{"id":1966,"date":"2021-11-14T09:52:07","date_gmt":"2021-11-14T00:52:07","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1966"},"modified":"2024-03-26T16:23:31","modified_gmt":"2024-03-26T07:23:31","slug":"tkr202106","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr202106\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2021\uff1a\u7b2c6\u554f"},"content":{"rendered":"\r\n

\u5b9a\u6570 \\(b , c , p , q , r\\) \u306b\u5bfe\u3057, \\[ x^4 +bx +c = ( x^2 +px +q ) ( x^2 -px +r ) \\] \u304c \\(x\\) \u306b\u3064\u3044\u3066\u306e\u6052\u7b49\u5f0f\u3067\u3042\u308b\u3068\u3059\u308b.<\/p>\r\n

    \r\n
  1. (1)<\/strong>\u3000\\(p \\neq 0\\) \u3067\u3042\u308b\u3068\u304d, \\(q , r\\) \u3092 \\(p , b\\) \u3067\u8868\u305b.<\/li>\r\n
  2. (2)<\/strong>\u3000\\(p \\neq 0\\) \u3068\u3059\u308b. \\(b , c\\) \u304c\u5b9a\u6570 \\(a\\) \u3092\u7528\u3044\u3066 \\[ b = ( a^2 +1 ) (a+2) , \\quad c = -\\left( a +\\dfrac{3}{4} \\right) ( a^2 +1 ) \\] \u3068\u8868\u3055\u308c\u3066\u3044\u308b\u3068\u304d, \u6709\u7406\u6570\u3092\u4fc2\u6570\u3068\u3059\u308b \\(t\\) \u306b\u3064\u3044\u3066\u306e\u6574\u5f0f \\(f(t)\\) \u3068 \\(g(t)\\) \u3067 \\[ \\{ p^2 -( a^2 +1 ) \\} \\{ p^4 +f(a) p^2 +g(a) \\} = 0 \\] \u3092\u6e80\u305f\u3059\u3082\u306e\u3092 \\(1\\) \u7d44\u6c42\u3081\u3088.<\/li>\r\n
  3. (3)<\/strong>\u3000\\(a\\) \u3092\u6574\u6570\u3068\u3059\u308b. \\(x\\) \u306e \\(4\\) \u6b21\u5f0f \\[ x^4 +( a^2 +1 ) (a+2) x -\\left( a +\\dfrac{3}{4} \\right) ( a^2 +1 ) \\] \u304c\u6709\u7406\u6570\u3092\u4fc2\u6570\u3068\u3059\u308b \\(2\\) \u6b21\u5f0f\u306e\u7a4d\u306b\u56e0\u6570\u5206\u89e3\u3067\u304d\u308b\u3088\u3046\u306a \\(a\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/li>\r\n<\/ol>\r\n\r\n\r\n\r\n\r\n\r\n
    \r\n
    \r\n\r\n

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

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

    \u4e0e\u5f0f\u306b\u3064\u3044\u3066 \\[ ( \\text{\u53f3\u8fba} ) = x^4 +( p +r -p^2 ) x^2 +p (r-q) x +qr \\] \u4e21\u8fba\u306e\u4fc2\u6570\u3092\u6bd4\u8f03\u3057\u3066 \\[ \\left\\{ \\begin{array}{ll} p +r -p^2 = 0 & ... [1] \\\\ p (r-q) = b & ... [2] \\\\ qr = c & ... [3] \\end{array} \\right. \\] [1] [2] \u3088\u308a \\[ q+r = p^2 \\ , \\ r-q = \\dfrac{b}{p} \\] \u8fba\u3005\u306e\u548c\u3068\u5dee\u3092\u3068\u3063\u3066 \\(2\\) \u3067\u5272\u308c\u3070 \\[ q = \\underline{\\dfrac{1}{2} \\left( p^2 -\\dfrac{b}{p} \\right)} \\ , \\ r = \\underline{\\dfrac{1}{2} \\left( p^2 +\\dfrac{b}{p} \\right)} \\]\r\n\r\n\r\n\r\n

