{"id":789,"date":"2023-05-07T21:19:58","date_gmt":"2023-05-07T13:19:58","guid":{"rendered":"https:\/\/www.muidar.com\/?p=789"},"modified":"2026-08-05T23:16:16","modified_gmt":"2026-08-05T15:16:16","slug":"notes-of-polynomial-factorization","status":"publish","type":"post","link":"https:\/\/muidar.com\/blog\/posts\/cmuidar\/notes-of-polynomial-factorization\/","title":{"rendered":"\u56e0\u5f0f\u5206\u89e3\u7b14\u8bb0"},"content":{"rendered":"\n

\u5b9a\u4e49<\/h2>\n\n\n\n

\u5c06\u4e00\u4e2a\u591a\u9879\u5f0f\u5728\u4e00\u5b9a\u8303\u56f4\u5316\u4e3a\u51e0\u4e2a\u6574\u5f0f\u7684\u79ef\u7684\u5f62\u5f0f,\u8fd9\u79cd\u5f0f\u5b50\u7684\u53d8\u5f62\u53eb\u505a\u56e0\u5f0f\u5206\u89e3\u3002<\/p>\n\n\n\n

\u5206\u89e3\u65b9\u6cd5<\/h2>\n\n\n\n

\u63d0\u516c\u56e0\u5f0f\u6cd5<\/h3>\n\n\n\n

\u63d0\u516c\u56e0\u5f0f\u6cd5\u65f6\u56e0\u5f0f\u5206\u89e3\u7684\u6700\u57fa\u672c\u65b9\u6cd5\uff0c\u4e5f\u5c31\u662f\u5c06\u591a\u9879\u5f0f\u4e2d\u6bcf\u4e2a\u9879\u90fd\u542b\u6709\u7684\u5b57\u6bcd\u3001\u5e38\u6570\u548c\u5f0f\u5b50\u63d0\u51fa\u6765\u3002\u5728\u4f7f\u7528\u4efb\u4f55\u5176\u4ed6\u65b9\u6cd5\u4e4b\u524d\uff0c\u5e94\u8be5\u5148\u7528\u63d0\u516c\u56e0\u5f0f\u6cd5\u628a\u5197\u4f59\u90e8\u5206\u63d0\u51fa\uff0c\u4ee5\u514d\u5f71\u54cd\u4e4b\u540e\u89e3\u9898\u3002<\/p>\n\n\n\n

\u516c\u5f0f\u6cd5<\/h3>\n\n\n\n

\u8fd0\u7528\u5176\u4ed6\u65b9\u6cd5\u65f6\uff0c\u5206\u89e3\u51fa\u9ad8\u6b21\u6570\u7684\u56e0\u5f0f\uff0c\u4e00\u5b9a\u8981\u6ce8\u610f\u662f\u5426\u5206\u89e3\u5b8c\u5168\uff0c\u770b\u770b\u53ef\u4e0d\u53ef\u4ee5\u7528\u516c\u5f0f\u6cd5\u6216\u5176\u4ed6\u65b9\u6cd5\u7ee7\u7eed\u5206\u89e3\u3002<\/mark><\/strong><\/p>\n\n\n\n

\u901a\u8fc7\u9006\u7528\u4e58\u6cd5\u516c\u5f0f\u5c06\u4e00\u4e2a\u591a\u9879\u5f0f\u5206\u89e3\u56e0\u5f0f\u7684\u65b9\u6cd5\u53eb\u505a\u516c\u5f0f\u6cd5
\u5e38\u7528\u7684\u516c\u5f0f\u6709\uff1a
1.\u5e73\u65b9\u5dee\u516c\u5f0f<\/p>\n\n\n\n

a^2-b^2=(a+b)(a-b)<\/pre><\/div>\n\n\n\n
2.\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f<\/p>\n\n\n\n
a^2\\pm 2ab+b^2=(a\\pm b)^2<\/pre><\/div>\n\n\n\n
3.\u7acb\u65b9\u548c\u4e0e\u7acb\u65b9\u5dee\u516c\u5f0f<\/p>\n\n\n\n
a^3\\pm b^3=(a\\pm b)(a^2\\mp ab+b^2)<\/pre><\/div>\n\n\n\n
4.\u4e09\u5143\u5b8c\u5168\u5e73\u65b9\u516c\u5f0f<\/p>\n\n\n\n
a^2+b^2+c^2+2ab+2bc+2ca=(a+b+c)^2<\/pre><\/div>\n\n\n\n
5.\u5b8c\u5168\u7acb\u65b9\u516c\u5f0f<\/p>\n\n\n\n
a^3+3a^2b+3ab^2+b^3=(a+b)^3<\/pre><\/div>\n\n\n\n
a^3-3a^2b+3ab^2-b^3=(a-b)^3<\/pre><\/div>\n\n\n\n
6.\u6b27\u62c9\u516c\u5f0f\u53ca\u5176\u63a8\u8bba<\/p>\n\n\n\n
