\\(5\\) \u6b21\u5f0f \\(f(x) =x^5 +px^4 +qx^3 +rx^2 +sx +t \\) \uff08 \\(p, q, r, s, t\\) \u306f\u5b9f\u6570\uff09\u306b\u3064\u3044\u3066\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u6570\u5217 \\(f(0) , f(1) , f(2) , f(3) , f(4)\\) \u304c\u7b49\u5dee\u6570\u5217\u3067\u3042\u308b\u3053\u3068\u3068,\r\n\\[\r\nf(x) =x(x-1)(x-2)(x-3)(x-4) +lx +m\n\\]\r\n\uff08 \\(l, m\\) \u306f\u5b9f\u6570\uff09\u3068\u66f8\u3051\u308b\u3053\u3068\u306f\u4e92\u3044\u306b\u540c\u5024\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(f(x)\\) \u306f (1)<\/strong> \u306e\u6761\u4ef6\u3092\u307f\u305f\u3059\u3082\u306e\u3068\u3059\u308b. \\(\\alpha\\) \u3092\u5b9f\u6570, \\(k\\) \u3092 \\(3\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b. \\(k\\) \u9805\u304b\u3089\u306a\u308b\u6570\u5217\r\n\\[\r\nf( \\alpha ) , f( \\alpha +1 ) , f( \\alpha +2 ) , \\cdots , f( \\alpha +k-1 )\n\\]\r\n\u304c\u7b49\u5dee\u6570\u5217\u3068\u306a\u308b\u3088\u3046\u306a \\(\\alpha , k\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u304a\u304f.<\/p>\r\n 1*<\/strong>\u3000\\(\\text{[Q]} \\Rightarrow \\text{[P]}\\) \u306e\u8a3c\u660e\r\n\\[\\begin{align}\r\nf(0) & = m , \\ f(1) = l+m , \\ f(2) = 2l+m , \\\\\r\nf(3) & = 3l+m , \\ f(4) = 4l+m\n\\end{align}\\]\r\n\u3088\u3063\u3066, [P]\u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n 2*<\/strong>\u3000\\(\\text{[P]} \\Rightarrow \\text{[Q]}\\) \u306e\u8a3c\u660e\r\n\u6761\u4ef6\u3088\u308a, \\(f(i) =Li +M \\ ( i =0, 1, 2, 3, 4 )\\) \u3068\u304a\u3051\u308b.\r\n\\(g(x) = f(x) -(Lx+M)\\) \u3068\u304a\u304f\u3068, \\(i =0, 1, 2, 3, 4\\) \u306b\u3064\u3044\u3066\r\n\\[\r\ng(i) = f(i) -(Li+M) =0\n\\]\r\n\u306a\u306e\u3067, \\(g(x)=0\\) \u306f \\(x =0 , 1 , 2 , 3 , 4\\) \u3092\u89e3\u306b\u3082\u3064.\r\n\\(g(x)\\) \u306f, \u6700\u9ad8\u6b21\u306e\u4fc2\u6570\u304c \\(1\\) \u306e \\(5\\) \u6b21\u5f0f\u306a\u306e\u3067\r\n\\[\\begin{align}\r\ng(x) & = x(x-1)(x-2)(x-3)(x-4) \\\\\r\n\\text{\u2234} \\quad f(x) & = x(x-1)(x-2)(x-3)(x-4) +Lx+M\n\\end{align}\\]\r\n\u3088\u3063\u3066, [Q]\u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n 1*<\/strong> 2*<\/strong>\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n (2)<\/strong><\/p>\r\n \\(( \\alpha , k ) \\ (k \\geqq 3)\\) \u306e\u7d44\u304c\u6761\u4ef6\u3092\u307f\u305f\u3059\u305f\u3081\u306b\u306f, \\(( \\alpha , 3 )\\) \u306e\u7d44\u304c\u6761\u4ef6\u3092\u307f\u305f\u3059\u5fc5\u8981\u304c\u3042\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n
\r\n\u307e\u305a, \\(k=3\\) \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u8003\u3048\u308b.
\r\n\u3053\u306e\u3068\u304d\r\n\\[\r\nf( \\alpha ) +f( \\alpha +2 ) = 2 f( \\alpha +1 )\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b.
\r\n(1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\alpha ( \\alpha -1 )( \\alpha -2 )( \\alpha -3 )( \\alpha -4 ) & +l \\alpha +m \\\\\r\n+ ( \\alpha +2 )( \\alpha +1 ) \\alpha & ( \\alpha -1 )( \\alpha -2 ) +l ( \\alpha +2 ) +m \\\\\r\n= 2 ( \\alpha +1 ) & \\alpha ( \\alpha -1 )( \\alpha -2 )( \\alpha -3 ) +2l ( \\alpha +1 ) +2m \\\\\r\n\\alpha ( \\alpha -1 )( \\alpha -2 )( -3 \\alpha +14 ) & = 0 \\\\\r\n20 \\alpha ( \\alpha -1)( \\alpha -2) & =0 \\\\\r\n\\text{\u2234} \\quad \\alpha & = 0, 1, 2\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nf(5) -f(4) & = (5!+5l+m) -(4l+m) \\\\\r\n& =120+l \\neq l\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(f(5)\\) \u304c \\(f(3) , f(4)\\) \u3068\u7b49\u5dee\u6570\u5217\u3092\u306a\u3059\u3053\u3068\u306f\u306a\u3044.
\r\n\u3088\u3063\u3066, \u7b49\u5dee\u6570\u5217\u3092\u306a\u3059\u306e\u306f \\(f(0)\\) \uff5e \\(f(4)\\) \u3067\u3042\u308a, \u6c42\u3081\u308b\u7d44\u306f\r\n\\[\r\n( \\alpha , k ) =\\underline{(0,3) , (0,4) , (0,5) , (1,3) , (1,4) , (2,3)}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(5\\) \u6b21\u5f0f \\(f(x) =x^5 +px^4 +qx^3 +rx^2 +sx +t \\) \uff08 \\(p, q, r, s, t\\) \u306f\u5b9f\u6570\uff09\u306b\u3064\u3044\u3066\u8003\u3048\u308b. \u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u6570\u5217 \\(f( … \u7d9a\u304d\u3092\u8aad\u3080