トップ› 名古屋大学› 1963年› 理系第4問

名古屋大学 1963年理系第4問解説

譁ｹ驥昴・蛻晄焔

蜑ｲ繧句ｼ上′ $3$ 谺｡蠑・$(x-1)^3$ 縺ｧ縺ゅｋ縺溘ａ縲∵ｱゅａ繧倶ｽ吶ｊ縺ｯ $2$ 谺｡莉･荳九・謨ｴ蠑上〒縺ゅｋ縲・ $(x-1)^3$ 縺ｧ蜑ｲ繧九％縺ｨ繧定・∴繧九◆繧√・x-1$ 縺ｮ蝗ｺ縺ｾ繧翫ｒ菴懊ｊ蜃ｺ縺吶％縺ｨ縺梧怏蜉ｹ縺ｧ縺ゅｋ縲・x-1 = t$ 縺ｨ縺翫＞縺ｦ莠碁・ｮ夂炊繧貞茜逕ｨ縺吶ｋ譁ｹ豕輔ｄ縲∝臆菴吶・螳夂炊縺ｨ蠕ｮ蛻・ｒ邨・∩蜷医ｏ縺帙ｋ譁ｹ豕輔↑縺ｩ縺瑚・∴繧峨ｌ繧九よ悽蝠上〒縺ｯ $n$ 縺ｮ謖・ｮ壹・縺ｪ縺・′縲・x$ 縺ｮ謨ｴ蠑上→縺・≧譚｡莉ｶ縺九ｉ $n$ 縺ｯ閾ｪ辟ｶ謨ｰ縺ｨ縺励※謇ｱ縺・・

隗｣豕・

$n$ 繧定・辟ｶ謨ｰ縺ｨ縺吶ｋ縲・x-1=t$ 縺ｨ縺翫￥縺ｨ $x=t+1$ 縺ｧ縺ゅｋ縲ゅ％繧後ｒ荳主ｼ上↓莉｣蜈･縺励※莠碁・ｮ夂炊繧堤畑縺・※螻暮幕縺吶ｋ縲・

$$ x^n - 1 = (t+1)^n - 1 $$

$n \geqq 3$ 縺ｮ縺ｨ縺阪∽ｺ碁・ｮ夂炊繧医ｊ

$$ \begin{aligned} (t+1)^n - 1 &= \left( 1 + {}_n\mathrm{C}_{1} t + {}_n\mathrm{C}_{2} t^2 + {}_n\mathrm{C}_{3} t^3 + \cdots + {}_n\mathrm{C}_{n} t^n \right) - 1 \\ &= n t + \frac{n(n-1)}{2} t^2 + t^3 \left( {}_n\mathrm{C}_{3} + \cdots + {}_n\mathrm{C}_{n} t^{n-3} \right) \end{aligned} $$

縺薙％縺ｧ縲・t = x-1$ 縺ｫ謌ｻ縺吶→

$$ x^n - 1 = n(x-1) + \frac{n(n-1)}{2} (x-1)^2 + (x-1)^3 \left\{ {}_n\mathrm{C}_{3} + \cdots + {}_n\mathrm{C}_{n} (x-1)^{n-3} \right\} $$

蜿ｳ霎ｺ縺ｮ隨ｬ $3$ 鬆・・ $(x-1)^3$ 繧貞屏謨ｰ縺ｫ繧ゅ▽縺溘ａ縲・(x-1)^3$ 縺ｧ蜑ｲ繧雁・繧後ｋ縲・縺励◆縺後▲縺ｦ縲∵ｱゅａ繧倶ｽ吶ｊ縺ｯ

$$ \begin{aligned} \frac{n(n-1)}{2} (x-1)^2 + n(x-1) &= \frac{n(n-1)}{2} (x^2 - 2x + 1) + nx - n \\ &= \frac{n(n-1)}{2} x^2 - n(n-1)x + nx + \frac{n(n-1)}{2} - n \\ &= \frac{n(n-1)}{2} x^2 - n(n-2)x + \frac{n(n-3)}{2} \end{aligned} $$

