{"id":33659,"date":"2021-03-28T13:00:33","date_gmt":"2021-03-28T13:00:33","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33659"},"modified":"2025-07-30T18:13:28","modified_gmt":"2025-07-30T18:13:28","slug":"algebra-polynomiis-numerorum-realium","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/","title":{"rendered":"Algebra Polynomiis Numerorum Realium"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px; text-align:center;\">\n<h1>Algebra Polynomiis Numerorum Realium<\/h1>\n<p>    <em><strong>Summarium:<\/strong><br \/>\n        In hac lectione, algebra polynomiis explorabitur: eius definitio, proprietates, et applicationes. Polynomia sunt pars fundamentalis mathematicae et amplas applicationes habent in variis disciplinis.<br \/>\n    <\/em><\/p>\n<p>    <strong>METAE DISCENDI<\/strong><\/p>\n<p>Post hanc lectionem peractam, discipulus poterit:<\/p>\n<p style=\"text-align:left;\">\n        1. Polynomia earumque proprietates definire atque intellegere.<br \/>\n        2. Gradum et coefficientes polynomii cognoscere.<br \/>\n        3. Operationes algebraicas cum polynomiis perficere et proprietates in contextibus mathematicis applicare.\n    <\/p>\n<p>    <strong>INDEX RERUM:<\/strong><\/p>\n<p>\n        <a href=\"#1\"><strong>1. Algebra Polynomiis: Definitiones<\/strong><\/a><br \/>\n        <a href=\"#2\"><strong>2. Genera Polynomiis<\/strong><\/a><br \/>\n        <a href=\"#3\"><strong>3. Algebra Polynomiis: Operationes<\/strong><\/a><br \/>\n        <a href=\"#4\"><strong>4. Factorisatio et Divisio Polynomiis<\/strong><\/a>\n    <\/p>\n<p>    <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ry4sKaS3RMc\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2><strong>1. Algebra Polynomiis: Definitiones<\/strong><\/h2>\n<p style=\"text-align: justify;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=139s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Ut algebra polynomiis intellegatur, primum scire oportet quid sint polynomia.<\/span><\/strong><\/a> Polynomia sunt functiones algebraicae. Si <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> est variabilis realis, tum functio <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> polynomium appellatur, si scribi potest forma:\n<\/p>\n<p style=\"text-align: center;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(x)= \\sum_{i=0}^n a_i x^i= a_0 + a_1x + a_2x^2 + a_3x^3 + \\cdots + a_nx^n,<\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n    ubi <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> est numerus integer non negativus et omnes <span class=\"katex-eq\" data-katex-display=\"false\">a_i<\/span>, cum <span class=\"katex-eq\" data-katex-display=\"false\">i\\in\\{1,2,3,\\cdots,n\\},<\/span> sunt coefficientes reales. Si existit <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> talis ut <span class=\"katex-eq\" data-katex-display=\"false\">a_k\\neq 0<\/span> et, cum <span class=\"katex-eq\" data-katex-display=\"false\">k\\lt i<\/span>, fit ut <span class=\"katex-eq\" data-katex-display=\"false\">a_i=0<\/span>, tunc dicitur talis valor <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> esse <strong>gradus polynomii.<\/strong> Aliter dicendum, gradus polynomii est maxima potentia cuius coefficientes non est nullus.\n<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2><strong>2. Genera Polynomiis<\/strong><\/h2>\n<p style=\"text-align: justify;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=340s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Polynomia secundum gradum suum classificantur;<\/span><\/strong><\/a> ideo, cum polynomium commemoratur, fere semper dicitur esse polynomium gradus <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span>, cum <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> sit maxima potentia <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> quae coefficientem non nullum comitatur in tali polynomio.\n<\/p>\n<h3>2.1. Polynomia Constans<\/h3>\n<p style=\"text-align: justify;\">Haec est familia quae omnia polynomia gradus nulli et polynomium nullum complectitur. Dicimus polynomium esse gradus nulli, si scribi potest forma <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=c,<\/span> cum <span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0.<\/span> Contra, polynomium nullum est formae <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 0<\/span> et huic gradus non definitur.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2><strong>3. Algebra Polynomiis: Operationes<\/strong><\/h2>\n<p style=\"text-align: justify;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=428s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Polynomia omnes suas proprietates ex algebra numerorum realium accipiunt.<\/span><\/strong><\/a> Praecipuae sunt proprietates distributivae et associativae.