{"id":33671,"date":"2021-03-30T13:00:36","date_gmt":"2021-03-30T13:00:36","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33671"},"modified":"2025-07-30T18:45:12","modified_gmt":"2025-07-30T18:45:12","slug":"factorisatio-polynomi-quadratici-et-2n-quadratici","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/factorisatio-polynomi-quadratici-et-2n-quadratici\/","title":{"rendered":"Factorisatio Polynomi Quadratici et (2n)-Quadratici"},"content":{"rendered":"<p><center><\/p>\n<h1>Factorisatio Polynomi Quadratici et (2n)-Quadratici<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\n   In hac lectione accuratius perpendemus processum factorisationis polynominum quadraticorum <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^2 + bx + c<\/span> necnon polynominum (2n)-quadraticorum <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2n} + bx^n + c<\/span>, eos in factores simplices resolventes. Procedendi rationes mathematico modo evolventur atque exempla practica demonstrabuntur.<\/em><\/p>\n<p>   <strong>Proposita Discendi<\/strong><\/p>\n<ol style=\"text-align: left;\">\n<li><strong>Discere<\/strong> factorisationem polynominum quadraticorum formae <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^2 + bx + c<\/span>.<\/li>\n<li><strong>Derivare<\/strong> et uti formula quadatica <span class=\"katex-eq\" data-katex-display=\"false\">x = \\displaystyle \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}<\/span> ad radices inveniendas.<\/li>\n<li><strong>Applicare<\/strong> technicas factorisationis ad polynomos (2n)-quadraticos formae <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2n} + bx^n + c<\/span>.<\/li>\n<li><strong>Agnoscere<\/strong> condiciones necessarias ad factorisationem polynominum quadraticorum.<\/li>\n<li><strong>Adhibere<\/strong> methodum complementi quadrati in processu factorisationis.<\/li>\n<\/li>\n<\/ol>\n<p>   <strong>INDEX MATERIAE:<\/strong><br \/>\n   <a href=\"#1\">Introductio<\/a><br \/>\n   <a href=\"#2\">Polynomi Quadraticus et Polynomi (2n)-Quadraticus<\/a><br \/>\n   <a href=\"#3\">Factorisatio Polynomi Quadratici<\/a><br \/>\n   <a href=\"#4\">Expansio ad factorisationem Polynomi Bi-Quadratici<\/a><br \/>\n   <a href=\"#5\">Exempla Exercitiorum<\/a>\n   <\/p>\n<p>   <\/center><\/p>\n<p>   <center><br \/>\n   <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ddTfUR7QBfY\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n   <a name=\"1\"><\/a><\/p>\n<h2>Introductio<\/h2>\n<p style=\"text-align: justify;\">Discere quomodo polynomi quadratici factorisentur est primus gradus ad multarum aliarum technicarum factorisationis studium incipiendum. Quam ob rem hanc technicam penitus investigabimus eiusque usum quam longissime extendemus. Peracto hoc studio, non solum polynomos quadratcos (gradus 2) factorisare poteris, sed etiam easdem technicas adhibebis ad polynomos (2n)-quadraticos factorisandos.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Polynomi Quadraticus et Polynomi (2n)-Quadraticus<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=96s\" target=\"_blank\" rel=\"noopener\"><strong>Polynomi quadraticus est polynomi gradus secundi.<\/strong><\/a> Ex hoc sequitur quod polynomi quadraticus est quaelibet functio formae<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2}+bx +c <\/span>\n<p style=\"text-align: justify;\">ubi <span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{R}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>. Studium nostrum tamen non solum in factorisatione talium polynominum versabitur, sed etiam ad formam generalizatam intendemus, cuius polynomi quadraticus casus tantum particularis est. Agitur de polynomo (2n)-quadratico. Haec generalizatio omnes polynomos comprehendit qui scribi possunt forma<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2n}+bx^n +c <\/span>\n<p style=\"text-align: justify;\">ubi praeterea supponimus <span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{R}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>, et <span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{N}<\/span> quemlibet accipimus. Exempla huius generis polynomi sunt:<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 3x^2 -x + 1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 7x^4 +5x^2 + 3<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">R(x) = -4x^6 +12x^3 + 2<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">S(x) = 21x^8 -75 x^4 -9<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">et sic generaliter.