    (2)<\/strong><\/p>\r\n\r\n\r\n\r\n[3] \u3068 (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a \\[ qr = \\dfrac{1}{4} \\left( p^4 -\\dfrac{b^2}{p^2} \\right) = c \\] \\(b , c\\) \u306e\u5f0f\u3092\u4ee3\u5165\u3057, \u4e21\u8fba\u3092 \\(4p^2\\) \u500d\u3057\u3066 \\[\\begin{align} p^6 -( a^2 +1 )^2 (a+2)^2 = -( 4a+3 ) ( a^2 +1 ) & p^2 \\\\ p^6 +( 4a+3 ) ( a^2 +1 ) p^2 -( a^2 +1 )^2 (a+2)^2 & = 0 \\\\ \\{ p^2 -( a^2 +1 ) \\} \\{ p^4 +( a^2 +1 ) p^2 +( a^2 +1 ) (a+2)^2 \\} & = 0 \\end{align}\\] \u3088\u3063\u3066, \u6c42\u3081\u308b\u7d44\u306f \\[ f(t) = \\underline{t^2 +1} \\ , \\ g(t) = \\underline{( t^2 +1 ) (a+2)^2} \\]\r\n\r\n\r\n\r\n

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

    (2)<\/strong> \u3067\u5b9a\u3081\u305f \\(b , c\\) \u306b\u5bfe\u3057\u3066, \u4e0e\u3048\u3089\u308c\u305f\u6052\u7b49\u5f0f\u3092\u307f\u305f\u3059\u3082\u306e\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.<\/p>\r\n\r\n\r\n\r\n

      \r\n
    1. 1*<\/strong>\u3000\\(p = 0\\) \u306e\u3068\u304d
      [2] \u3088\u308a, \\(b = 0\\) \u306a\u306e\u3067 \\[\\begin{align} b & = ( a^2 +1 ) (a+2) = 0 \\\\ & \\text{\u2234} \\quad a = -2 \\end{align}\\] \u307e\u305f, [1] \u3088\u308a \\(r = -q\\) \u3067, [3] \u3088\u308a \\(c = -q^2\\) \u306a\u306e\u3067 \\[ c = -\\left( -2 +\\dfrac{3}{4} \\right) \\cdot 5 = \\dfrac{25}{4} = -q^2 \\] \u3053\u308c\u3092\u307f\u305f\u3059 \\(q\\) \u306f\u5b58\u5728\u3057\u306a\u3044\u306e\u3067, \u4e0d\u9069.<\/li>\r\n\r\n\r\n\r\n
    2. 2*<\/strong>\u3000\\(p \\neq 0\\) \u306e\u3068\u304d
      (2)<\/strong> \u306e\u5f0f\u304c\u6210\u7acb\u3057, \\(f(a) \\gt 0\\) , \\(g(a) \\gt 0\\) \u306a\u306e\u3067, (2)<\/strong> \u306e\u5f0f\u3092\u89e3\u3051\u3070 \\[ p^2 -a^2 = (p+a) (p-a) = 1 \\] \\(a\\) \u306f\u6574\u6570\u306a\u306e\u3067 \\(p\\) \u3082\u6574\u6570\u3067\u3042\u308a, \u3053\u308c\u3092\u3068\u304f\u3068 \\[ ( p , a ) = ( \\pm 1 , 0 ) \\] \u3053\u306e\u3068\u304d \\[ b = 2 \\ , \\ c = \\dfrac{3}{4} \\] \u307e\u305f, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a \\[\\begin{align} q & = \\dfrac{1}{2} \\left( 1 \\pm 2 \\right) = \\dfrac{3}{2} , -\\dfrac{1}{2} \\ , \\\\ r & = \\dfrac{1}{2} \\left( 1 \\mp 2 \\right) = -\\dfrac{1}{2} , \\dfrac{3}{2} \\quad ( \\ \\text{\u8907\u53f7\u540c\u9806} \\ ) \\end{align}\\] \u306a\u306e\u3067, \\(p , q\\) \u3082\u6709\u7406\u6570\u3068\u306a\u308b.
      \u203b\u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u56e0\u6570\u5206\u89e3\u3067\u304d\u308b. \\[ x^4 +2x -\\dfrac{3}{4} = \\left( x^2 -x +\\dfrac{3}{2} \\right) \\left( x^2 +x -\\dfrac{1}{2} \\right) \\]<\/li>\r\n<\/ol>\r\n\r\n\r\n\r\n

      \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u5024\u306f \\[ a = \\underline{0} \\]\r\n<\/div>\r\n\r\n\r\n\r\n

      <\/div>\r\n<\/div>\r\n\r\n","protected":false},"excerpt":{"rendered":"\u5b9a\u6570 \\(b , c , p , q , r\\) \u306b\u5bfe\u3057, \\[ x^4 +bx +c = ( x^2 +px +q ) ( x^2 -px +r ) \\] \u304c \\(x\\) \u306b\u3064\u3044\u3066\u306e\u6052\u7b49\u5f0f\u3067\u3042\u308b\u3068\u3059\u308b. (1)\u3000\\(p … \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":[164],"tags":[139,165],"class_list":["post-1966","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2021","tag-tokyo_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1966","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=1966"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1966\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1966"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1966"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1966"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}