縺ｾ縺溘・n=1$ 縺ｮ縺ｨ縺阪∽ｸ雁ｼ上↓ $n=1$ 繧剃ｻ｣蜈･縺吶ｋ縺ｨ

$$ 0 \cdot x^2 - 1 \cdot (-1) x + \frac{1 \cdot (-2)}{2} = x - 1 $$

縺ｨ縺ｪ繧翫√％繧後・ $x-1$ 繧・$(x-1)^3$ 縺ｧ蜑ｲ縺｣縺滉ｽ吶ｊ $x-1$ 縺ｨ荳閾ｴ縺吶ｋ縲・

$n=2$ 縺ｮ縺ｨ縺阪∽ｸ雁ｼ上↓ $n=2$ 繧剃ｻ｣蜈･縺吶ｋ縺ｨ

$$ 1 \cdot x^2 - 2 \cdot 0 \cdot x + \frac{2 \cdot (-1)}{2} = x^2 - 1 $$

縺ｨ縺ｪ繧翫√％繧後・ $x^2-1$ 繧・$(x-1)^3$ 縺ｧ蜑ｲ縺｣縺滉ｽ吶ｊ $x^2-1$ 縺ｨ荳閾ｴ縺吶ｋ縲・繧医▲縺ｦ縲√☆縺ｹ縺ｦ縺ｮ閾ｪ辟ｶ謨ｰ $n$ 縺ｫ縺､縺・※謌舌ｊ遶九▽縲・

隗｣豕・

蜑ｲ繧句ｼ・$(x-1)^3$ 縺ｯ $3$ 谺｡蠑上〒縺ゅｋ縺九ｉ縲∵ｱゅａ繧倶ｽ吶ｊ縺ｯ $2$ 谺｡莉･荳九・謨ｴ蠑上〒縺ゅｋ縲・蝠・ｒ $Q(x)$縲∽ｽ吶ｊ繧・$ax^2+bx+c$ ・・a, b, c$ 縺ｯ螳壽焚・峨→縺翫￥縺ｨ

$$ x^n - 1 = (x-1)^3 Q(x) + ax^2 + bx + c \quad \cdots \text{竭} $$

縺ｨ陦ｨ縺帙ｋ縲・竭縺ｮ荳｡霎ｺ縺ｫ $x=1$ 繧剃ｻ｣蜈･縺励※

$$ 0 = a + b + c \quad \cdots \text{竭｡} $$

竭縺ｮ荳｡霎ｺ繧・$x$ 縺ｫ縺､縺・※蠕ｮ蛻・＠縺ｦ

$$ nx^{n-1} = 3(x-1)^2 Q(x) + (x-1)^3 Q'(x) + 2ax + b \quad \cdots \text{竭｢} $$

竭｢縺ｮ荳｡霎ｺ縺ｫ $x=1$ 繧剃ｻ｣蜈･縺励※

$$ n = 2a + b \quad \cdots \text{竭｣} $$

縺輔ｉ縺ｫ竭｢縺ｮ荳｡霎ｺ繧・$x$ 縺ｫ縺､縺・※蠕ｮ蛻・＠縺ｦ

$$ n(n-1)x^{n-2} = 6(x-1)Q(x) + 6(x-1)^2 Q'(x) + (x-1)^3 Q''(x) + 2a \quad \cdots \text{竭､} $$

竭､縺ｮ荳｡霎ｺ縺ｫ $x=1$ 繧剃ｻ｣蜈･縺励※

$$ n(n-1) = 2a $$

縺薙ｌ繧医ｊ

$$ a = \frac{n(n-1)}{2} $$

縺薙ｌ繧停促縺ｫ莉｣蜈･縺励※

$$ b = n - 2a = n - n(n-1) = n(1 - n + 1) = -n(n-2) $$

$a, b$ 繧停贈縺ｫ莉｣蜈･縺励※

$$ c = -a - b = -\frac{n(n-1)}{2} + n(n-2) = \frac{-n^2+n+2n^2-4n}{2} = \frac{n(n-3)}{2} $$

縺励◆縺後▲縺ｦ縲∵ｱゅａ繧倶ｽ吶ｊ縺ｯ

$$ \frac{n(n-1)}{2} x^2 - n(n-2)x + \frac{n(n-3)}{2} $$

隗｣豕・

蝗謨ｰ蛻・ｧ｣繧堤畑縺・※蠑上ｒ螟牙ｽ｢縺励※縺・￥縲・

$$ x^n - 1 = (x-1)(x^{n-1} + x^{n-2} + \cdots + x + 1) $$

縺薙％縺ｧ縲・P(x) = x^{n-1} + x^{n-2} + \cdots + x + 1$ 縺ｨ縺翫￥縲・P(x)$ 繧・$x-1$ 縺ｧ蜑ｲ縺｣縺溘→縺阪・蝠・ｒ $Q(x)$ 縺ｨ縺吶ｋ縺ｨ縲∝臆菴吶・螳夂炊繧医ｊ菴吶ｊ縺ｯ $P(1) = n$ 縺ｧ縺ゅｋ縲ゅｈ縺｣縺ｦ

