{"id":12411,"date":"2010-10-15T07:18:38","date_gmt":"2010-10-14T23:18:38","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=12411"},"modified":"2021-10-06T16:31:10","modified_gmt":"2021-10-06T08:31:10","slug":"%e5%a4%9a%e9%a0%85%e5%bc%8f%e9%99%a4%e6%b3%95polynomial-quotients-and-remainders","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%a4%9a%e9%a0%85%e5%bc%8f%e9%99%a4%e6%b3%95polynomial-quotients-and-remainders\/","title":{"rendered":"\u591a\u9805\u5f0f\u9664\u6cd5"},"content":{"rendered":"

\u591a\u9805\u5f0f\u9664\u6cd5 (Polynomial Quotients and Remainders)<\/span>
\n\u570b\u7acb\u4e2d\u592e\u5927\u5b78\u6578\u5b78\u7cfb\u55ae\u7dad\u5f70\u526f\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/p>\n

\u591a\u9805\u5f0f\u7684\u300c\u5143\u300d\u5177\u5099\u5982\u5be6\u6578\u4e00\u822c\u7684\u904b\u7b97\u6027\u8cea\uff0c\u4f46\u662f\u591a\u9805\u5f0f\u672c\u8eab\u7684\u904b\u7b97\u6027\u8cea\u537b\u50cf\u6b63\u6574\u6578\uff1a\u5169\u500b\u591a\u9805\u5f0f\u76f8\u9664\uff0c\u6703\u5f97\u5230\u5546\u5f0f\u4e26\u7522\u751f\u9918\u5f0f\u3002\u985e\u4f3c\u65bc\u6b63\u6574\u6578\u7684\u9664\u6cd5\u539f\u7406\uff0c\u82e5\u591a\u9805\u5f0f $$P$$\u3001$$K$$\u3001$$Q$$\u3001$$R$$ \u6eff\u8db3 $$\\mathrm{deg}~R <\\mathrm{deg}~K$$ \u4e14 $$P=QK+R$$\uff0c\u4f8b\u5982 $$x^2=(x+1)(x-1)+1$$ \u5247\u5b83\u5011\u4e5f\u53ef\u4ee5\u5beb\u6210\u591a\u9805\u5f0f\u9664\u6cd5\u5f62\u5f0f\uff1a<\/p>\n

$$P\\div{K}=Q…R$$<\/p>\n

\u4f8b\u5982 $$x^2\\div(x-1)=(x+1)…1$$\u3002\u6211\u5011\u7a31 $$P$$ \u70ba\u88ab\u9664\u5f0f<\/strong>\u3001$$K$$ \u70ba\u9664\u5f0f<\/strong>\u3001$$Q$$ \u70ba\u5546\u5f0f<\/strong>\u3001$$R$$ \u70ba\u9918\u5f0f<\/strong>\uff0c\u800c $$P=QK+R$$ \u548c $$P\\div{K}=Q…R$$\u7684\u7b49\u50f9\u95dc\u4fc2\uff0c\u7a31\u70ba\u591a\u9805\u5f0f\u7684\u9664\u6cd5\u539f\u7406<\/strong>\u3002<\/p>\n

\u985e\u4f3c\u591a\u9805\u5f0f\u7684\u52a0\u3001\u6e1b\u3001\u4e58\u53ef\u4ee5\u5beb\u6210\u76f4\u5f0f\uff0c\u591a\u9805\u5f0f\u76f8\u9664\u4e5f\u6709\u76f4\u5f0f\uff0c\u7a31\u70ba\u9577\u9664\u6cd5<\/strong>\uff1b\u6211\u5011\u4ee5 $$x^2$$ \u9664\u4ee5 $$x-1$$ \u70ba\u4f8b\u793a\u7bc4\u5982\u4e0b\uff1a<\/p>\n

\"\"<\/a><\/p>\n

\u5728\u5f97\u5230\u4e86\u7d05\u8272\u7684\u5546\u5f0f\u8207\u7da0\u8272\u7684\u9918\u5f0f\u4e4b\u5f8c\uff0c\u6211\u5011\u53ef\u4ee5\u5c07\u7b54\u6848\u8a18\u6210<\/p>\n

$$x^2\\div(x-1)=(x+1)…1$$<\/p>\n

\u540c\u6a23\u5730\uff0c\u6211\u5011\u4e5f\u80fd\u4f7f\u7528\u5206\u96e2\u4fc2\u6578\u6cd5\uff0c\u4ee5 $$6x^3-x+3$$ \u9664\u4ee5 $$2x-1$$ \u70ba\u4f8b\u793a\u7bc4\u5982\u4e0b\uff1a<\/p>\n