\n<\/p>\n<h3>3.1. Additio et Subtractio<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=470s\" target=\"_blank\" rel=\"noopener\"> <strong><span style=\"color: #ff0000;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> sunt duo polynomia gradus<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">m<\/span>, respective, cum<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">m=n+k<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">0\\leq k,<\/span>\n<p style=\"text-align: justify;\">tunc habebitur:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle P(x) \\pm Q(x) &amp;=\\displaystyle \\sum_{i=0}^n a_i x^i \\pm \\sum_{i=0}^m b_i x^i \\\\ \\\\\n\n &amp;\\displaystyle = \\sum_{i=0}^n a_i x^i \\pm \\left( \\sum_{i=0}^n b_i x^i + \\sum_{i=n+1}^{n+k} b_i x^i \\right) \\\\ \\\\\n\n&amp;\\displaystyle = \\sum_{i=0}^n (a_i \\pm b_i) x^i + \\sum_{i=n+1}^m b_i x^i\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: justify;\">Id est, coefficientes eidem potentiis <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> adsignati adduntur aut subtrahuntur, prout res postulaverit.<\/p>\n<p style=\"text-align: justify;\"><span style=\"color: #000080;\">EXEMPLUM:<\/span><br \/>\nSi <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 3+5x+2x^2<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 6x-3x^2 +23x^5<\/span>, tunc:\n<\/p>\n<p style=\"text-align: justify;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">P(x) + Q(x) = \\cdots \\\\ = (3+5x+2x^2) + (6x-3x^2 +23x^5) \\\\ = 3 + (5+6)x + (2-3)x^2 + 23x^5 \\\\ = 3 + 11x - x^2 + 23x^5 <\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">P(x) - Q(x) = \\cdots \\\\ = (3+5x+2x^2) - (6x-3x^2 +23x^5) \\\\ = 3 + (5-6)x + (2+3)x^2 - 23x^5 \\\\ = 3 - x + 5x^2 - 23x^5 <\/span>\n<\/p>\n<h3>3.2. Multiplicatio<\/h3>\n<p style=\"text-align: justify;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=894s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Eodem contextu ac in additione et subtractione polynomiis,<\/span><\/strong><\/a> productio polynomiis sic evolvetur:\n<\/p>\n<p style=\"text-align: justify;\">\n    Primum distinguimus multiplicationem per scalar. Si <span class=\"katex-eq\" data-katex-display=\"false\">c \\in \\mathbb{R},<\/span> tunc habetur:\n<\/p>\n<p style=\"text-align: center;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle c P(x) = c \\sum_{i=0}^n a_i x^i =\\sum_{i=0}^n c a_i x^i <\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n    Deinde habemus multiplicationem inter polynomia:\n<\/p>\n<p style=\"text-align: center;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\displaystyle P(x) Q(x) &amp;\\displaystyle = \\left( \\sum_{i=0}^n a_i x^i \\right) \\left(\\sum_{j=0}^m b_j x^j\\right) \\\\ \\\\\n\n&amp;=\\displaystyle \\left[\\sum_{j=0}^m \\left( \\sum_{i=0}^n a_i x^i \\right) b_j x^j\\right] \\\\ \\\\\n\n&amp;=\\displaystyle \\sum_{j=0}^m \\left( \\sum_{i=0}^n a_ib_j x^{i+j} \\right) \\\\ \\\\\n\n&amp;=\\displaystyle \\sum_{i,j=0}^{n,m} a_ib_j x^{i+j}\n\n\\end{array}<\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n    Hoc est quod per compendium dicimus: \u00absumma productorum omnium cum omnibus\u00bb.\n<\/p>\n<p style=\"text-align: justify;\">\n    <span style=\"color: #000080;\">EXEMPLUM:<\/span><br \/>\n    Si <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 4x+ 2x^2-x^4<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 5 - x + x^2-7x^3,<\/span> tunc:\n<\/p>\n<p style=\"text-align: justify;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">P(x)Q(x) =\\cdots \\\\ {} \\\\= (4x+ 2x^2-x^4)(5 - x + x^2-7x^3) \\\\ {} \\\\ = 4x(5 - x + x^2-7x^3) \\\\ + 2x^2 (5 - x + x^2-7x^3) \\\\ - x^4 (5 - x + x^2-7x^3) \\\\ {} \\\\ = 20x - 4x^2 + 4x^3 - 28x^4 \\\\ + 10x^2 - 2x^3 + 2x^4 - 14x^5 \\\\ -5x^4 + x^5 - x^6 + 7x^7 \\\\ {} \\\\ = 20x + 6x^2 + 2x^3 - 31x^4 - 13x^5 - x^6 + 7x^7<\/span>\n<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2><strong>4. Factorisatio et Divisio Polynomiis<\/strong><\/h2>\n<p style=\"text-align: justify;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1375s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Cum duo polynomia multiplicamus, transimus a duobus simplicibus ad unum difficiliorem (maioris gradus).<\/span><\/strong><\/a> Cum polynomium factorisamus, processum inversum sequimur: transformamus polynomium difficilem in productum duorum vel plurium polynomiis minoris gradus.\n<\/p>\n<p style=\"text-align: justify;\">\n    Ut polynomium <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> factorisetur, necesse est invenire valores <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> qui polynomium annulant; si tales valores existunt, tunc polynomium est factorisabile. De existentia loqui est facile, sed eos invenire alia res est. Hoc argumentum diligentius considerabimus cum factorisationes polynomiis quadraticorum et (2n)quadraticorum tractabimus.