a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)<\/pre><\/div>\n\n\n\n
\u63a8\u8bba\uff1a1.\u82e5a+b+c=0<\/span>\uff0c\u5219a^3+b^3+c^3=3abc<\/span>
2.\u82e5a^3+b^3+c^3=3abc<\/span>\uff0c\u5219a+b+c=0<\/span>\u6216a=b=c<\/span><\/p>\n\n\n\n
\u8fd9\u4e2a\u65b9\u6cd5\u6ca1\u6709\u4ec0\u4e48\u4f8b\u5b50\u597d\u4e3e\uff0c\u5957\u516c\u5f0f\u5c31\u884c\u3002<\/p>\n\n\n\n
\u5341\u5b57\u76f8\u4e58\u6cd5<\/h3>\n\n\n\n
\u5b9a\u4e49\uff1a\u7528\u5341\u5b57\u4ea4\u53c9\u5148\u6765\u5206\u89e3\u7cfb\u6570\uff0c\u628a\u4e8c\u6b21\u4e09\u9879\u5f0f\u5206\u89e3\u56e0\u5f0f\u7684\u65b9\u6cd5\u53eb\u505a\u5341\u5b57\u76f8\u4e58\u6cd5\u3002
\u53e3\u8bc0\uff1a\u9996\u5c3e\u5206\u89e3\uff0c\u4ea4\u53c9\u76f8\u4e58\uff0c\u6c42\u548c\u51d1\u4e2d\u3002<\/p>\n\n\n\n
\u5341\u5b57\u76f8\u4e58\u6cd5\u5c31\u662f\u901a\u8fc7\u753b\u5341\u5b57\u7ebf\u7684\u65b9\u6cd5\u628a\u5f62\u5982abx\u00b2+(an+bm)x+mn\u548cabx\u00b2+(an+bm)+mny\u00b2\u5206\u522b\u5206\u89e3\u4e3a(ax+m)(bx+n)\u548c(ax+my)(bx+ny)
<\/p>\n\n\n\n
\"\"<\/div><\/figure>\n\n\n\n
\u770b\u5b8c\u8fd9\u4e2a\u4f8b\u5b50\uff0c\u4f60\u5e94\u8be5\u77e5\u9053\uff0c\u5341\u5b57\u76f8\u4e58\u5c31\u662f\u5c06\u672b\u5c3e\u7684\u5e38\u6570\u9879\u6216\u4e8c\u6b21\u9879\u8fdb\u884c\u62c6\u5206(\u5148\u8003\u8651\u5dee\u503c\u8f83\u5c0f\u7684)\uff0c\u8ba9\u4ed6\u4eec\u53ef\u4ee5\u51d1\u51fa\u4e2d\u95f4\u9879\u7684\u7cfb\u6570\u3002<\/p>\n\n\n\n
\u5982\u679cx\u00b2\u9879\u7684\u7cfb\u6570\u4e0d\u4e3a\u4e00\uff0c\u90a3\u6211\u4eec\u53ef\u4ee5\u5148\u628ax\u00b2\u9879\u62c6\u6210\u4e24\u4e2a\u542bx\u9879\u76f8\u4e58\u7684\u5f62\u5f0f\uff0c\u5982\uff1a4x\u00b2=2x\u00b72x=x\u00b74x\u4e4b\u7c7b\u7684\uff0c\u8981\u8003\u8651\u7684\u5c31\u6bd4\u8f83\u591a\u4e86\u3002\u5982\u679c\u6709\u4e00\u79cd\u884c\u4e0d\u901a\uff0c\u4e0d\u8981\u6000\u7591\u81ea\u5df1\uff0c\u5f88\u53ef\u80fd\u5c31\u662f\u8fd9\u4e2a\u65b9\u6cd5\u505a\u4e0d\u51fa\u6765\u3002<\/p>\n\n\n\n
\"\"<\/div><\/figure>\n\n\n\n
\u5206\u7ec4\u6cd5<\/h3>\n\n\n\n
\u5b9a\u4e49\uff1a\u7528\u5206\u7ec4\u6765\u5206\u89e3\u56e0\u5f0f\u7684\u65b9\u6cd5\u53eb\u505a\u5206\u7ec4\u6cd5
\u5982\uff1aax+ay+bx+by=(ax+ay)+(bx+by)=a(x+y)+b(x+y)=(a+b)(x+y)
\u5206\u7ec4\u65b9\u6cd5\u5e76\u4e0d\u552f\u4e00\uff1aax+ay+bx+by=(ax+bx)+(ay+by)=x(a+b)+y(a+b)=(x+y)(a+b)<\/p>\n\n\n\n
\u5206\u89e3\u6b65\u9aa4\uff1a
1) \u5c06\u591a\u9879\u5f0f\u6309\u7167\u5b57\u6bcd\u3001\u7cfb\u6570\u3001\u6b21\u6570\u7b49\u6765\u5206\u7ec4\uff0c\u5206\u522b\u5bf9\u6bcf\u7ec4\u56e0\u5f0f\u5206\u89e3\uff0c\u53ef\u6301\u7eed\u53d1\u5c55\u3002