\"\"<\/a><\/p>\n

\u6211\u5011\u4e00\u6a23\u53ef\u4ee5\u5c07\u9664\u6cd5\u8a18\u9304\u70ba $$\\displaystyle(6x^3-x+3)\\div(2x-1)=(3x^2+{\\frac{3}{2}}x+\\frac{1}{4})…\\frac{15}{4}$$\u3002<\/p>\n

\u6839\u64da\u9664\u6cd5\u539f\u7406\uff0c\u5982\u679c $$P\\div{K}=Q…R$$ \u5247 $$P-R={QK}$$\u3002<\/p>\n

\u4f8b\u5982 $$\\displaystyle 6x^3-x+3-{\\frac{15}{4}}=(3x^2+{\\frac{3}{2}}x+\\frac{1}{4})(2x-1)$$\u3002<\/p>\n

\u5982\u679c\u9918\u5f0f\u662f $$0$$\uff0c\u4ea6\u5373 $$P\\div{K}=Q$$ \u6216 $$P=QK$$ \uff0c\u5247 $$Q$$ \u548c $$K$$ \u90fd\u7a31\u70ba $$P$$ \u7684\u56e0\u5f0f<\/strong>\u3002<\/p>\n

\u4f8b\u5982 $$\\displaystyle 3x^2+\\frac{3}{2}x+\\frac{1}{4}$$ \u548c $$2x-1$$ \u90fd\u662f $$\\displaystyle 6x^3-x-\\frac{3}{4}$$ \u7684\u56e0\u5f0f\u3002<\/p>\n

\u986f\u7136\uff0c\u5982\u679c $$P=QK$$ \u5247 $$P\\div{K}=Q$$\uff0c\u9019\u662f\u591a\u9805\u5f0f\u7684\u4e58\u9664\u4e92\u9006<\/strong>\u6027\u8cea\u3002<\/p>\n

\u5c07\u591a\u9805\u5f0f $$P$$ \u5beb\u6210\u56e0\u5f0f\u76f8\u4e58\u7684\u5f62\u5f0f\uff0c\u5982 $$P=QK$$\uff0c\u7a31\u70ba $$P$$ \u7684\u56e0\u5f0f\u5206\u89e3\u3002\u56e0\u70ba\u975e\u96f6\u5be6\u6578\u4e5f\u5c31\u662f\u96f6\u6b21\u591a\u9805\u5f0f\uff0c\u6240\u4ee5\u591a\u9805\u5f0f\u7e3d\u6709\uff08\u7121\u804a\u7684\uff09\u56e0\u5f0f\u5206\u89e3\uff0c\u4f8b\u5982 $$\\displaystyle x^2+1=\\frac{1}{5}(5x^2+5)$$\u3002<\/p>\n

\u540c\u7406\uff0c\u56e0\u5f0f\u5206\u89e3\u4e26\u4e0d\u552f\u4e00\uff0c\u4f8b\u5982<\/p>\n

$$\\displaystyle x^2-1=(x+1)(x-1)=(2x+2)(\\frac{1}{2}x-\\frac{1}{2})$$\u3002<\/p>\n

\u591a\u9805\u5f0f\u9664\u6cd5\u4e2d\uff0c\u82e5\u9664\u5f0f\u6b63\u597d\u70ba $$x-c$$ \u7684\u5f62\u5f0f\uff0c\u5247\u6709\u4e00\u7a2e\u66f4\u7c21\u4fbf\u7684\u65b9\u6cd5\u53ef\u4ee5\u505a\u9664\u6cd5\uff0c\u7a31\u70ba\u7d9c\u5408\u9664\u6cd5\u3002\u8b93\u6211\u5011\u89c0\u5bdf $$(x^3-2x^2-x-4)\\div(x-3)$$\uff0c\u5728\u5c07\u4e4b\u4ee5\u5206\u96e2\u4fc2\u6578\u6cd5\u8868\u793a\u5f8c\uff0c\u53ef\u4ee5\u767c\u73fe\u6709\u8a31\u591a\u6578\u5b57\u90fd\u5728\u91cd\u8986\uff0c\u5728\u4e0b\u65b9\u7528\u76f8\u540c\u984f\u8272\u6a19\u8a18\uff1a<\/p>\n

