{"id":56,"date":"2023-09-17T07:25:02","date_gmt":"2023-09-17T07:25:02","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e4%bd%99%e6%95%b0%e5%ae%9a%e7%90%86%e7%a4%ba%e4%be%8b%e5%92%8c%e4%b9%a0%e9%a2%98%e5%b7%b2%e8%a7%a3%e5%86%b3\/"},"modified":"2023-09-17T07:25:02","modified_gmt":"2023-09-17T07:25:02","slug":"%e4%bd%99%e6%95%b0%e5%ae%9a%e7%90%86%e7%a4%ba%e4%be%8b%e5%92%8c%e4%b9%a0%e9%a2%98%e5%b7%b2%e8%a7%a3%e5%86%b3","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e4%bd%99%e6%95%b0%e5%ae%9a%e7%90%86%e7%a4%ba%e4%be%8b%e5%92%8c%e4%b9%a0%e9%a2%98%e5%b7%b2%e8%a7%a3%e5%86%b3\/","title":{"rendered":"\u4f59\u6570\uff08\u6216\u7559\u6570\uff09\u5b9a\u7406"},"content":{"rendered":"<p>\u5728\u8fd9\u91cc\uff0c\u60a8\u5c06\u627e\u5230\u5173\u4e8e\u4f59\u6570\u5b9a\u7406\uff08\u6216\u4f59\u6570\u5b9a\u7406\uff09\u662f\u4ec0\u4e48\u4ee5\u53ca\u5982\u4f55\u5c06\u5176\u5e94\u7528\u4e8e\u591a\u9879\u5f0f\u7684\u89e3\u91ca\u3002\u60a8\u8fd8\u53ef\u4ee5\u67e5\u770b\u793a\u4f8b\uff0c\u5e76\u901a\u8fc7\u9010\u6b65\u89e3\u51b3\u4f59\u6570\u5b9a\u7406\u7684\u7ec3\u4e60\u8fdb\u884c\u7ec3\u4e60\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-es-el-teorema-del-resto\"><\/span>\u4ec0\u4e48\u662f\u4f59\u6570\u5b9a\u7406\uff1f <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffebee\">\u5728\u6570\u5b66\u4e2d\uff0c<strong>\u4f59\u6570\u5b9a\u7406<\/strong>\u8868\u793a\u4efb\u4f55\u591a\u9879\u5f0f P(x) \u9664\u4ee5\u53e6\u4e00\u4e2a (xa) \u5f62\u5f0f\u7684\u591a\u9879\u5f0f\u7684\u4f59\u6570\u7b49\u4e8e\u591a\u9879\u5f0f P(x) \u7684\u6570\u503c\uff0c\u5176\u4e2d\u503c x=a\uff0c\u6362\u53e5\u8bdd\u8bf4\uff0c\u9664\u6cd5 P(x):(xa) \u7684\u4f59\u6570\u7b49\u4e8e P(a)\u3002 <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/theoreme-des-restes.jpg\" alt=\"\u4f59\u6570\u5b9a\u7406\" class=\"wp-image-1041\" width=\"202\" height=\"203\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplo-del-teorema-del-resto\"><\/span>\u4f59\u6570\u5b9a\u7406\u7684\u4f8b\u5b50<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u4e86\u89e3\u4e86\u4f59\u6570\u5b9a\u7406\u662f\u4ec0\u4e48\u540e\uff0c\u8ba9\u6211\u4eec\u770b\u4e00\u4e0b\u5176\u5e94\u7528\u7684\u5b9e\u9645\u793a\u4f8b\uff1a<\/p>\n<ul>\n<li>\u8ba1\u7b97\u4ee5\u4e0b\u4e24\u4e2a\u591a\u9879\u5f0f\u9664\u6cd5\u7684\u4f59\u6570\uff1a<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eba049cd57efaf6f586d13f45f82f98b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = x^3+2x^2-4x+3 \\qquad \\qquad Q(x)=x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"371\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4fc4381966c56691db34f3b902a9fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{P(x)}{Q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"38\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p>\u4e3a\u4e86\u627e\u5230\u591a\u9879\u5f0f\u9664\u6cd5\u7684\u4f59\u6570\uff08\u6216\u6b8b\u5dee\uff09\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4f59\u6570\u5b9a\u7406\uff0c\u56e0\u4e3a\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u9664\u6cd5\u591a\u9879\u5f0f\u7684\u5f62\u5f0f\u4e3a (xa)\uff0c\u4e5f\u5c31\u662f\u8bf4\u5b83\u662f\u4e00\u6b21\uff0c\u7cfb\u6570\u53d8\u91cf x \u4e3a 1 \u5e76\u4e14\u6709\u4e00\u4e2a\u72ec\u7acb\u9879\u3002<\/p>\n<p>\u56e0\u6b64\uff0c\u6211\u4eec\u5e94\u7528\u4f59\u6570\u5b9a\u7406\uff0c\u8be5\u5b9a\u7406\u8868\u660e\uff0c\u50cf\u8fd9\u6837\u7684\u9664\u6cd5\u7684\u4f59\u6570\u7b49\u4e8e\u5728\u53d8\u53f7\u9664\u6570\u591a\u9879\u5f0f\u7684\u72ec\u7acb\u9879\u4e2d\u8ba1\u7b97\u7684\u88ab\u9664\u6570\u591a\u9879\u5f0f\u7684\u6570\u503c\uff0c\u5373 P (1)\u3002 <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/theoreme-du-reste-et-du-facteur-pdf.jpg\" alt=\"\u4f59\u6570\u548c\u56e0\u6570\u5b9a\u7406.pdf\" class=\"wp-image-1048\" width=\"476\" height=\"118\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p>\u56e0\u6b64\uff0c\u4e3a\u4e86\u627e\u5230\u9664\u6cd5\u7684\u4f59\u6570\uff0c\u6211\u4eec\u9700\u8981\u8ba1\u7b97 x=1 \u5904\u7684\u591a\u9879\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ff03f53066d698ee3d76e0024f3b51ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(1) &amp;= 1^3+2\\cdot 1^2-4\\cdot 1+3\\\\[2ex] &amp;= 1+2\\cdot 1-4 \\cdot 1+3  \\\\[2ex] &amp; = 1+2-4+3 \\\\[2ex] &amp; =\\bm{2}  \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"219\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p><strong>\u56e0\u6b64\uff0c\u591a\u9879\u5f0f\u9664\u6cd5\u7684\u4f59\u6570\u4e3a 2<\/strong> \u3002<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u7528<strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/cn\/\u89c4\u5219\u5df2\u89e3\u51b3\u793a\u4f8b\u9c81\u83f2\u5c3c\u7ec3\u4e60\/\">\u9c81\u83f2\u5c3c\u591a\u9879\u5f0f\u9664\u6cd5\u89c4\u5219<\/a><\/span><\/strong>\u6765\u68c0\u67e5\u4f59\u6570\u662f\u5426\u4e0e\u6211\u4eec\u53d1\u73b0\u7684\u7ed3\u679c\u4e00\u81f4\uff1a <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/theoreme-du-reste-de-ruffini.jpg\" alt=\"\u9c81\u83f2\u5c3c\u4f59\u6570\u5b9a\u7406\" class=\"wp-image-1043\" width=\"252\" height=\"133\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p>\u6b63\u5982\u60a8\u6240\u770b\u5230\u7684\uff0c\u4f7f\u7528\u4f59\u6570\u5b9a\u7406\u786e\u5b9a\u591a\u9879\u5f0f\u9664\u4ee5\u4e8c\u9879\u5f0f\u7684\u4f59\u6570\u6bd4\u4f7f\u7528\u9c81\u83f2\u5c3c\u89c4\u5219\u66f4\u5feb\u3001\u66f4\u5bb9\u6613\uff0c\u56e0\u4e3a\u6267\u884c\u7684\u8ba1\u7b97\u8981\u5c11\u5f97\u591a\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Teorema-del-resto-y-del-factor\"><\/span>\u4f59\u6570\u548c\u56e0\u6570\u5b9a\u7406<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u4ece\u4f59\u6570\u5b9a\u7406\u548c\u591a\u9879\u5f0f\u7684\u6839\uff08\u6216\u96f6\uff09\u7684\u5b9a\u4e49\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u56e0\u5b50\u5b9a\u7406\u3002\u56e0\u6b64\uff0c\u56e0\u5b50\u5b9a\u7406\u610f\u5473\u7740\uff1a<\/p>\n<p class=\"has-background\" style=\"background-color:#ffebee\"><strong>\u56e0\u5b50\u5b9a\u7406<\/strong>\u8868\u660e\uff0c\u5f53\u4e14\u4ec5\u5f53 P(a)=0 \u65f6\uff0c\u591a\u9879\u5f0f P(x) \u624d\u80fd\u88ab (xa) \u5f62\u5f0f\u7684\u53e6\u4e00\u4e2a\u591a\u9879\u5f0f\u6574\u9664\u3002\u5e76\u4e14\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8fd9\u610f\u5473\u7740 a \u662f\u591a\u9879\u5f0f P(x) \u7684\u6839\u6216\u96f6\u3002<\/p>\n<p>\u6b64\u5916\uff0c\u6839\u636e\u4f59\u6570\u5b9a\u7406\uff0c\u8fd9\u610f\u5473\u7740\u5982\u679c\u4e00\u4e2a\u591a\u9879\u5f0f\u53ef\u88ab\u53e6\u4e00\u4e2a\u591a\u9879\u5f0f\u6574\u9664\uff0c\u5219\u6240\u8ff0\u9664\u6cd5\u7684\u4f59\u6570\u4e3a\u96f6\uff0c\u56e0\u4e3aP(a)=0\u3002<\/p>\n<p>\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u6709\u67d0\u4e2a\u591a\u9879\u5f0f\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3d378a4916cbd7a268ed60de50589d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^2+2x-8\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"151\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p>\u8be5\u591a\u9879\u5f0f\u53ef\u88ab\u4e8c\u9879\u5f0f (x-2) \u6574\u9664\uff0c\u56e0\u4e3a P(2)=0\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e90c14ff06cdfa041299e016051b1dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(2) &amp;= 2^2+2\\cdot 2-8\\\\[2ex] &amp;= 4+4-8 \\\\[2ex] &amp; =\\bm{0} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"100\" width=\"159\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>\u7531\u4e8ex=2\u62b5\u6d88\u4e86\u591a\u9879\u5f0fP(x)\uff0c\u8fd9\u610f\u5473\u7740x=2\u662f\u6240\u8ff0\u591a\u9879\u5f0f\u7684\u6839\u3002<\/p>\n<p>\u8fdb\u4e00\u6b65\uff0c\u7531\u4e8e P(2)=0\uff0c\u6839\u636e\u4f59\u6570\u5b9a\u7406\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\u9664\u6cd5\u7684\u4f59\u6570<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ba6849f9b139a364e36b1e7f461969c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x^2+2x-8}{x-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"90\" style=\"vertical-align: -12px;\"><\/p>\n<p>\u7b49\u4e8e 0\u3002 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-del-teorema-del-resto\"><\/span>\u4f59\u6570\u5b9a\u7406\u7684\u89e3\u7b54\u7ec3\u4e60<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u4e3a\u4e86\u5b8c\u6210\u4f59\u6570\u5b9a\u7406\u7684\u7406\u89e3\uff0c\u6211\u4eec\u51c6\u5907\u4e86\u4e00\u4e9b\u9010\u6b65\u89e3\u51b3\u7684\u7ec3\u4e60\uff0c\u4ee5\u4fbf\u60a8\u7ec3\u4e60\u3002\u6211\u4eec\u5efa\u8bae\u60a8\u5148\u4eb2\u81ea\u5c1d\u8bd5\u4e00\u4e0b\u8fd9\u4e2a\u7ec3\u4e60\uff0c\u7136\u540e\u68c0\u67e5\u60a8\u662f\u5426\u505a\u5bf9\u4e86\u3002<\/p>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e601<\/h3>\n<p>\u901a\u8fc7\u4f59\u6570\u5b9a\u7406\u6c42\u591a\u9879\u5f0f\u9664\u6cd5\u7684\u4f59\u6570<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4fc4381966c56691db34f3b902a9fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{P(x)}{Q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"38\" style=\"vertical-align: -17px;\"><\/p>\n<p>\uff0c\u662f\u53c2\u4e0e\u8fd0\u7b97\u7684\u591a\u9879\u5f0f\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4201895897fc514d2ec7ef49b43ca580_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =x^3+4x^2-2x+1\\qquad \\qquad Q(x)=x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"371\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u9664\u6570\u591a\u9879\u5f0f\u4ec5\u7531\u4e00\u6b21\u9879\u548c\u72ec\u7acb\u9879\u7ec4\u6210\uff0c\u800c\u4e14\u4e00\u6b21\u9879\u7684\u7cfb\u6570\u4e3a1\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4f59\u5f0f\u5b9a\u7406\u3002<\/p>\n<p class=\"has-text-align-left\">\u4e3a\u4e86\u5e94\u7528\u4f59\u6570\u5b9a\u7406\uff0c\u7528\u9664\u6570\u591a\u9879\u5f0f\u6539\u53d8\u7b26\u53f7\u7684\u72ec\u7acb\u9879\u6765\u8ba1\u7b97\u88ab\u9664\u591a\u9879\u5f0f\u5c31\u8db3\u591f\u4e86\uff0c\u6216\u8005\u6362\u53e5\u8bdd\u8bf4\uff0c\u6211\u4eec\u5fc5\u987b\u8ba1\u7b97P(2)\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23790b78a8463a23a7b8202ab544ade9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(2) &amp;= 