2) \u5206\u7ec4\u540e\uff0c\u7528\u5176\u4ed6\u65b9\u6cd5\u8fdb\u4e00\u6b65\u56e0\u5f0f\u5206\u89e3\u3002<\/p>\n\n\n\n
\u4f8b1\uff1a\u56e0\u5f0f\u5206\u89e3\uff1a4xy+2x+2y+1
     \u539f\u5f0f=(4xy+2x)+(2y+1)
     =2x(2y+1)+(2y+1)
     =(2x+1)(2y+1)<\/p>\n\n\n\n

\u4f8b2\uff1a\u56e0\u5f0f\u5206\u89e3\uff1aac\u00b2+bd\u00b2-ad\u00b2-bc\u00b2
     \u601d\u8def:\u628a\u542b\u6709c\u00b2\u7684\u9879\u653e\u5728\u4e00\u8d77\uff0c\u628a\u542b\u6709d\u00b2\u7684\u9879\u653e\u5728\u4e00\u8d77\uff0c\u518d\u5206\u522b\u63d0\u516c\u56e0\u5f0f\u3002
     \u539f\u5f0f=(ac\u00b2-bc\u00b2)+(bd\u00b2-ad\u00b2)
     =c\u00b2(a-b)-d\u00b2(a-b)
     =(a-b)(c\u00b2-d\u00b2) \/\/\u6ce8\u610f\u6709\u6ca1\u6709\u5206\u89e3\u5b8c\u5168
     =(a-b)(c+d)(c-d)
<\/p>\n\n\n\n
\u4f8b3\uff1a\u56e0\u5f0f\u5206\u89e3\uff1ax\u2075+x\u2074+x\u00b3+x\u00b2+x+1
     \u601d\u8def\uff1a\u6309\u7167\u6b21\u6570\u5206\u7ec4\uff0c\u4e09\u4e2a\u4e00\u7ec4\uff0c\u518d\u5206\u522b\u63d0\u516c\u56e0\u5f0f\u3002
     \u539f\u5f0f=(x\u2075+x\u2074+x\u00b3)+(x\u00b2+x+1)
     =x\u00b3(x\u00b2+x+1)+(x\u00b2+x+1)
     =(x+1)(x\u00b2-x+1)(x\u00b2+x+1)<\/p>\n\n\n\n
\u4e3b\u5143\u6cd5<\/h3>\n\n\n\n
\u5b9a\u4e49: \u4e3b\u5143\u6cd5\u5c31\u662f\u518d\u5206\u89e3\u542b\u6709\u591a\u4e2a\u5b57\u6bcd\u7684\u4ee3\u6570\u5f0f\u65f6\uff0c\u9009\u53d6\u5176\u4e2d\u4e00\u4e2a\u5b57\u6bcd\u4e3a\u4e3b\u5143(\u5143\u5c31\u662f\u672a\u77e5\u6570)\uff0c\u5c06\u5176\u770b\u4e3a\u5e38\u719f\uff0c\u628a\u4ee3\u6570\u5f0f\u6574\u7406\u4e3a\u5173\u4e8e\u4e3b\u5143\u7684\u964d\u5e42\u6392\u5217\u6216\u5347\u5e42\u6392\u5217\u7684\u591a\u9879\u5f0f\uff0c\u518d\u5c1d\u8bd5\u7528\u5176\u4ed6\u65b9\u6cd5\u7ecf\u884c\u5206\u89e3\u3002<\/p>\n\n\n\n
\u5206\u89e3\u6b65\u9aa4: 
1)\u9009\u62e9\u4e00\u4e2a\u4e3b\u5143\uff0c\u8fd9\u91cc\u7528x\uff0c\u6574\u7406\u4e3a\u5f62\u5982ax\u00b2+bx+c\u7684\u5f62\u5f0f\u3002
2)\u5bf9\u6574\u7406\u597d\u7684\u5f0f\u5b50\u7528\u5176\u4ed6\u65b9\u6cd5\uff0c\u5982\u5341\u5b57\u76f8\u4e58\u6cd5\uff0c\u7ee7\u7eed\u5206\u89e3\u3002<\/p>\n\n\n\n
\u4f8b1\uff1a\u56e0\u5f0f\u5206\u89e3: x\u00b2-mx\u00b2+mx-3x+2
     \u601d\u8def\uff1a\u8fd9\u91cc\u4e3b\u5143m\u7684\u8bdd\u5c31\u6ca1\u6709\u4e8c\u6b21\u9879\u4e86\uff0c\u6bd4\u4e3b\u5143x\u8981\u7b80\u5355\u4e00\u70b9\u3002
     \u539f\u5f0f=(-x\u00b2+x)m+x\u00b2-3x+2
     =-x(x-1)m+(x-1)(x-2)
     =(x-1)(x-mx-2)<\/p>\n\n\n\n
\u4f8b2\uff1a\u56e0\u5f0f\u5206\u89e3\uff1ax\u00b2-a(3x-2a+b)-b\u00b2
     \u4e3b\u5143a
