{"id":2330,"date":"2021-08-14T09:10:05","date_gmt":"2021-08-14T13:10:05","guid":{"rendered":"https:\/\/josmfs.net\/?p=2330"},"modified":"2021-08-23T12:24:09","modified_gmt":"2021-08-23T16:24:09","slug":"polynomial-division-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2021\/08\/14\/polynomial-division-problem\/","title":{"rendered":"Polynomial Division Problem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-2326\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/08\/Polynomial-Division-Prob-Fig-2.jpg\" alt=\"\" width=\"200\" height=\"240\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/08\/Polynomial-Division-Prob-Fig-2.jpg 402w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/08\/Polynomial-Division-Prob-Fig-2-250x300.jpg 250w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/>Here is a challenging problem from the Polish Mathematical Olympiads published in 1960.<\/p>\n<p><strong>\u201c22.<\/strong> Prove that the polynomial<\/p>\n<p style=\"text-align: center;\"><em>x<\/em><sup>44<\/sup> + <em>x<\/em><sup>33<\/sup> + <em>x<\/em><sup>22<\/sup> +<em> x<\/em><sup>11<\/sup> + 1<\/p>\n<p>is divisible by the polynomial<\/p>\n<p style=\"text-align: center;\"><em>x<\/em><sup>4<\/sup> + <em>x<\/em><sup>3<\/sup> + <em>x<\/em><sup>2<\/sup> +<em> x<\/em> + 1.\u201d<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2021\/08\/Polynomial-Division-Problem-210811.pdf\">Polynomial Division Problem<\/a><\/p>\n<p><strong>(Update 8\/23\/2021)<\/strong>\u00a0 The idea expressed in this post that mathematicians are &#8220;lazy&#8221; and seek short-cuts to solving questions and problems, as I did in this one, was recently the subject of a <em>Numberphile<\/em> post by Marcus du Sautoy: &#8220;<a href=\"https:\/\/www.youtube.com\/watch?v=BdEWCxt8C0M\">Mathematics is all about SHORTCUTS<\/a>&#8220;.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is a challenging problem from the Polish Mathematical Olympiads published in 1960. \u201c22. Prove that the polynomial x44 + x33 + x22 + x11 + 1 is divisible by the polynomial x4 + x3 + x2 + x + 1.\u201d See the Polynomial Division Problem (Update 8\/23\/2021)\u00a0 The idea expressed in this post that [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[133,222],"class_list":["post-2330","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-algebra","tag-polish-math-olympiads"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2330","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=2330"}],"version-history":[{"count":5,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2330\/revisions"}],"predecessor-version":[{"id":2350,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2330\/revisions\/2350"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=2330"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=2330"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=2330"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}