{"id":864,"date":"2013-08-04T16:43:37","date_gmt":"2013-08-04T07:43:37","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=864"},"modified":"2021-09-23T09:31:33","modified_gmt":"2021-09-23T00:31:33","slug":"thr201301","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201301\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2013\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(k\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \\(3\\) \u6b21\u5f0f \\(f(x) = x^3-kx^2-1\\) \u306b\u5bfe\u3057, \u65b9\u7a0b\u5f0f \\(f(x) = 0\\) \u306e \\(3\\) \u3064\u306e\u89e3\u3092 \\(\\alpha , \\beta , \\gamma\\) \u3068\u3059\u308b.\r\n\\(g(x)\\) \u306f \\(x^3\\) \u306e\u4fc2\u6570\u304c \\(1\\) \u3067\u3042\u308b \\(3\\) \u6b21\u5f0f\u3067, \u65b9\u7a0b\u5f0f \\(g(x) = 0\\) \u306e \\(3\\) \u3064\u306e\u89e3\u304c \\(\\alpha \\beta , \\beta \\gamma , \\gamma \\alpha\\) \u3067\u3042\u308b\u3082\u306e\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(g(x)\\) \u3092 \\(k\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(2\\) \u3064\u306e\u65b9\u7a0b\u5f0f \\(f(x) = 0\\) \u3068 \\(g(x) = 0\\) \u304c\u5171\u901a\u306e\u89e3\u3092\u3082\u3064\u3088\u3046\u306a \\(k\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

    \\(f(x) = 0\\) \u304c \\(\\alpha , \\beta , \\gamma\\) \u3092\u89e3\u306b\u3082\u3064\u306e\u3067, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\n\\left\\{\\begin{array}{l} \\alpha +\\beta +\\gamma = 0 \\\\ \\alpha \\beta +\\beta \\gamma +\\gamma \\alpha = -k \\\\ \\alpha \\beta \\gamma = 1 \\end{array} \\right. \\quad ... [1] \\ .\r\n\\]\r\n\\(\\alpha' = \\alpha \\beta\\) , \\(\\beta' = \\beta \\gamma\\) , \\(\\gamma' = \\gamma \\alpha\\) \u3068\u304a\u304f\u3068, [1] \u3092\u7528\u3044\u3066\r\n\\[\r\n\\left\\{\\begin{array}{l} \\alpha' +\\beta' +\\gamma' = \\alpha \\beta +\\beta \\gamma +\\gamma \\alpha = -k \\\\\r\n\\alpha' \\beta' +\\beta' \\gamma' +\\gamma' \\alpha' = \\alpha \\beta \\gamma \\left( \\alpha +\\beta +\\gamma \\right) = 0 \\\\\r\n\\alpha' \\beta' \\gamma' = \\left( \\alpha \\beta \\gamma \\right)^2 = 1 \\end{array} \\right. \\ .\r\n\\]\r\n\u3088\u3063\u3066, \\(g(x) = 0\\) \u306f \\(\\alpha' , \\beta' , \\gamma'\\) \u3092\u89e3\u306b\u3082\u3061, \\(x^3\\) \u306e\u4fc2\u6570\u304c \\(1\\) \u306a\u306e\u3067, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\ng(x) = \\underline{x^3+kx^2-1} \\ .\r\n\\]\r\n

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

    \u5171\u901a\u89e3\u3092 \\(a\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\left\\{\\begin{array}{l} f(a) = a^3-ka-1 = 0 \\\\ g(a) = a^3+ka-1 = 0 \\end{array} \\right. \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\na^3-ka-1 & = a^3+ka-1 \\\\\r\nka(a+1) & = 0 \\\\\r\n\\text{\u2234} \\quad k = 0 , \\ a & = 0 , -1 \\ .\r\n\\end{align}\\]\r\n\u3067\u3042\u308b\u3053\u3068\u304c\u5fc5\u8981\u3067\u3042\u308b.
    \r\n\u305d\u308c\u305e\u308c\u306e\u5834\u5408\u306b\u3064\u3044\u3066, \u5171\u901a\u89e3\u304c\u3042\u308b\u304b\u3092\u78ba\u8a8d\u3059\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(k = 0\\) \u306e\u3068\u304d\r\n\\[\r\nf(x) = g(x) = x^3-1 \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u5171\u901a\u89e3\u3092\u3082\u3064.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(a = 0\\) \u306e\u3068\u304d\r\n\\[\r\nf(0) = g(0) = -1 \\neq 0 \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u5171\u901a\u89e3\u306f\u5b58\u5728\u3057\u306a\u3044.<\/p><\/li>\r\n

    3. 3*<\/strong>\u3000\\(a = -1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nf(-1) = g(-1) & = k-2 = 0 \\\\\r\n\\text{\u2234} \\quad k & = 2 \\ .\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d, \u78ba\u304b\u306b \\(x = -1\\) \u304c\u5171\u901a\u89e3\u3068\u306a\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b \\(k\\) \u306e\u5024\u306f\r\n\\[\r\nk = \\underline{0 , 2} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(k\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \\(3\\) \u6b21\u5f0f \\(f(x) = x^3-kx^2-1\\) \u306b\u5bfe\u3057, \u65b9\u7a0b\u5f0f \\(f(x) = 0\\) \u306e \\(3\\) \u3064\u306e\u89e3\u3092 \\(\\alpha , \\beta , \\gamma\\) \u3068\u3059 … \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":[96],"tags":[148,111],"class_list":["post-864","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2013","tag-tohoku_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/864","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=864"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/864\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=864"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=864"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=864"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}