{"id":2556,"date":"2025-08-08T14:00:00","date_gmt":"2025-08-08T05:00:00","guid":{"rendered":"https:\/\/math-friend.com\/?p=2556"},"modified":"2025-08-07T01:52:06","modified_gmt":"2025-08-06T16:52:06","slug":"%e3%80%90%e5%8c%97%e6%b5%b7%e9%81%93%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e3%81%82%e3%82%8b%e6%9d%a1%e4%bb%b6%e3%82%92%e6%ba%80%e3%81%9f%e3%81%99%e6%95%b4%e5%bc%8f%e3%81%ae%e6%80%a7%e8%b3%aa2","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=2556","title":{"rendered":"\u3010\u5317\u6d77\u9053\u5927\u5b66\u5165\u8a66\u3011\u3042\u308b\u6761\u4ef6\u3092\u6e80\u305f\u3059\u6574\u5f0f\u306e\u6027\u8cea(2023)"},"content":{"rendered":"\n

\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n

\\(P(x)\\)\u306f\\(x\\)\u306e\u6574\u5f0f\u3067\u3042\u308a, \\(x\\)\u306b\u3064\u3044\u3066\u306e\u6052\u7b49\u5f0f\\(\\displaystyle P(x)P(-x)=P(x^2)\\)\u304c\u6210\u308a\u7acb\u3064. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.

(1) \\(P(0)=0\\)\u307e\u305f\u306f\\(P(0)=1\\)\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.
(2) \\(P(x)\\)\u304c\\(x-1\\)\u3067\u5272\u308a\u5207\u308c\u306a\u3044\u306a\u3089\u3070, \\(P(x)-1\\)\u306f\\(x+1\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u3092\u793a\u305b.
(3) \u6b21\u6570\u304c\\(2\\)\u3067\u3042\u308b\\(P(x)\\)\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.
(2023 \u5317\u6d77\u9053\u5927\u5b66 \u6587\u7cfb [1])<\/span><\/p>\n\n\n\n

\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n

\n

(1) \u6052\u7b49\u5f0f\\(\\displaystyle P(x)P(-x)=P(x^2)\\)\u306b\u3066\\(x=0\\)\u3068\u3059\u308b\u3068,
$$
\\begin{align}
&{P(0)}^2=P(0)\\\\[1.5ex]
\\iff & P(0)\\left(P(0)-1\\right)=0\\\\[1.5ex]
\\iff & P(0)=0\\,\\,\\,\u307e\u305f\u306f\\,\\,\\,P(0)=1
\\end{align}
$$\u3068\u306a\u308a, \u793a\u3055\u308c\u305f.<\/p>\n\n\n\n

(2) \u6052\u7b49\u5f0f\\(\\displaystyle P(x)P(-x)=P(x^2)\\)\u306b\u3066\\(x=1\\)\u3068\u3059\u308b\u3068,$$
\\begin{align}
& P(1)P(-1)=P(1)\\\\[1.5ex]
\\iff & P(1)\\left\\{P(-1)-1\\right\\}=0\\\\[1.5ex]
\\iff & P(1)=0\\,\\,\\,\u307e\u305f\u306f\\,\\,\\,P(-1)=1
\\end{align}
$$\u3068\u306a\u308b\u304c, \\(P(1)=0\\)\u3068\u3059\u308b\u3068, \u5270\u4f59\u306e\u5b9a\u7406\u3088\u308a\\(P(x)\\)\u306f\\(x-1\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u306b\u306a\u308a\u524d\u63d0\u306b\u77db\u76fe\u3059\u308b\u306e\u3067, \\(P(1)\\neq 0\\)\u3067\u3042\u308b. \u3088\u3063\u3066, \\(P(-1)=1\\)\u304c\u308f\u304b\u308b.