\n<\/p>\n<h3>4.1. Producta Notabilia<\/h3>\n<p style=\"text-align: justify;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1654s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Exstant tamen casus in quibus factorizatio simpliciter obtinetur,<\/span><\/strong><br \/>\n    <\/a> ut in productis notabilibus. Quaedam ex his resultatis sunt haec:\n<\/p>\n<p style=\"text-align: justify;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">x^2 - y^2 = (x-y)(x+y)<\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">(x\\pm y)^2 = x^2 \\pm 2xy + y^2<\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">(x \\pm y)^3 = x^3 \\pm 3x^2y + 3xy^2 \\pm y^3<\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">x^3-y^3=(x-y)(x^2+xy+y^2)<\/span>\n<\/p>\n<p style=\"text-align: justify;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">x^3+y^3=(x+y)(x^2-xy+y^2)<\/span>\n<\/p>\n<h3>4.2. Algorithmus Divisionis<\/h3>\n<p style=\"text-align: justify;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1854s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Sicut multiplicando numeros integros numeri compositi fiunt et divisio per algorithmum divisionis permittit factorizationem si residuum est nullum,<\/span><\/strong><\/a> simile fit cum polynomiis. Algorithmi divisionis explicatio per textum aliquantum difficilis esse potest; multo facilius est intelligere directe intuendo quomodo fiat et in quibus casibus algorithmus ad factorizationem ducat. Ad hoc assequendum, exempli gratia quaedam considerabimus.\n<\/p>\n<p style=\"text-align: justify;\">\n    <span style=\"color: #000080;\">EXEMPLUM:<\/span> Computare <span class=\"katex-eq\" data-katex-display=\"false\">P(x):Q(x)<\/span> in casibus sequentibus:\n<\/p>\n<ol style=\"text-align: justify;\">\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=2 x^3 + x^2 - 2 x - 1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x-1<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1930s\" target=\"_blank\" rel=\"noopener\"> <strong><span style=\"color: #ff0000;\">[SOLUTIO]<\/span><\/strong> <\/a>\n    <\/li>\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=x^4+2x^3-x+1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x^2-4<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=2120s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[SOLUTIO]<\/strong><\/span> <\/a>\n    <\/li>\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=3 x^4 - 2 x^3 - x^2 - 4 x + 1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x^2+x+1<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=2331s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[SOLUTIO]<\/strong><\/span> <\/a>\n    <\/li>\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=x^7+5x^4+5x^2-3x+1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x^3-2x^2+1<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=2464s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[SOLUTIO]<\/strong><\/span> <\/a>\n    <\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Algebra Polynomiis Numerorum Realium Summarium: In hac lectione, algebra polynomiis explorabitur: eius definitio, proprietates, et applicationes. Polynomia sunt pars fundamentalis mathematicae et amplas applicationes habent in variis disciplinis. METAE DISCENDI Post hanc lectionem peractam, discipulus poterit: 1. Polynomia earumque proprietates definire atque intellegere. 2. Gradum et coefficientes polynomii cognoscere. 3. Operationes algebraicas cum polynomiis perficere [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27134,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":8,"footnotes":""},"categories":[1304,1298],"tags":[],"class_list":["post-33659","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-et-geometria","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Algebra Polynomiis Numerorum Realium - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce de algebra polynomiis: definitionibus, generibus, operationibus, productis, factorisatione et divisione.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Algebra Polynomiis Numerorum Realium\" \/>\n<meta property=\"og:description\" content=\"Disce de algebra polynomiis: definitionibus, generibus, operationibus, productis, factorisatione et divisione.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-28T13:00:33+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-07-30T18:13:28+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/polinomios.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Algebra Polynomiis Numerorum Realium\" \/>\n<meta name=\"twitter:description\" content=\"Disce de algebra polynomiis: definitionibus, generibus, operationibus, productis, factorisatione et divisione.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/polinomios.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Algebra Polynomiis Numerorum Realium\",\"datePublished\":\"2021-03-28T13:00:33+00:00\",\"dateModified\":\"2025-07-30T18:13:28+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/\"},\"wordCount\":1000,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/polinomios.jpg\",\"articleSection\":[\"Algebra et Geometria\",\"Mathematica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/\",\"url\":\"http:\/\/toposuranos.com\/material\/la\/algebra-polynomiis-numerorum-realium\/\",\"name\":\"Algebra Polynomiis Numerorum Realium - 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