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Factorisatio Polynomi Quadratici<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=193s\" target=\"_blank\" rel=\"noopener\"><strong>Ut iam vidimus, polynomi gradus 2 formam generalem habet<\/strong><\/a><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2}+bx +c \\;\\; , \\;\\; a\\neq 0 <\/span>\n<p style=\"text-align: justify;\">Factorisatio est processus quo polynomi complexus separatur in productum duorum polynominum simpliciorum. Itaque, si factorisatio fieri potest, tunc exsistunt constantes <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta,\\gamma,\\delta \\in\\mathbb{R}<\/span>, cum <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\gamma \\neq 0<\/span> tales ut:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^2 + bx + c <\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\">= (\\alpha x + \\beta)(\\gamma x + \\delta) <\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\">= \\alpha \\gamma \\left(x +\\displaystyle \\frac{\\beta}{\\alpha}\\right)\\left(x + \\frac{\\delta}{\\gamma}\\right) <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Cum aequalitas inter sinistram et dextram partem habeatur, sequitur ut, si una pars annuletur, altera quoque necessario annuletur. Pars dextra annullatur cum <span class=\"katex-eq\" data-katex-display=\"false\">x=-\\beta\/\\alpha<\/span> aut <span class=\"katex-eq\" data-katex-display=\"false\">x=-\\delta\/\\gamma<\/span>. Nunc videamus quibus valoribus pars sinistra huius aequalitatis annuletur. Habebimus quod<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">ax^2 + bx + c<\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = 0<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">ax^2 + bx <\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = -c<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">x^2 + \\displaystyle \\frac{b}{a}x <\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = - \\displaystyle \\frac{c}{a}<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right; background-color: #ffc0c0;\"><span class=\"katex-eq\" data-katex-display=\"false\">x^2 + \\displaystyle \\frac{b}{a}x + \\frac{b^2}{4a^2}<\/span><\/td>\n<td style=\"text-align: left; background-color: #ffc0c0;\"><span class=\"katex-eq\" data-katex-display=\"false\"> =\\displaystyle \\frac{b^2}{4a^2} -\\frac{c}{a} = \\frac{ab^2 - 4a^2 c}{4a^3} = \\frac{b^2 - 4ac }{4a^2}<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(x + \\displaystyle \\frac{b}{2a}\\right)^2<\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = \\displaystyle \\frac{b^2 - 4ac }{4a^2} <\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\"> x + \\displaystyle \\frac{b}{2a} <\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = \\pm \\sqrt{\\displaystyle \\frac{b^2 - 4ac }{4a^2}} = \\frac{\\pm\\sqrt{b^2 - 4ac }}{2a} <\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right; background-color: #a0ffa0;\"><span class=\"katex-eq\" data-katex-display=\"false\"> x <\/span><\/td>\n<td style=\"text-align: left; background-color: #a0ffa0;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = \\displaystyle \\frac{-b \\pm\\sqrt{b^2 - 4ac }}{2a} <\/span> \u2705<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Ex hoc ratiocinio sequitur quod constantes litteris Graecis expressae in factorisatione debent (absque amissione generalitatis) sequentia requisita complere:<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\gamma = a<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\beta}{\\alpha} = - \\left(\\frac{-b + \\sqrt{b^2 - 4ac }}{2a} \\right)<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\delta}{\\gamma} = - \\left(\\frac{-b - \\sqrt{b^2 - 4ac }}{2a} \\right)<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Quibus positis, iam habemus methodum quae sinit nos quemlibet polynomium gradus secundi factorisare. Sin autem factorisatio non sit possibilis, hoc per numerum sub radice ostendetur: si numerus negativus est, tunc factorisatio (inter numeros reales) fieri non potest. Totum hoc possumus compendiare introducendo signa notationis huiusmodi:<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">x_1 =\\displaystyle \\frac{-b + \\sqrt{b^2 - 4ac }}{2a} <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">x_2 =\\displaystyle \\frac{-b - \\sqrt{b^2 - 4ac }}{2a} <\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Quae denique in formula vetere et fidelissima comprehenduntur:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{x_{1,2} = \\displaystyle \\frac{-b \\pm \\sqrt{b^2 - 4ac }}{2a}}<\/span> \u2705<\/p>\n<p style=\"text-align: justify;\">Unde factorisatio demum talis erit:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{P(x) = ax^2 +bx + c = a(x-x_1)(x - x_2)}<\/span>\u2705<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Expansio ad Factorisationem Polynomi Bi-Quadratici<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=997s\" target=\"_blank\" rel=\"noopener\"><strong>Haec technica adhiberi potest etiam ad polynomium bi-quadraticum factorisandum<\/strong><\/a> hoc