2^3+4\\cdot 2^2-2\\cdot 2+1\\\\[2ex] &amp;=8+4\\cdot 4-2\\cdot 2+1  \\\\[2ex] &amp; = 8+16-4+1 \\\\[2ex] &amp; =\\bm{21}  \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"218\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"><strong>\u56e0\u6b64\uff0c\u4e24\u4e2a\u591a\u9879\u5f0f\u76f8\u9664\u7684\u4f59\u6570\u4e3a 21<\/strong> \u3002<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e602<\/h3>\n<p>\u7ed9\u5b9a\u591a\u9879\u5f0f<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-82d5f84d7abf17919f73133b0624418d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^4-2x^3+5x^2-3x+4 ,\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"251\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u6c42\u9664\u4ee5\u4e0b\u5217\u5404\u9879\u591a\u9879\u5f0f\u6240\u5f97\u7684\u4f59\u6570\uff1a <\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8d8cd22b173b522307e57fd84bcd5cf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\left(x-1 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2dd9c5036ff9ff34b1ea263893f64e24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\left(x+1 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-65e5cacdc716b7c9a064099e0ee78439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\left(x+2 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2a7b8e26b769665de5041f278d15686_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\left(x-3 \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u7531\u4e8e\u6240\u6709\u9664\u6cd5\u591a\u9879\u5f0f\u90fd\u6ee1\u8db3\u4f59\u6570\u5b9a\u7406\u7684\u6761\u4ef6\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u8be5\u5b9a\u7406\u6765\u786e\u5b9a\u6bcf\u6b21\u9664\u6cd5\u7684\u4f59\u6570\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f8ae7d7c667bf9ca6bd7417356756447_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\mathbf{A}\\bm{)} \\ P(1) &amp;= 1^4-2\\cdot 1^3+5\\cdot 1^2-3\\cdot 1+4\\\\[2ex] &amp;=1-2\\cdot 1+5\\cdot 1 -3 \\cdot 1+4 \\\\[2ex] &amp; = 1-2+5-3+4 \\\\[2ex] &amp; =\\bm{5} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"307\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e2a31e7c1334f8d1a24ba246d0459e4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\mathbf{B}\\bm{)} \\ P(-1) &amp;= (-1)^4-2\\cdot (-1)^3+5\\cdot (-1)^2-3\\cdot (-1)+4\\\\[2ex] &amp;=1-2\\cdot (-1)+5\\cdot 1 -3 \\cdot (-1)+4 \\\\[2ex] &amp; = 1+2+5+3+4 \\\\[2ex] &amp; =\\bm{15} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"431\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e8191f6ce490a0786515d84efaf45ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\mathbf{C}\\bm{)} \\ P(-2) &amp;= (-2)^4-2\\cdot (-2)^3+5\\cdot (-2)^2-3\\cdot (-2)+4\\\\[2ex] &amp;=16-2\\cdot (-8)+5\\cdot 4 -3 \\cdot (-2)+4 \\\\[2ex] &amp; = 16+16+20+6+4 \\\\[2ex] &amp; =\\bm{62} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"430\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2d2e1e17bbcf91abb36d8cad24cadf0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\mathbf{D}\\bm{)} \\ P(3) &amp;= 3^4-2\\cdot 3^3+5\\cdot 3^2-3\\cdot 3+4\\\\[2ex] &amp;=81-2\\cdot 27+5\\cdot 9 -3 \\cdot 3+4 \\\\[2ex] &amp; = 81-54+45-9+4 \\\\[2ex] &amp; =\\bm{67} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"308\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e603<\/h3>\n<p>\u8ba1\u7b97\u53c2\u6570\u5e94\u8be5\u503c\u591a\u5c11<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u8fd9\u6837\u591a\u9879\u5f0f\u9664\u6cd5\u7684\u4f59\u6570<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4fc4381966c56691db34f3b902a9fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{P(x)}{Q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"38\" style=\"vertical-align: -17px;\"><\/p>\n<p>\u7b49\u4e8e 3\uff0c\u5747\u4e3a\u591a\u9879\u5f0f\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-39aa975b852cd190f864491d029e3d58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =x^3-5x^2-mx+9 \\qquad \\qquad Q(x)=x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"379\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u5728\u8fd9\u79cd\u7279\u6b8a\u60c5\u51b5\u4e0b\uff0c\u9664\u6cd5\u591a\u9879\u5f0f\u7531\u4e00\u6b21\u5355\u9879\u5f0f\u548c\u4e00\u4e2a\u72ec\u7acb\u9879\u7ec4\u6210\uff0c\u800c\u4e14\u4e00\u6b21\u5355\u9879\u5f0f\u7684\u7cfb\u6570\u4e3a1\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4f59\u6570\u5b9a\u7406\u3002<\/p>\n<p class=\"has-text-align-left\">\u8981\u4f7f\u7528\u4f59\u6570\u5b9a\u7406\uff0c\u53ea\u9700\u7528\u7b26\u53f7\u53d8\u5316\u66ff\u6362\u9664\u6cd5\u591a\u9879\u5f0f\u7684\u72ec\u7acb\u9879\uff0c\u5176\u4e2d\u9664\u6cd5\u591a\u9879\u5f0f\u4e2d\u6709 x\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u6c42\u89e3 P(-3)\u3002<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-693cd65a618f884d0dd1be2f20594229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(-3) &amp;=(-3)^3-5\\cdot (-3)^2-m\\cdot (-3)+9\\\\[2ex] &amp;=-27-5\\cdot 9 -m\\cdot (-3)+9 \\\\[2ex] &amp; = -27-45+3m+9 \\\\[2ex] &amp; =3m-63 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4f46\u663e\u7136\u6211\u4eec\u5f97\u5230\u7684\u7ed3\u679c\u662f\u57fa\u4e8e\u672a\u77e5\u6570<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4a447f470995624a31e9fc621325af2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p>\u7136\u800c\uff0c\u95ee\u9898\u9648\u8ff0\u544a\u8bc9\u6211\u4eec\u4f59\u6570\u5fc5\u987b\u7b49\u4e8e 3\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5c06\u627e\u5230\u7684\u4f59\u6570\u8bbe\u7f6e\u4e3a\u7b49\u4e8e 3\uff1a<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c683d547456ad6fc7d9a9c123be3ce3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3m-63 = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"96\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u6700\u540e\uff0c\u6211\u4eec\u6c42\u89e3\u65b9\u7a0b\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dd907c2a7cc446f45435dd543851062a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3m = 3+63\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"97\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-89214c7e36a0809ce927b0de14579ff7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3m = 66\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"66\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be9d6df593c2b5cd5333f8450a1e2afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = \\cfrac{66}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"59\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8631ca446a606a9511ad9a1bbf07bf1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{m = 