     \u539f\u5f0f=x\u00b2-3xa+2a\u00b2-ba-b\u00b2\/\/\u5efa\u8bae\u628a\u4e3b\u5143\u653e\u5728\u6bcf\u4e2a\u9879\u7684\u672b\u5c3e\uff0c\u8fd9\u6837\u6bd4\u8f83\u6e05\u6670
     =2a\u00b2-(3x+b)a+x\u00b2-b\u00b2
     =2a\u00b2-(3x+b)a+(x+b)(x-b)<\/p>\n\n\n\n
\"\"<\/div><\/figure>\n\n\n\n
\u2234\u539f\u5f0f=(a-x-b)(2a-x+b)<\/p>\n\n\n\n
\u62c6\u586b\u9879<\/h3>\n\n\n\n
\u6982\u5ff5<\/h4>\n\n\n\n
\u62c6\u9879\uff1a\u6307\u7684\u662f\u628a\u4ee3\u6570\u5f0f\u62c6\u6210\u82e5\u5e72\u9879\u4ee3\u6570\u7684\u548c\uff0c\u5982a^2=2a^2-a^2<\/span>
\u586b\u9879\uff1a\u5728\u4ee3\u6570\u5f0f\u4e2d\u586b\u4e0a\u4e24\u4e2a\u76f8\u53cd\u7684\u9879\uff0c\u53eb\u505a\u586b\u9879\uff0c\u5982a^2+1=a^2+2a+1-2a<\/span><\/p>\n\n\n\n
\u5728\u5bf9\u6240\u7ed9\u591a\u9879\u5f0f\u56e0\u5f0f\u5206\u89e3\u65f6\uff0c\u82e5\u96be\u4ee5\u76f4\u63a5\u5206\u7ec4\u5206\u89e3\u65f6\uff0c\u5e38\u53ef\u4ee5\u7528\u62c6\u586b\u9879\u7684\u53d8\u5f62\u521b\u9020\u51fa\u53ef\u4ee5\u63d0\u53d6\u516c\u56e0\u5f0f\u6216\u8fd0\u7528\u516c\u5f0f\u6cd5\u7ecf\u884c\u5206\u89e3\u7684\u6761\u4ef6\uff0c\u4f7f\u5f97\u539f\u5f0f\u67d0\u4e9b\u9879\u53ef\u4ee5\u5efa\u7acb\u7ec3\u4e60\uff0c\u4fbf\u4e8e\u91c7\u7528\u5206\u7ec4\u6cd5\u7ecf\u884c\u56e0\u5f0f\u5206\u89e3\u3002<\/p>\n\n\n\n
\u4f8b\u9898<\/h4>\n\n\n\n
\u4f8b1\uff1a\u56e0\u5f0f\u5206\u89e3\uff1aa^3-4a+3<\/span>
\u89e3\uff1a
\u6cd5\u4e00 (\u586b\u9879)\uff1a
\u539f\u5f0f=a^3-a^2+a^2-4a+3<\/span>
=a^2(a-1)+(a-1)(a-3)<\/span>
=(a-1)(a^2+a-3)<\/span>
\u6cd5\u4e8c (\u62c6\u9879)\uff1a
\u539f\u5f0f=a^3-a-3a+3<\/span>
=a(a+1)(a-1)-3(a-1)<\/span>
=(a-1)(a^2+a-3)<\/span><\/p>\n\n\n\n
\u4f8b2\uff1a\u56e0\u5f0f\u5206\u89e3\uff1aa^4+2a^3b+3a^2b^2+2ab^3+b^4<\/span>
\u89e3\uff1a\u539f\u5f0f=(a^4+a^3b+a^2b^2)+(a^3b+a^2b^2+ab^3)+(a^2b^2+ab^3+b^4)<\/span>
=a^2(a^2+ab+b^2)+ab(a^2+ab+b^2)+b^2(a^2+ab^2)<\/span>
=(a^2+ab+b^2)^2<\/span><\/p>\n\n\n\n
\u6362\u5143\u6cd5<\/h3>\n\n\n\n
\u6362\u5143\u6307\u7684\u662f\u5728\u56e0\u5f0f\u5206\u89e3\u8fc7\u7a0b\u4e2d\uff0c\u9047\u5230\u76f8\u540c\u6216\u51e0\u4e2a\u6709\u5173\u8054\u7684\u5f0f\u5b50\u7684\u65f6\u5019\uff0c\u7528\u5176\u4ed6\u7684\u672a\u77e5\u6570(\u5143)\u6765\u8868\u793a\u3002\u8fd9\u6837\u505a\u5f80\u5f80\u4f1a\u4f7f\u539f\u5f0f\u66f4\u52a0\u6e05\u6670\uff0c\u8ba9\u591a\u9879\u5f0f\u7684\u9879\u6570\u51cf\u5c11\uff0c\u53ef\u4ee5\u4f7f\u590d\u6742\u7684\u95ee\u9898\u7b80\u5355\u5316\u3002<\/p>\n\n\n\n
\u4f8b\uff1a\u56e0\u5f0f\u5206\u89e3\uff1a(x\u00b2+x+1)(x\u00b2+x+2)-12
\u89e3\uff1a\u4ee4t=x\u00b2+x,
\u5219\u539f\u5f0f=(t+1)(t+2)-12
=t\u00b2+3t+2-12
=t\u00b2+3t-10
=(t-2)(t+5)
=(x\u00b2+x-2)(x\u00b2+x+5)
