,"_55":256,"_57":257,"_174":175,"_176":258,"_147":259,"_149":259,"_45":308,"_151":257},{"_179":261,"_181":88,"_182":262},{"_43":310,"_55":311,"_57":312,"_174":175,"_176":313,"_147":190,"_149":190,"_45":314,"_151":317},"861ad829-750d-4a44-9569-2f18b430b4df","Engenharia Mecatrônica","engenharia-mecatronica","eb9315c5-bd04-4116-aa46-02c4fde320c0",{"_179":315,"_181":88,"_182":316},5,30614,"engenharia-mecatronica-1",{"_43":319,"_55":79,"_57":80,"_174":175,"_176":320,"_147":321,"_149":321,"_45":322,"_151":323},"8850e507-a691-4d45-a7c7-16221e9c5af2","88fe96b3-2a17-4198-90d5-ddda3d6f5d45","2025-04-29T04:00:13.215003+00:00",{"_179":88,"_181":88,"_182":200},"resistencia-dos-materiais-6",{"_43":320,"_55":188,"_57":189,"_174":175,"_176":310,"_147":259,"_149":259,"_45":325,"_151":327},{"_179":315,"_181":88,"_182":326},25741,"mecanica-3",{"_43":171,"_55":172,"_57":173,"_174":175,"_176":-5,"_147":177,"_149":177,"_45":329,"_151":173},{"_179":180,"_181":88,"_182":183},{"_43":197,"_55":79,"_57":80,"_174":175,"_176":187,"_147":198,"_149":198,"_45":331,"_151":201},{"_179":88,"_181":88,"_182":200},{"_43":333,"_55":334,"_57":335,"_174":175,"_176":313,"_147":190,"_149":190,"_45":336,"_151":339},"b6e37f32-f8e5-4025-bf0f-69ffc00fb331","Engenharia Civil","engenharia-civil",{"_179":337,"_181":88,"_182":338},21,45865,"engenharia-civil-3",{"_43":258,"_55":341,"_57":342,"_174":175,"_176":313,"_147":190,"_149":190,"_45":343,"_151":344},"Engenharia de Petróleo","engenharia-de-petroleo",{"_179":315,"_181":88,"_182":262},"engenharia-de-petroleo-1",{"_43":346,"_55":347,"_57":348,"_174":175,"_176":333,"_147":349,"_149":349,"_45":350,"_151":352},"c5bf57d4-c3d8-4365-b53e-085fa14a3a95","Estruturas","estruturas","2025-04-29T03:44:32.303369+00:00",{"_179":315,"_181":88,"_182":351},8852,"estruturas-1",{"_43":187,"_55":188,"_57":189,"_174":175,"_176":171,"_147":190,"_149":190,"_45":354,"_151":194},{"_179":192,"_181":88,"_182":193},{"_43":356,"_55":79,"_57":80,"_174":175,"_176":255,"_147":357,"_149":357,"_45":358,"_151":359},"d1e4567b-f68a-49e1-b3e5-3888343d7871","2025-04-29T04:00:14.680867+00:00",{"_179":88,"_181":88,"_182":200},"resistencia-dos-materiais-7",{"_43":361,"_55":362,"_57":80,"_174":175,"_176":346,"_147":363,"_149":363,"_45":364,"_151":365},"da8d2ecf-e949-4afc-966f-98b4b367fac1","Resistencia dos Materiais","2025-04-29T04:00:06.9766+00:00",{"_179":88,"_181":88,"_182":200},"resistencia-dos-materiais-5",{"_43":313,"_55":367,"_57":368,"_174":175,"_176":-5,"_147":369,"_149":369,"_45":370,"_151":368},"Engenharia","engenharia","2025-04-29T01:55:38.635246+00:00",{"_179":371,"_181":88,"_182":372},29,351586,"tagsTree",[375,384],{"_43":171,"_55":172,"_57":173,"_174":175,"_176":-5,"_147":177,"_149":177,"_45":376,"_151":173,"_184":377},{"_179":180,"_181":88,"_182":183},[378],{"_43":187,"_55":188,"_57":189,"_174":175,"_176":171,"_147":190,"_149":190,"_45":379,"_151":194,"_184":380},{"_179":192,"_181":88,"_182":193},[381],{"_43":197,"_55":79,"_57":80,"_174":175,"_176":187,"_147":198,"_149":198,"_45":382,"_151":201,"_184":383},{"_179":88,"_181":88,"_182":200},[],{"_43":313,"_55":367,"_57":368,"_174":175,"_176":-5,"_147":369,"_149":369,"_45":385,"_151":368,"_184":386},{"_179":371,"_181":88,"_182":372},[387,396,405],{"_43":310,"_55":311,"_57":312,"_174":175,"_176":313,"_147":190,"_149":190,"_45":388,"_151":317,"_184":389},{"_179":315,"_181":88,"_182":316},[390],{"_43":320,"_55":188,"_57":189,"_174":175,"_176":310,"_147":259,"_149":259,"_45":391,"_151":327,"_184":392},{"_179":315,"_181":88,"_182":326},[393],{"_43":319,"_55":79,"_57":80,"_174":175,"_176":320,"_147":321,"_149":321,"_45":394,"_151":323,"_184":395},{"_179":88,"_181":88,"_182":200},[],{"_43":333,"_55":334,"_57":335,"_174":175,"_176":313,"_147":190,"_149":190,"_45":397,"_151":339,"_184":398},{"_179":337,"_181":88,"_182":338},[399],{"_43":346,"_55":347,"_57":348,"_174":175,"_176":333,"_147":349,"_149":349,"_45":400,"_151":352,"_184":401},{"_179":315,"_181":88,"_182":351},[402],{"_43":361,"_55":362,"_57":80,"_174":175,"_176":346,"_147":363,"_149":363,"_45":403,"_151":365,"_184":404},{"_179":88,"_181":88,"_182":200},[],{"_43":258,"_55":341,"_57":342,"_174":175,"_176":313,"_147":190,"_149":190,"_45":406,"_151":344,"_184":407},{"_179":315,"_181":88,"_182":262},[408],{"_43":255,"_55":256,"_57":257,"_174":175,"_176":258,"_147":259,"_149":259,"_45":409,"_151":257,"_184":410},{"_179":261,"_181":88,"_182":262},[411],{"_43":356,"_55":79,"_57":80,"_174":175,"_176":255,"_147":357,"_149":357,"_45":412,"_151":359,"_184":413},{"_179":88,"_181":88,"_182":200},[],"videoDislikes",{},"videoLikes",{},"actionData","errors"]

Comentários