{"id":5096,"date":"2025-05-31T09:22:00","date_gmt":"2025-05-31T13:22:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=5096"},"modified":"2025-08-18T09:23:32","modified_gmt":"2025-08-18T13:23:32","slug":"algebra-challenge-2025-05-31","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=5096","title":{"rendered":"Algebra Challenge &#8211; 2025\/05\/31"},"content":{"rendered":"\n<p>Let $P(x)$ is a polynomial with integer coefficients so that $P(d)=\\dfrac{2025}{d}$, where $d$ is a positive divisor of $2025$. Find $P(x)$.<a href=\"javascript:toggle_visibility('sol_alge_chall_2025_05_31')\">\ud83d\udd11<\/a><\/p>\n<div id=\"sol_alge_chall_2025_05_31\" style=\"display:none\">\n\n\n\n<p><strong>Claim<\/strong>: There is no $P(x)$ to satisfy $P(d)=\\dfrac{2025}{d}$, where $d$ is a positive divisor of $2025$.<\/p>\n\n\n\n<p><strong>Lemma<\/strong>: If $P(x)$ is a polynomial with integer coefficients, and $a$ and $b$ are different integers. Then $P(a)-P(b)$ is a multiple of $a-b$.<\/p>\n\n\n\n<p>Let $P(x)=\\sum_{i=0}^{n}a_ix^i$. Then $P(a)-P(b)=\\sum_{i=0}^{n}a_i(a^i-b^i)$. Since $a^i-b^i$ is a multiple of $a-b$, and $a_i$ are integers, therefore $P(a)-P(b)$ is a multiple of $a-b$.<\/p>\n\n\n\n<p><strong>Proof<\/strong>: Since $2025=3^4\\cdot 5^2$, therefore $d=1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025$.<\/p>\n\n\n\n<p>Let $a=135$, $b=81$, we have $$P(a)=P(135)=\\dfrac{2025}{135}=15$$ $$P(b)=P(81)=\\dfrac{2025}{81}=25$$ $$P(a)-P(b)=15-25=10$$<\/p>\n\n\n\n<p>However, $a-b=135-81=54$, and $10$ is not a multiple of $54$. Therefore, $P(x)$ does not exist.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Let $P(x)$ is a polynomial with integer coefficients so that $P(d)=\\dfrac{2025}{d}$, where $d$ is a positive divisor of $2025$. Find $P(x)$.\ud83d\udd11 Claim: There is no $P(x)$ to satisfy $P(d)=\\dfrac{2025}{d}$, where $d$ is a positive divisor of $2025$. Lemma: If $P(x)$ &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=5096\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5096"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5096"}],"version-history":[{"count":13,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5096\/revisions"}],"predecessor-version":[{"id":5126,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5096\/revisions\/5126"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5096"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5096"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5096"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}