22}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u7ec3\u4e604<\/h3>\n<p>\u7528\u56e0\u5b50\u548c\u4f59\u6570\u5b9a\u7406\u786e\u5b9a\u591a\u9879\u5f0f\u662f\u5426<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u53ef\u88ab\u591a\u9879\u5f0f\u6574\u9664<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-12d5846d8a96763047fb4c9f458420f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16c539cdb7c8de8425a9d943f4ffa4df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =-2x^3-5x^2-x+2 \\qquad \\qquad Q(x)=x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"385\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"><strong>\u67e5\u770b\u89e3\u51b3\u65b9\u6848<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\">\u4f7f\u5f97\u591a\u9879\u5f0f<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u53ef\u88ab\u591a\u9879\u5f0f\u6574\u9664<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8061e215a5d055a2cf14c44c4febfad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u8fd9\u4e24\u4e2a\u591a\u9879\u5f0f\u4e4b\u95f4\u7684\u9664\u6cd5\u5fc5\u987b\u7cbe\u786e\uff0c\u56e0\u6b64\u4f59\u6570\u5fc5\u987b\u4e3a\u96f6\u3002<\/p>\n<p class=\"has-text-align-left\">\u90a3\u4e48\uff0c\u7531\u4e8e\u9664\u6570\u591a\u9879\u5f0f\u662f<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f8aa3ce3e7e742e73a1b997a758126f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+2),\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u7531\u56e0\u5b50\u5b9a\u7406\u548c\u4f59\u6570\u5b9a\u7406\uff0c\u6211\u4eec\u77e5\u9053\u591a\u9879\u5f0f<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u5c06\u88ab\u591a\u9879\u5f0f\u6574\u9664<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8061e215a5d055a2cf14c44c4febfad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u5982\u679c\u5df2\u6ee1<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ba038b936f6d3cecf0aac3be3a9fbcbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2)=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u56e0\u6b64\uff0c\u6211\u4eec\u5fc5\u987b\u770b\u770b\u8fd9\u4e2a\u7b49\u5f0f\u662f\u5426\u5f97\u5230\u9a8c\u8bc1\uff1a <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-921817de49081656ab2dd5a8fc6a97ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2)=0 \\quad \\color{blue} \\bm{?}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5e8793835809000092b15eaec3877c18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} P(-2) &amp;=-2\\cdot (-2)^3-5\\cdot (-2)^2-(-2)+2\\\\[2ex] &amp;=-2 \\cdot (-8) -5 \\cdot 4+2 +2\\\\[2ex] &amp; =16-20+2+2 \\\\[2ex] &amp; =0\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"329\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\">\u4e8b\u5b9e\u4e0a\uff0c\u8be5\u90e8\u95e8\u7684\u5176\u4f59\u90e8\u5206<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4fc4381966c56691db34f3b902a9fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{P(x)}{Q(x)}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"38\" style=\"vertical-align: -17px;\"><\/p>\n<p>\u7b49\u4e8e 0\uff0c\u6240\u4ee5\u591a\u9879\u5f0f<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p>\u662f\u7684\uff0c\u5b83\u53ef\u4ee5\u88ab\u53e6\u4e00\u4e2a\u591a\u9879\u5f0f\u6574\u9664<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-12d5846d8a96763047fb4c9f458420f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Q(x).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end 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