\"\"<\/a><\/p>\n

\u5982\u6211\u5011\u6240\u77e5\uff0c\u9918\u6578\u5b58\u5728\u65bc\u9577\u9664\u6cd5\u7684\u6700\u5f8c\u4e00\u5c64\uff0c\u800c\u7c89\u7d05\u8272\u8207\u7da0\u8272\u7684\u90e8\u5206\u5176\u5be6\u4e5f\u8868\u73fe\u51fa\u4e86\u5546\u5f0f\uff0c\u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u7701\u7565\u5546\u5f0f\uff0c\u4e5f\u7701\u7565\u9577\u9664\u6cd5\u4e2d\u91cd\u8986\u7684\u6578\u5b57\uff0c\u5982\u4e0b\uff1a<\/p>\n

\"\"<\/p>\n

\u6b64\u6642\u7d05\u8272\u70ba\u5546\u5f0f\uff0c\u85cd\u8272\u70ba\u9918\u5f0f\u3002\u5c07\u9577\u9664\u6cd5\u4e2d\u7b2c\u4e8c\u5c64\u3001\u7b2c\u4e09\u5c64\u7684\u6578\u5b57\u5168\u90e8\u5408\u4f75\u4e0a\u4f86\uff0c\u53ef\u4ee5\u9032\u4e00\u6b65\u7e2e\u6e1b\u76f4\u5f0f\u7684\u9ad8\u5ea6\uff0c\u5982\u4e0b\uff1a<\/p>\n

\"\"<\/a><\/p>\n

\u7531\u65bc\u7d9c\u5408\u9664\u6cd5\u50c5\u9069\u7528\u65bc$$x-c$$ \u7684\u5f62\u5f0f\uff0c$$x$$ \u9805\u7684\u4fc2\u6578 $$1$$ \u4e5f\u53ef\u4ee5\u7701\u7565\u3002\u7d9c\u5408\u6240\u6709\u7684\u7701\u7565\u548c\u5408\u4f75\uff0c\u6211\u5011\u4fbf\u80fd\u5beb\u6210\u4ee5\u4e0b\u5f62\u5f0f\uff1a<\/p>\n

\"\"<\/a><\/p>\n

\u5176\u4e2d\u7d05\u8272\u7684\u90e8\u5206\u662f\u5546\u5f0f\uff0c\u85cd\u8272\u662f\u9918\u5f0f\uff08\u6b64\u6642\u7684\u9918\u5f0f\u5fc5\u70ba\u4e00\u500b\u5be6\u6578\uff09\u3002\u6211\u5011\u53c8\u6ce8\u610f\u5230\uff0c\u5728\u4ee5\u4e0a\u8a08\u7b97\u904e\u7a0b\u4e2d\uff0c\u7e3d\u662f\u91cd\u8907\u5730\u505a $$a-(-c)b$$ \u5f62\u5f0f\u7684\u8a08\u7b97\uff0c\u4f8b\u5982\u7b2c\u4e8c\u884c\u505a $$(-2)-(-3)$$\uff0c\u7b2c\u4e09\u884c\u505a $$(-1)-(-3)$$\uff0c\u7b2c\u56db\u884c\u505a $$(-4)-(-6)$$\u3002\u65e2\u7136\u5982\u6b64\uff0c\u4f55\u4e0d\u505a $$a+cb$$ \u5c31\u597d\u4e86\uff1f<\/p>\n

\u91cd\u65b0\u6574\u7406\u4ee5\u4e0a\u7684\u767c\u73fe\uff0c\u5c31\u662f\u7d9c\u5408\u9664\u6cd5\uff0c\u793a\u7bc4\u5982\u4e0b\u3002\u6211\u5011\u5148\u5217\u51fa\u88ab\u9664\u5f0f\u7684\u4fc2\u6578\uff08\u964d\u51aa\u6392\u5217\uff0c\u7f3a\u9805\u88dc\u96f6\uff09\uff0c\u4f8b\u5982\u5c07 $$x^3-2x^2-x-4$$ \u7684\u4fc2\u6578\u5217\u70ba $$1~~-2~~-1~~-4$$\uff1b\u5982\u679c\u9664\u5f0f\u662f $$x-c$$\uff0c\u5c07 $$c$$\uff08\u800c\u4e0d\u662f $$-c$$\uff09\u5beb\u5230\u5f0f\u5b50\u7684\u53f3\u5074\uff0c\u4f8b\u5982\u7576\u9664\u5f0f\u662f $$x-3$$\uff0c\u5beb $$3$$ \u5728\u53f3\u5074\u3002\u756b\u4e00\u689d\u6298\u7dda\u9694\u958b\uff1a<\/p>\n