modo:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = ax^4 + bx^2 + c = a(x^2)^2 + bx^2 + c =a (x^2 - x_1^2)(x^2-x_2^2) <\/span>\n<p style=\"text-align: justify;\">Ubi <span class=\"katex-eq\" data-katex-display=\"false\"> x^2_{1,2} = \\displaystyle \\dfrac{-b \\pm \\sqrt{b^2 - 4ac }}{2a}<\/span>. Hac ratione, nunc poteris scribere<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = ax^4 + bx^2 + c = a\\left(x^2 - \\displaystyle \\dfrac{-b + \\sqrt{b^2 - 4ac }}{2a}\\right) \\left(x^2- \\dfrac{-b - \\sqrt{b^2 - 4ac }}{2a}\\right) <\/span>\n<p style=\"text-align: justify;\">Hoc loco attendendum est, nam quae sequuntur certas habent restrictiones. Si <span class=\"katex-eq\" data-katex-display=\"false\">x_1^2<\/span> numerus positivus est, tunc poteris uti regula summae et differentiae ad separandum <span class=\"katex-eq\" data-katex-display=\"false\">(x^2 - x_1^2) = (x-x_1)(x + x_1)<\/span>; alioquin numeri imaginarii emergent, et factorisatio inter reales iam non erit possibilis. Si radices bene definitae sunt, tum scribere poteris:<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nQ(x) &amp;= ax^4 + bx^2 + c \\\\ \\\\\n\n     &amp; = a \\left(x -\\displaystyle \\sqrt{\\frac{-b + \\sqrt{b^2 - 4ac }}{2a}}\\right) \\left(x + \\displaystyle \\sqrt{\\frac{-b + \\sqrt{b^2 - 4ac }}{2a}}\\right) \\\\ \\\\\n\n&amp; \\left(x- \\displaystyle \\sqrt{\\frac{-b - \\sqrt{b^2 - 4ac }}{2a}}\\right) \\left(x+ \\sqrt{\\displaystyle \\frac{-b - \\sqrt{b^2 - 4ac }}{2a}}\\right)\n\n\\end{array}<\/span>\n<p style=\"text-align: justify;\">Aliter, in passo superiore sistendum erit.<\/p>\n<h3>Generalizatio ad Factorisationem Polynomi (2n)-Quadratici<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=1521s\" target=\"_blank\" rel=\"noopener\"><strong>Iam ex hoc intellegitur quo spectet methodus: ad polynomium (2n)-quadraticum factorisandum<\/strong><\/a> sufficit modum scribendi reformulare atque priores methodos adhibere, dummodo radices bene definitae maneant. Itaque habebimus:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">R(x) = a(x^n)^{2}+b (x^n) +c = a(x^n-x_1^n)(x^n-x_2^n) <\/span>\n<p style=\"text-align: justify;\">Ubi <span class=\"katex-eq\" data-katex-display=\"false\">x^n_{1,2} =\\displaystyle \\frac{-b \\pm \\sqrt{b^2 - 4ac }}{2a}<\/span>. Deinde ex hoc, separatio per summam et differentiam fiet, nisi numeri imaginarii occurrant.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercitia Exemplaria:<\/h2>\n<p style=\"text-align: justify;\">Nunc tuum est has technicas per exercitia experiri. Polynomi infra positi omnino fortuito electi sunt, itaque aptissimi erunt ad agnoscendas difficultates quae in factorisatione huiusmodi expressionum occurrere possunt.<\/p>\n<h3>Primus Cursus<\/h3>\n<p style=\"text-align: justify;\">Hi sunt polynomi quos in initio huius scripti ad exemplum adduximus:<\/p>\n<ol style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 3x^2 -x + 1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 7x^4 +5x^2 + 3<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">R(x) = -4x^6 +12x^3 + 2<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">S(x) = 21x^8 -75 x^4 -9<\/span><\/li>\n<\/ol>\n<h3>Secundus Cursus<\/h3>\n<p style=\"text-align: justify;\">Hi autem sunt aliqui alii paulo difficiliores:<\/p>\n<ol style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 78x^2 -21x - 13<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 27x^4 +5x^2 - 14<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">R(x) = 9x^6 +12x^3 - 16<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">S(x) = -9x^8 -2 x^4 + 10<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">T(x) = 5x^{12} -2 x^6 - 15<\/span><\/li>\n<\/ol>\n<h3>Solutio Exercitiorum<\/h3>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ilNTFyF7Hmo\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Factorisatio Polynomi Quadratici et (2n)-Quadratici Summarium: In hac lectione accuratius perpendemus processum factorisationis polynominum quadraticorum necnon polynominum (2n)-quadraticorum , eos in factores simplices resolventes. Procedendi rationes mathematico modo evolventur atque exempla practica demonstrabuntur. Proposita Discendi Discere factorisationem polynominum quadraticorum formae . Derivare et uti formula quadatica ad radices inveniendas. Applicare technicas factorisationis ad polynomos (2n)-quadraticos [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28831,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":3,"footnotes":""},"categories":[1304,1298],"tags":[],"class_list":["post-33671","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Factorisatio Polynomi Quadratici et (2n)-Quadratici - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce quomodo polynomium quadraticum simpliciter factorisetur. 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