\\(P(-1)=1\\)\u304b\u3089, \u5270\u4f59\u306e\u5b9a\u7406\u3088\u308a, \\(P(x)\\)\u3092\\(x+1\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f\\(1\\)\u306b\u306a\u308b. \u3088\u3063\u3066, \\(P(x)\\)\u3092\\(x+1\\)\u3067\u5272\u3063\u305f\u3068\u304d\u306e\u5546\u3092\\(Q(x)\\)\u3068\u7f6e\u304f\u3068, $$
P(x)=(x+1)Q(x)+1
$$\u3068\u306a\u308a, \u3053\u308c\u304b\u3089, $$
P(x)-1=(x+1)Q(x)
$$\u3068\u306a\u308b. \u3088\u3063\u3066, \\(P(x)-1\\)\u306f\\(x+1\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n

(3) \\(P(x)\\)\u306f\\(2\\)\u6b21\u306e\u6574\u5f0f\u306a\u306e\u3067, \u5b9f\u6570\\(a\\), \\(b\\), \\(c\\)\u3092\u7528\u3044\u3066,$$
P(x)=ax^2+bx+c,\\,\\,\\,(a\\neq 0)
$$\u3068\u304a\u3051\u308b. \u3053\u308c\u304b\u3089, $$
\\begin{align}
P(x)P(-x)&=(ax^2+bx+c)(ax^2-bx+c)\\\\[1.5ex]
&=a^2x^4+(2ac-b^2)x^2+c^2
\\end{align}
$$\u3068\u306a\u308b. \u3053\u308c\u304c, $$
P(x^2)=ax^4+bx^2+c
$$\u306b\u7b49\u3057\u3044\u306e\u3067, \u4fc2\u6570\u3092\u6bd4\u3079\u3066\u4ee5\u4e0b\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u5f97\u308b. $$
\\left\\{
\\begin{aligned}
& a=a^2\\,\\,\\,\u30fb\u30fb\u30fb\u2460\\\\[1.5ex]
& b=2ac-b^2\\,\\,\\,\u30fb\u30fb\u30fb\u2461 \\\\[1.5ex]
& c=c^2 \\,\\,\\,\u30fb\u30fb\u30fb\u2462
\\end{aligned}
\\right.
$$

\u2460\u304b\u3089, \\(a(a-1)=0\\)\u3068\u306a\u308b\u304c, \\(a\\neq 0\\)\u3088\u308a, \\(a=1\\)\u304c\u78ba\u5b9a\u3059\u308b. \u307e\u305f\u2462\u304b\u3089\\(c(c-1)=0\\)\u3068\u306a\u308a, \\(c=0\\) \u307e\u305f\u306f, \\(c=1\\) \u3067\u3042\u308b.

\u30a2) \\(c=0\\)\u306e\u3068\u304d
\u2461\u3088\u308a, \\(b=-b^2\\) \u3068\u306a\u308a, \\(b(b+1)=0\\)\u304b\u3089, \\(b=0\\) \u307e\u305f\u306f, \\(b=-1\\)\u3067\u3042\u308b.

\u30a4) \\(c=1\\)\u306e\u3068\u304d
\u2461\u3088\u308a, \\(b=2-b^2\\) \u3068\u306a\u308a, \\((b+2)(b-1)=0\\)\u304b\u3089, \\(b=-2\\) \u307e\u305f\u306f, \\(b=1\\)\u3067\u3042\u308b.

\u4ee5\u4e0a\u304b\u3089, \u9023\u7acb\u65b9\u7a0b\u5f0f\u306e\u89e3\\((a, b, c)\\)\u306f,$$
(a, b, c)=(1, 0, 0), (1, -1, 0), (1, -2, 1), (1, 1, 1)
$$\u3068\u306a\u308a, \u5bfe\u5fdc\u3059\u308b\\(P(x)\\)\u306f, $$
P(x)=x^2, x^2-x, x^2-2x+1, x^2+x+1
$$\u306e4\u3064\u3068\u306a\u308b.<\/p>\n<\/div><\/div>\n\n\n\n

\u89e3\u7b54\u4e2d\u3067\u5270\u4f59\u306e\u5b9a\u7406<\/strong>\u3092\u7528\u3044\u307e\u3057\u305f. \u5ff5\u306e\u70ba, \u5270\u4f59\u306e\u5b9a\u7406\u3092\u63b2\u8f09\u3057\u3066\u304a\u304d\u307e\u3059.<\/p>\n\n\n\n

\u5270\u4f59\u306e\u5b9a\u7406<\/strong>
\u6574\u5f0f\\(P(x)\\)\u3092\\(x-a)\\)\u3067\u5272\u3063\u305f\u4f59\u308a\u306f\\(P(a)\\)\u3067\u3042\u308b.<\/p>\n\n\n\n

youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n