{"id":297,"date":"2012-02-11T18:20:11","date_gmt":"2012-02-11T09:20:11","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=297"},"modified":"2021-09-16T06:43:28","modified_gmt":"2021-09-15T21:43:28","slug":"ngr200801","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr200801\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2008\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(a , b , c\\) \u3092\u5b9f\u6570\u3068\u3057\u3066 \\(f(x) = x^4 +ax^3 +bx^2 +cx +2\\) \u3068\u3059\u308b.\r\n\u884c\u5217 \\(A = \\left( \\begin{array}{cc} -1 & -1 \\\\ 1 & -1 \\end{array} \\right)\\) \u3068\u5358\u4f4d\u884c\u5217 \\(E\\) \u306b\u5bfe\u3057\u3066, \\(A^4 +aA^3 +bA^2 +cA +2E = O\\) \uff08\u305f\u3060\u3057 \\(O\\) \u306f\u96f6\u884c\u5217\uff09\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(b , c\\) \u3092 \\(a\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u65b9\u7a0b\u5f0f \\(f(x) = 0\\) \u304c\u5c11\u306a\u304f\u3068\u3082 \\(1\\) \u3064\u6b63\u306e\u89e3\u3092\u6301\u3064\u3068\u304d, \\(a\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \u30b1\u30fc\u30ea\u30fc\u30fb\u30cf\u30df\u30eb\u30c8\u30f3\u306e\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{align}\r\nA^2 -(-2) A +\\left\\{ -1 \\cdot (-1) -1 \\cdot (-1) \\right\\} E & = O \\\\\r\n\\text{\u2234} \\quad A^2 -2A +2E & = O \\quad ... [1]\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nf(A) & = \\left\\{ A^2 +(a-2) A +(b-2a+2) E \\right\\} (A^2 -2A +2E) \\\\\r\n& \\qquad + (2a-2b+c) A +(4a-2b-2) E\n\\end{align}\\]\r\n[1] \u3088\u308a, \\(f(A) = O\\) \u3067\u3042\u308a, \\(A \\neq k E\\) \uff08 \\(k\\) \u306f\u5b9f\u6570\uff09\u306a\u306e\u3067\r\n\\[\\begin{gather}\r\n2a-2b+c =0 , \\ 4a-2b-2 =0 \\\\\r\n\\text{\u2234} \\quad b= \\underline{2a-1} , \\ c= \\underline{2a-2}\n\\end{gather}\\]\r\n

    (2)<\/strong><\/p>\r\n

    (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nf(x) = (x^2+2x+2) \\left\\{ x^2+(a-2)x+1 \\right\\}\n\\]\r\n\\(x^2+2x+2 =(x+1)^2+1 \\gt 0\\) \u306a\u306e\u3067, \\(g(x) = x^2-(a-2)x+1 = 0\\) \u304c\u5c11\u306a\u304f\u3068\u3082 \\(1\\) \u3064\u306e\u6b63\u306e\u89e3\u3092\u6301\u3064\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
    \r\n\\(g(0) \\gt 0\\) \u306a\u306e\u3067, \u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nD & = (a-2)^2 -4 = a(a-4) \\geqq 0 \\\\\r\n\\text{\u2234} \\quad a & \\leqq 0 , 4 \\leqq a \\quad ... [2]\n\\end{align}\\]\r\n\u307e\u305f, \u8ef8\u306e\u4f4d\u7f6e\u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n-\\dfrac{a-2}{2} & \\gt 0 \\\\\r\n\\text{\u2234} \\quad a & \\lt 2 \\quad ... [3]\n\\end{align}\\]\r\n\u3088\u3063\u3066, [2] [3] \u3088\u308a\r\n\\[\r\n\\underline{a \\leqq 0}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a , b , c\\) \u3092\u5b9f\u6570\u3068\u3057\u3066 \\(f(x) = x^4 +ax^3 +bx^2 +cx +2\\) \u3068\u3059\u308b. \u884c\u5217 \\(A = \\left( \\begin{array}{cc} -1 & -1 \\\\ 1 & - … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[53],"tags":[143,16],"class_list":["post-297","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2008","tag-nagoya_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/297","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=297"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/297\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=297"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=297"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}