=(x-1)(x+2)(x\u00b2+x+5)<\/p>\n\n\n\n
\u8bd5\u6839\u6cd5<\/h3>\n\n\n\n
\u4ecb\u7ecd\u8bd5\u6839\u6cd5\u524d\uff0c\u9700\u8981\u5148\u77e5\u9053\u4ee5\u4e0b\u51e0\u4e2a\u6982\u5ff5\uff1a\u5173\u4e8ex\u7684\u4ee3\u6570\u5f0f\u3001\u4f59\u6570\u5b9a\u7406\u3001\u56e0\u5f0f\u5b9a\u7406\u3002<\/p>\n\n\n\n
\u5173\u4e8ex\u7684\u4ee3\u6570\u5f0f<\/h4>\n\n\n\n
\u5173\u4e8e\u4e0b\u7684\u4ee3\u6570\u5f0f\u53ef\u4ee5\u7528f(x)\u6216g(x)\u7b49\u8868\u793a\u3002
\u8fd9\u91cc\u7528f\u8868\u793a\u5bf9\u62ec\u53f7\u5185\u7684\u5b57\u6bcd\u7ecf\u884c\u4e00\u7cfb\u5217\u7684\u8fd0\u7b97\uff0cf(3)\u8868\u793a\u5f53x=3\u65f6\uff0c\u4ee3\u6570\u5f0ff(x)\u7684\u503c\u3002<\/p>\n\n\n\n
\u4f59\u6570\u5b9a\u7406\u4e0e\u56e0\u5f0f\u5b9a\u7406<\/h4>\n\n\n\n
\u591a\u9879\u5f0ff(x)\u9664\u4ee5\u591a\u9879\u5f0fg(x)\u7684\u5546\u5f0f\u4e3aq(x),\u4f59\u5f0f\u4e3ar(x)\u3002
f(x) \\div g(x)=q(x) \\cdots r(x)<\/span>,\u53ef\u4ee5\u5199\u4e3af(x)=g(x)q(x)=r(x)\u3002
\u5f53g(x)\u4e3a\u4e00\u6b21\u5f0f(ax+b)\u65f6\uff0c\u5219\u4f59\u5f0fr(x)\u53ea\u80fd\u4e3a\u5e38\u6570\uff0c\u6b64\u65f6\u4f59\u5f0f\u53ef\u4ee5\u88ab\u79f0\u4e3a\u4f59\u6570\uff0c\u8bb0\u4f5cr\uff0c\u5219\u53ef\u4ee5\u5199\u4e3af(x)=(ax+b)\u00b7q(x)+r<\/p>\n\n\n\n
\u4f59\u6570\u5b9a\u7406\uff1a\u591a\u9879\u5f0ff(x)\u9664\u4ee5(ax+b)\u7684\u4f59\u6570\u7b49\u4e8ef(-\\frac{b}{a})<\/span>
\u56e0\u5f0f\u5b9a\u7406\uff1a\u82e5\u591a\u9879\u5f0ff(x)\u6709\u4e00\u4e2a\u56e0\u5f0f(ax+b)\uff0c\u5219f(-\\frac{b}{a})=0<\/span>\uff0c\u53cd\u4e4b\u4ea6\u7136<\/p>\n\n\n\n
\u7531\u6b64\u53ef\u4ee5\u770b\u51fa\uff0c\u56e0\u5f0f\u5b9a\u7406\u65f6\u4f59\u6570\u5b9a\u7406\u5728r=0\u65f6\u7684\u4e00\u79cd\u7279\u6b8a\u60c5\u51b5\u3002<\/p>\n\n\n\n
\u5b9a\u4e49<\/h4>\n\n\n\n
\u901a\u8fc7\u56e0\u5f0f\u5b9a\u7406\u548c\u957f\u9664\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u53e6\u4e00\u4e2a\u56e0\u5f0f\u5206\u89e3\u7684\u65b9\u6cd5\u2014\u2014\u8bd5\u6839\u6cd5\u3002<\/p>\n\n\n\n
(1)\u5bf9\u4e8e\u6574\u7cfb\u6570\u591a\u9879\u5f0f f(x)=a_0x^{n}+a_1x^{n-1}+\u2026+a_{n-1}x+a_n<\/span>\uff1a
\u6027\u8d281\uff1a\u82e5\u9996\u9879\u7cfb\u6570a_0=1<\/span>\uff0c\u4e14\u5b83\u6709\u56e0\u5f0f(x-q)\uff08q\u4e3a\u6574\u6570\uff09\uff0c\u5219q\u4e00\u5b9a\u662f\u5e38\u6570\u9879a_n<\/span>\u7684\u56e0\u6570
\u6027\u8d282\uff1a\u82e5\u9996\u9879\u7cfb\u6570a_0\\ne1<\/span>\uff0c\u4e14\u5b83\u6709\u56e0\u5f0f(px-q)\uff08p\u3001q\u4e92\u7d20\uff09\uff0c\u5219p\u4e00\u5b9a\u662f\u9996\u9879\u7cfb\u6570a_0<\/span>\u7684\u56e0\u6570\uff0cq\u4e00\u5b9a\u65f6\u5e38\u6570\u9879a_n<\/span>\u7684\u56e0\u6570