$$ P(x) = (x-1)Q(x) + n $$

縺薙ｌ繧貞・縺ｮ蠑上↓莉｣蜈･縺吶ｋ縺ｨ

$$ x^n - 1 = (x-1) \{ (x-1)Q(x) + n \} = (x-1)^2 Q(x) + n(x-1) $$

谺｡縺ｫ縲・Q(x)$ 繧呈ｱゅａ繧九・

$$ \begin{aligned} (x-1)Q(x) &= P(x) - n \\ &= (x^{n-1} - 1) + (x^{n-2} - 1) + \cdots + (x - 1) + (1 - 1) \end{aligned} $$

蜷・・ｒ $x-1$ 縺ｧ蜑ｲ繧九→

$$ Q(x) = (x^{n-2} + \cdots + 1) + (x^{n-3} + \cdots + 1) + \cdots + 1 $$

縺輔ｉ縺ｫ $Q(x)$ 繧・$x-1$ 縺ｧ蜑ｲ縺｣縺溘→縺阪・蝠・ｒ $R(x)$ 縺ｨ縺吶ｋ縺ｨ縲∽ｽ吶ｊ縺ｯ $Q(1)$ 縺ｧ縺ゅｋ縲・

$$ Q(1) = (n-1) + (n-2) + \cdots + 1 = \frac{n(n-1)}{2} $$

繧医▲縺ｦ

$$ Q(x) = (x-1)R(x) + \frac{n(n-1)}{2} $$

縺薙ｌ繧貞・縺ｻ縺ｩ縺ｮ蠑上↓莉｣蜈･縺吶ｋ縺ｨ

$$ \begin{aligned} x^n - 1 &= (x-1)^2 \left\{ (x-1)R(x) + \frac{n(n-1)}{2} \right\} + n(x-1) \\ &= (x-1)^3 R(x) + \frac{n(n-1)}{2} (x-1)^2 + n(x-1) \end{aligned} $$

$(x-1)^3 R(x)$ 縺ｯ $(x-1)^3$ 縺ｮ蛟肴焚縺ｧ縺ゅｋ縺九ｉ縲∵ｱゅａ繧倶ｽ吶ｊ縺ｯ

$$ \begin{aligned} \frac{n(n-1)}{2} (x-1)^2 + n(x-1) &= \frac{n(n-1)}{2} (x^2 - 2x + 1) + nx - n \\ &= \frac{n(n-1)}{2} x^2 - n(n-2)x + \frac{n(n-3)}{2} \end{aligned} $$

隗｣隱ｬ

$(x-\alpha)^k$ 縺ｧ蜑ｲ縺｣縺溘→縺阪・菴吶ｊ繧呈ｱゅａ繧句撫鬘後〒縺ｯ縲∵悽蝠上・繧医≧縺ｫ $(x-\alpha) = t$ 縺ｨ鄂ｮ謠帙＠縺ｦ莠碁・ｮ夂炊繧堤畑縺・ｋ謇区ｳ輔′髱槫ｸｸ縺ｫ蠑ｷ蜉帙°縺､險育ｮ励Α繧ｹ繧呈ｸ帙ｉ縺励ｄ縺吶＞縲りｧ｣豕・縺ｮ蠕ｮ蛻・・蛻ｩ逕ｨ縺ｯ縲∝､夐㍾譬ｹ繧偵ｂ縺､蠑上〒蜑ｲ繧矩圀縺ｮ蜈ｸ蝙狗噪縺ｪ繧｢繝励Ο繝ｼ繝√〒縺ゅｋ縲よ焚蟄ｦ竇｡縺ｮ遽・峇縺ｧ繧ょ､夐・ｼ上・蠕ｮ蛻・→縺励※謇ｱ縺・％縺ｨ縺後〒縺阪ｋ縺後∬ｨ倩ｿｰ蠑上〒縺ｯ隗｣豕・繧・ｧ｣豕・縺ｮ繧医≧縺ｪ諱堤ｭ牙ｼ上・螟牙ｽ｢繧偵・繝ｼ繧ｹ縺ｫ縺吶ｋ縺ｻ縺・′辟｡髮｣縺ｧ縺ゅｋ縲・