\"\"<\/a><\/p>\n

\u5c07\u9996\u9805\u4fc2\u6578\u6284\u5230\u4e0b\u5c64\uff1a<\/p>\n

\"\"<\/a><\/p>\n

\u4e58\u4ee5\u53f3\u908a\u7684 $$3$$\uff0c\u5beb\u5728\u7b2c\u4e8c\u884c\u7684\u4e2d\u5c64\uff1b\u5c07\u4e0a\u3001\u4e2d\u5c64\u7684\u5169\u6578\u76f8\u52a0\u5beb\u5230\u4e0b\u5c64\uff1a<\/p>\n

\"\"<\/a>\u00a0\u518d\u4e58\u4ee5\u53f3\u908a\u7684 $$3$$\uff0c\u5beb\u5728\u7b2c\u4e09\u884c\u7684\u4e2d\u5c64\uff1b\u5c07\u4e0a\u3001\u4e2d\u5c64\u7684\u5169\u6578\u76f8\u52a0\u5beb\u5230\u4e0b\u5c64\uff1a<\/p>\n

\"\"<\/a><\/p>\n

\u518d\u4e58\u4ee5\u53f3\u908a\u76843\uff0c\u5beb\u5728\u7b2c\u56db\u884c\u7684\u4e2d\u5c64\uff1b\u5c07\u4e0a\u3001\u4e2d\u5c64\u7684\u5169\u6578\u76f8\u52a0\u5beb\u5230\u4e0b\u5c64\uff1a<\/p>\n

\"\"<\/a><\/p>\n

\u5247\u4e0b\u5c64\u7684\u6700\u5f8c\u4e00\u500b\u6578\u5c31\u662f\u9918\u5f0f\uff0c\u800c\u5176\u4ed6\u6578\u5247\u662f\u964d\u51aa\u6392\u5217\u7684\u5546\u5f0f\u4fc2\u6578\u3002\u6240\u4ee5\u5546\u5f0f\u662f $$x^2+x+2$$\uff0c\u9918\u5f0f\u662f $$2$$\u3002<\/p>\n

\u5411\u524d\u9023\u7d50\uff1a<\/strong>\u591a\u9805\u5f0f<\/a>\uff0c\u81ea\u7136\u6578\u4e58\u9664
\n\u5411\u5f8c\u9023\u7d50\uff1a<\/strong>\u970d\u5167\u6f14\u7b97\u6cd5<\/span><\/a><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u95e1\u8ff0\u591a\u9805\u5f0f\u76f8\u9664\u5f97\u5230\u5546\u8207\u9918\u5f0f\u7684\u95dc\u4fc2\uff0c\u53ef\u985e\u6bd4\u65bc\u6b63\u6574\u6578\u7684\u9664\u6cd5\u6027\u8cea\uff0c\u4e26\u4ee5\u76f8\u540c\u5f62\u5f0f\u5beb\u6210\u9664\u6cd5\u539f\u7406\u3002\u7576\u9664\u5f0f\u70ba\u4e00\u6b21\u591a\u9805\u5f0f\u6642\uff0c\u4ecb\u7d39\u7279\u6b8a\u7684\u9664\u6cd5\u7b97\u5247\u3002<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[219,220,111],"tags":[313,311,292,1618],"class_list":["post-12411","post","type-post","status-publish","format-standard","hentry","category-math03","category-math03-01","category-mathematics00","tag-313","tag-311","tag-292","tag-1618","loop-entry","cat-219","cat-220","cat-111","no-thumbnail"],"views":31108,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/12411","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=12411"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/12411\/revisions"}],"predecessor-version":[{"id":89454,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/12411\/revisions\/89454"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=12411"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=12411"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=12411"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}