(2)\u8bd5\u6839\u6cd5\uff1a\u5bf9\u6574\u7cfb\u6570\u591a\u9879\u5f0ff(x)<\/span>\u7ecf\u884c\u56e0\u5f0f\u5206\u89e3\u65f6\uff0c\u6839\u636e\u56e0\u5f0f\u5b9a\u7406\uff0c\u82e5\u80fd\u627e\u5230\u65b9\u7a0bf(x)=0<\/span>\u7684\u6839x=q<\/span>\u6216x=\\frac{q}{p}<\/span>\u3002\u5269\u4e0b\u90e8\u5206\u53ef\u7528\u957f\u9664\u6cd5\u6765\u6c42\u5f97\uff0c\u53ef\u4e5f\u8981\u6ce8\u610f\u662f\u5426\u5206\u89e3\u5b8c\u5168\u3002<\/p>\n\n\n\n
\u4f46\u8981\u6ce8\u610f\u7684\u662f\uff1a\u5373\u4f7f\u8bd5\u4e0d\u51fa\u6709\u7406\u6839\uff0c\u4e5f\u4e0d\u80fd\u8bf4\u660e\u539f\u5f0f\u4e0d\u80fd\u5728\u6709\u7406\u6570\u8303\u56f4\u5185\u56e0\u5f0f\u5206\u89e3\uff0c\u53ea\u80fd\u8bf4\u660e\u6ca1\u6709\u4e00\u6b21\u56e0\u5f0f\u3002<\/strong><\/mark><\/p>\n\n\n\n
\u4f8b\u9898<\/h4>\n\n\n\n
\u4f8b1\uff1a\u56e0\u5f0f\u5206\u89e3\uff1ax\u2075+5x\u2074-5x\u00b3-25x\u00b2+4x+20
\u89e3\uff1a\u8bd5\u6839\uff1a\u5f53x=\u00b11\u65f6\uff0c\u539f\u5f0f\u4e3a0 \u2234\u539f\u5f0f\u5fc5\u6709\u56e0\u5f0f(x+1)(x-1)
\u7531\u957f\u9664\u6cd5\uff0c
\u539f\u5f0f=(x+1)(x-1)(x\u00b3+5x\u00b2-4x-20)
=(x+1)(x-1)[x\u00b2(x+5)-4(x+5)]
=(x+1)(x-1)(x+5)(x\u00b2-4)
=(x+1)(x-1)(x+2)(x-2)(x+5)<\/p>\n\n\n\n
\u4f8b2\uff1a\u56e0\u5f0f\u5206\u89e3\uff1a6x^4+5x^3+3x^2-3x-2<\/span>
\u8bd5\u6839\uff1a\u5f53x=-\\frac{1}{2},x=\\frac{2}{3}<\/span>\u65f6\uff0c\u539f\u5f0f=0 \u2234\u539f\u5f0f\u5fc5\u6709\u56e0\u5f0f(2x+1)(3x-2)
\u7531\u957f\u9664\u6cd5\uff0c\u539f\u5f0f=(2x+1)(3x-2)(x^2+x+1)<\/span><\/p>\n\n\n\n
\u4f8b3\uff1a\u56e0\u5f0f\u5206\u89e3\uff1a3x^3-5x^2y+x^2y+x^3<\/span>
\u8bd5\u6839\uff0c\u5f53x=y\u65f6\uff0c\u539f\u5f0f=0 \u2234\u539f\u5f0f\u5fc5\u6709\u56e0\u5f0f(x-y)
\u7531\u957f\u9664\u6cd5\uff0c
\u539f\u5f0f=(x-y)(3x^2-2xy-y^2)<\/span>
=(x-y)^2(3x+y)<\/span><\/p>\n\n\n\n
\u8f6e\u6362\u5bf9\u79f0\u6cd5<\/h3>\n\n\n\n
\u6982\u5ff5<\/h4>\n\n\n\n
1.\u5bf9\u79f0\u5f0f\u6307\u7684\u662f\u6709\u82e5\u5e72\u5143\u7684\u591a\u9879\u5f0f\uff0c\u4efb\u610f\u4ea4\u6362\u4e24\u4e2a\u5143\u7684\u4f4d\u7f6e\uff0c\u539f\u591a\u9879\u5f0f\u4e0d\u53d8\u3002
\u8f6e\u6362\u5f0f\u6307\u7684\u662f\u5982\u679c\u4e00\u4e2a\u591a\u9879\u5f0f\u4e2d\u5143\u6309\u7167\u4efb\u4f55\u6b21\u5e8f\u8f6e\u6362\u540e\uff0c\u539f\u591a\u9879\u5f0f\u4e0d\u53d8\u3002 
2.\u9f50\u6b21\u5f0f\u6307\u4e00\u4e2a\u591a\u9879\u5f0f\uff0c\u5b83\u6240\u6709\u7684\u9879\u90fd\u5177\u6709\u76f8\u540c\u7684\u6b21\u6570n\uff0c\u5219\u79f0\u8fd9\u6837\u7684\u591a\u9879\u5f0f\u4e3an\u6b21\u9f50\u6b21\u5f0f\u3002<\/p>\n\n\n\n
\n
