\u591a\u9805\u5f0f \\(P(x)\\) \u3092\r\n\\[\r\nP(x) = \\dfrac{(x+i)^7 -(x-i)^7}{2i}\r\n\\]\r\n\u306b\u3088\u308a\u5b9a\u3081\u308b. \u305f\u3060\u3057, \\(i\\) \u306f\u865a\u6570\u5358\u4f4d\u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\r\n\\[\\begin{align}\r\nP(x) & = a _ 0 x^7 +a _ 1 x^6 +a _ 2 x^5 +a _ 3 x^4 \\\\\r\n& \\qquad +a _ 4 x^3 +a _ 5 x^2 +a _ 6 x +a _ 7\r\n\\end{align}\\]\r\n\u3068\u3059\u308b\u3068\u304d, \u4fc2\u6570 \\(a _ 0 , \\cdots , a _ 7\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(0 \\lt \\theta \\lt \\pi\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\nP \\left( \\dfrac{\\cos \\theta}{\\sin \\theta} \\right) = \\dfrac{\\sin 7 \\theta}{\\sin^7 \\theta}\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000(1)<\/strong> \u3067\u6c42\u3081\u305f \\(a _ 1 , a _ 3 , a _ 5 , a _ 7\\) \u3092\u7528\u3044\u3066, \u591a\u9805\u5f0f \\(Q (x) = a _ 1 x^3 +a _ 3 x^2 +a _ 5 x^2 +a _ 7\\) \u3092\u8003\u3048\u308b.\r\n\\(\\theta = \\dfrac{\\pi}{7}\\) \u3068\u3057\u3066, \\(k = 1, 2, 3\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nx _ k = \\dfrac{\\cos^2 k \\theta}{\\sin^2 k \\theta}\r\n\\]\r\n\u3068\u304a\u304f. \u3053\u306e\u3068\u304d, \\(Q ( x _ k ) = 0\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3057, \\(x _ 1 + x _ 2 + x _ 3\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\n( x+i )^7 & = x^7 +{} _ {7}\\text{C} {} _ {1} i x^6 +{} _ {7}\\text{C} {} _ {2} i^2 x^5 +{} _ {7}\\text{C} {} _ {3} i^3 x^4 \\\\\r\n& \\qquad +{} _ {7}\\text{C} {} _ {4} i^4 x^3 +{} _ {7}\\text{C} {} _ {5} i^5 x^2 +{} _ {7}\\text{C} {} _ {6} i^6 x +i^7 \\\\\r\n& = x^7 +7i x^6 -21 x^5 -35i x^4 \\\\\r\n& \\qquad +35x^3 +21i x^2 -7x -i \\quad ... [1] \\ .\r\n\\end{align}\\]\r\n[1] \u306e\u5404\u9805\u306e\u4fc2\u6570\u3092\u7528\u3044\u3066, \u7b26\u53f7\u306b\u6ce8\u610f\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n( x-i )^7 & = x^7 -7i x^6 -21 x^5 +35i x^4 \\\\\r\n& \\qquad +35x^3 -21i x^2 -7x +i \\quad ... [2] \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nP(x) & = \\dfrac{[1] -[2]}{2i} \\\\\r\n& = 7 x^6 -35 x^4 +21 x^2 -1 \\quad ... [3] \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n( a_0 , a_1 , a_2 , a_3 , a_4 , a_5 , a_6 . a_7 ) = \\underline{( 0 , 7 , 0 , -35 , 0 , 21 , 0 , -1 )} \\ .\r\n\\]\r\n (2)<\/strong><\/p>\r\n \u30c9\u30fb\u30e2\u30a2\u30d6\u30eb\u306e\u5b9a\u7406\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nP \\left( \\dfrac{\\cos \\theta}{\\sin \\theta} \\right) & = \\dfrac{( \\cos \\theta +i \\sin \\theta )^7 -( \\cos \\theta -i \\sin \\theta )^7}{2i \\sin^7 \\theta} \\\\\r\n& = \\dfrac{( \\cos 7 \\theta +i \\sin 7 \\theta ) -( \\cos 7 \\theta -i \\sin 7 \\theta )^7}{2i \\sin^7 \\theta} \\\\\r\n& = \\dfrac{\\sin 7 \\theta}{\\sin^7 \\theta}\r\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n \\[\r\nQ(x) = 7 x^3 -35 x^2 +21 x -1 \\ .\r\n\\]\r\n[3] \u3068\u6bd4\u8f03\u3059\u308c\u3070\r\n\\[\r\nQ( x^2 ) = P(x) \\ .\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\nQ( x_k ) & = Q \\left( \\dfrac{\\cos^2 k \\theta}{\\sin^2 k \\theta} \\right) = P \\left( \\dfrac{\\cos k \\theta}{\\sin k \\theta} \\right) \\\\\r\n& = \\dfrac{\\sin 7 k \\theta}{\\sin^7 k \\theta} \\qquad( \\ \\text{\u2235} \\ \\text{(2)\u306e\u7d50\u679c} \\ ) \\\\\r\n& = \\dfrac{\\sin k \\pi}{\\sin^7 \\frac{k \\pi}{7}} \\qquad ( \\text{\u2235} \\ \\theta = \\dfrac{\\pi}{7} ) \\\\\r\n& = 0 \\ .\r\n\\end{align}\\]\r\n\u95a2\u6570 \\(\\dfrac{1}{\\tan^2 u}\\) \u304c \\(0 \\lt u \\lt \\dfrac{\\pi}{2}\\) \u306b\u304a\u3044\u3066\u5358\u8abf\u6e1b\u5c11\u3059\u308b\u306e\u3067, \\(x_k = \\dfrac{\\cos^2 k \\theta}{\\sin^2 k \\theta} = \\dfrac{1}{\\tan^2 \\frac{k \\pi}{7}} \\ ( k = 1, 2, 3 )\\) \u306f, \u305d\u308c\u305e\u308c\u7570\u306a\u308b\u5b9f\u6570\u3067\u3042\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(Q(x) = 0\\) \u306f \\(x\\) \u306e \\(3\\) \u6b21\u65b9\u7a0b\u5f0f\u3067, \u9ad8\u3005 \\(3\\) \u3064\u306e\u5b9f\u6570\u89e3\u3057\u304b\u6301\u305f\u306a\u3044\u306e\u3067, \\(x_1 , x_2 , x_3\\) \u304c \\(Q(x) = 0\\) \u306e \\(3\\) \u3064\u306e\u89e3\u3067\u3042\u308b.
\r\n\u3088\u3063\u3066, \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u3088\u308a\r\n\\[\r\nx_1 +x_2 +x_3 = -\\dfrac{(-35)}{7} = \\underline{5} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u591a\u9805\u5f0f \\(P(x)\\) \u3092 \\[ P(x) = \\dfrac{(x+i)^7 -(x-i)^7}{2i} \\] \u306b\u3088\u308a\u5b9a\u3081\u308b. \u305f\u3060\u3057, \\(i\\) \u306f\u865a\u6570\u5358\u4f4d\u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000 \\[\\begin … \u7d9a\u304d\u3092\u8aad\u3080