\u8f6e\u6362\u5f0f\u548c\u5bf9\u79f0\u5f0f\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a
1.\u5bf9\u79f0\u5f0f\u4e00\u5b9a\u662f\u8f6e\u6362\u5f0f\uff0c\u800c\u8f6e\u6362\u5f0f\u4e0d\u4e00\u5b9a\u662f\u5bf9\u79f0\u5f0f\u3002
2.\u5173\u4e8e\u76f8\u540c\u5b57\u6bcd\u7684\u8f6e\u6362\u5f0f\u6216\u5bf9\u79f0\u5f0f\u7684\u548c\u3001\u5dee\u3001\u79ef\u3001\u5546(\u9664\u5f0f\u4e0d\u4e3a0)\u4efb\u7136\u4e3a\u8f6e\u6362\u5f0f\u6216\u5bf9\u79f0\u5f0f\u3002
3.\u82e5\u8f6e\u6362\u5f0f\u6216\u5bf9\u79f0\u5f0f\u4e2d\u542b\u6709\u67d0\u79cd\u5f62\u5f0f\u7684\u5f0f\u5b50\uff0c\u5219\u5fc5\u6709\u76f8\u540c\u7c7b\u578b\u7684\u5f0f\u5b50\u3002
  \u5982\u4e00\u4e2a\u5173\u4e8ex,y,z\u7684\u8f6e\u6362\u5bf9\u79f0\u5f0f\uff0c\u82e5\u6709ax\u00b2\u9879\uff0c\u5219\u5fc5\u6709ay\u00b2\u548caz\u00b2\u9879\uff1b\u82e5\u542b\u6709bxy\u9879\uff0c\u5219\u5fc5\u6709byz\u548cbzx\u9879\u3002<\/p>\n<\/div><\/div>\n\n\n\n
\u89e3\u9898\u6b65\u9aa4<\/h4>\n\n\n\n
\n
1)\u8bd5\u6839<\/p>\n\n\n\n
\u8bd5\u6839\u5c31\u662f\u4f7f\u539f\u5f0f=0\uff0c\u89c1\u4e0a\u6587\u201c\u8bd5\u6839\u6cd5\u201d\u3002
\u8bd5\u6839\u662f\u6709\u8303\u56f4\u7684\uff0c\u89c1\u4e0b\u8868\uff1a<\/p>\n\n\n\n
\u8bd5\u6839<\/td>\u56e0\u5f0f<\/td><\/tr>
x=0<\/td>xyz<\/td><\/tr>
x=y<\/td>(x-y)(y-z)(z-x)<\/td><\/tr>
x=-y<\/td>(x+y)(y+z)(z+x)<\/td><\/tr>
x=y+z<\/td>(x-y-z)(y-z-x)(z-x-y)<\/td><\/tr>
x=-(y+z)<\/td>(x+y+z)<\/td><\/tr><\/tbody><\/table>
\u5f53\u539f\u5f0f\u4ee5x,y,z\u4e3a\u4e3b\u5143\u65f6<\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n
\n
2)\u6bd4\u8f83\u6b21\u6570<\/p>\n\n\n\n
\u7528\u539f\u5f0f\u7684\u6b21\u6570\u51cf\u53bb\u56e0\u5f0f\u5fc5\u6709\u6b21\u6570\uff0c\u7136\u540e\u518d\u7528\u8bd5\u6839\u51fa\u6765\u7684\u56e0\u5f0f\u4e58\u4e0a\u76f8\u5dee\u6b21\u6570\u5bf9\u5e94\u7684\u56e0\u5f0f\u3002
\u5bf9\u5e94\u6b21\u6570\u7684\u56e0\u5f0f\u89c1\u4e0b\u8868:<\/p>\n\n\n\n
\u6b21\u6570<\/td>\u56e0\u5f0f<\/td><\/tr>
\u4e00\u6b21<\/td>a(x+y+z)<\/td><\/tr>
\u4e8c\u6b21<\/td>a(x\u00b2+y\u00b2+z\u00b2)+b(xy+yz+zx)<\/td><\/tr>
\u4e09\u6b21<\/td>a(x\u00b3+y\u00b3+z\u00b3)+b\u2081(x\u00b2y+y\u00b2z+z\u00b2x)b\u2082(xy\u00b2+yz\u00b2+zx\u00b2)+cxyz<\/td><\/tr><\/tbody><\/table>
\u53ea\u5217\u51fa\u4e86\u5e38\u89c1\u7684<\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n
\n
3) \u5bf9\u6bd4\u7cfb\u6570<\/p>\n\n\n\n
\u901a\u8fc7\u8d4b\u503c\u6cd5\u548c\u6bd4\u8f83\u7cfb\u6570\u6cd5\u4e4b\u7c7b\u7684\u65b9\u6cd5\uff0c\u8ba1\u7b97\u51fa\u56e0\u5f0f\u7684\u7cfb\u6570\uff0c\u5373\u53ef\u5206\u89e3\u5b8c\u6210\u3002<\/p>\n<\/div><\/div>\n\n\n\n
\u4f8b\u9898<\/h4>\n\n\n
\u4f8b1: \u56e0\u5f0f\u5206\u89e3: (a+b+c)^4-(a+b)^4-(b+c)^4-(c+a)^4+a^4+b^4+c^4<\/span>
\n\u89e3: \u8bd5\u6839\uff1a\u5f53a=0\u65f6\uff0c\u539f\u5f0f=0
\n    \u6839\u636e\u56e0\u5f0f\u5b9a\u7406\u53ca\u8f6e\u6362\u5f0f\u6027\u8d28\uff0c\u5fc5\u6709\u56e0\u5f0fabc
\n    \u6bd4\u8f83\u6b21\u6570\uff0c\u8bbe\u539f\u5f0f=kabc(a+b+c)
\n    \u4ee4a=b=1,c=-1\u5f97 1-16-0+1+1+1=-k
\n    \u89e3\u5f97k=12
\n    \u2234\u539f\u5f0f=12abc(a+b+c)<\/p>\n\n\n
\u4f8b2\uff1a\u56e0\u5f0f\u5206\u89e3\uff1aa^2b+ab^2+b^2c+bc^2+c^2a+ca^2+2abc<\/span>
\n\u89e3\uff1a\u8bd5\u6839\uff1a\u5f53a=-b\u65f6\uff0c\u539f\u5f0f=0
\n\u7531\u56e0\u5f0f\u5b9a\u7406\u4e0e\u8f6e\u6362\u5f0f\u6027\u8d28\uff0c\u539f\u5f0f\u5fc5\u6709\u56e0\u5f0f(a+b)(b+c)(c+a)
\n\u6bd4\u8f83\u6b21\u6570\uff0c\u8bbe\u539f\u5f0f=k(a+b)(b+c)(c+a)
\n\u8bbea=0,b=1,c=2 4+2=6k
\n\u2234k=1
\n\u2234\u539f\u5f0f=(a+b)(b+c)(c+a)<\/p>\n\n\n\n
\u5f85\u5b9a\u7cfb\u6570\u6cd5<\/h3>\n\n\n\n
\u8fd9\u4e2a\u65b9\u6cd5\u4e0d\u662f\u5f88\u91cd\u8981\uff0c\u8fc7\u7a0b\u5f88\u591a\u3002\u800c\u56e0\u5f0f\u5206\u89e3\u662f\u89e3\u51b3\u5b9e\u9645\u95ee\u9898\u7684\u4e00\u4e2a\u5c0f\u6b65\u9aa4\uff0c\u5982\u679c\u7528\u5f85\u5b9a\u7cfb\u6570\u6cd5\u4f1a\u628a\u95ee\u9898\u590d\u6742\u5316\uff0c\u6545\u7565\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u5b9a\u4e49 \u5c06\u4e00\u4e2a\u591a\u9879\u5f0f\u5728\u4e00\u5b9a\u8303\u56f4\u5316\u4e3a\u51e0\u4e2a\u6574\u5f0f\u7684\u79ef\u7684\u5f62\u5f0f,\u8fd9\u79cd\u5f0f\u5b50\u7684\u53d8\u5f62\u53eb\u505a\u56e0\u5f0f\u5206\u89e3\u3002 \u5206\u89e3\u65b9\u6cd5 \u63d0\u516c\u56e0\u5f0f\u6cd5 \u63d0\u516c\u56e0 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[72],"tags":[70],"class_list":["post-789","post","type-post","status-publish","format-standard","hentry","category-maths","tag-70"],"_links":{"self":[{"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/posts\/789","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/comments?post=789"}],"version-history":[{"count":2,"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/posts\/789\/revisions"}],"predecessor-version":[{"id":1790,"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/posts\/789\/revisions\/1790"}],"wp:attachment":[{"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/media?parent=789"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/categories?post=789"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/muidar.com\/blog\/wp-json\/wp\/v2\/tags?post=789"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}