{"id":364,"date":"2020-09-17T16:46:58","date_gmt":"2020-09-17T21:46:58","guid":{"rendered":"http:\/\/blogs.acu.edu\/mu_sigma\/?p=364"},"modified":"2020-09-26T17:00:42","modified_gmt":"2020-09-26T22:00:42","slug":"problem-1-fall-2020","status":"publish","type":"post","link":"https:\/\/blogs.acu.edu\/mu_sigma\/2020\/09\/17\/problem-1-fall-2020\/","title":{"rendered":"Problem 1 – Fall 2020 – Symmetric Functions I"},"content":{"rendered":"

Symmetric Functions I<\/b><\/span><\/p>\n

This problem is from Alexander Karabegov.<\/i><\/span><\/p>\n

Given a quadratic polynomial x<\/i>2<\/sup> + p x<\/i> + q<\/i> with two real roots s<\/i> and t<\/i>, express the following functions of s<\/i> and t<\/i> as polynomials in p<\/i> and q<\/i>.<\/p>\n

    \n
  1. s<\/i> + t<\/i><\/li>\n
  2. s<\/i>t<\/i><\/li>\n
  3. s<\/i>2<\/sup> + t<\/i>2<\/sup><\/li>\n
  4. s<\/i>3<\/sup> + t<\/i>3<\/sup><\/li>\n<\/ol>\n

    Please submit all problem solutions to mathpotw@acu.edu before 5:00 PM on the next Thursday.\u00a0<\/em><\/p>\n

    Solution to Symmetric Functions I<\/b><\/span><\/p>\n

    Correct solutions were submitted by:\u00a0 Ricky Kagoro Lumala, Wyatt Witemeyer<\/i><\/span><\/p>\n

    Since s<\/i> and t<\/i> are roots of the polynomial x<\/i>2<\/sup> + p x<\/i> + q<\/i>, then<\/p>\n

    x<\/i>2<\/sup> + p x<\/i> + q<\/i> = (x<\/i>–s<\/i>)(x<\/i>–t<\/i>).Multiplication of the right side yields<\/p>\n

    x<\/i>2<\/sup> + p x<\/i> + q<\/i> = x<\/i>2<\/sup> – (s<\/i>+t<\/i>)x<\/i> + s t<\/i>.Thus the coefficients of the x<\/i> term give the first result: s<\/i> + t<\/i> = –p<\/i>.<\/p>\n

    The constant terms produce the second: s<\/i>t<\/i> = q<\/i>.<\/p>\n

    The third part can be obtained by observing that s<\/i>2<\/sup> + t<\/i>2<\/sup> = (s<\/i>+t<\/i>)2<\/sup> – 2s<\/i>t<\/i>, and using the two previous results.
    \nTherefore, s<\/i>2<\/sup> + t<\/i>2<\/sup> = (-p<\/i>)2<\/sup> – 2q<\/i> = p<\/i>2<\/sup> – 2q<\/i>.<\/p>\n

    Similarly, s<\/i>3<\/sup> + t<\/i>3<\/sup>
    \n= (s<\/i>+t<\/i>)3<\/sup> – 3s<\/i>2<\/sup>t<\/i> – 3s<\/i>t<\/i>2<\/sup>
    \n= (s<\/i>+t<\/i>)3<\/sup> – 3s<\/i>t<\/i>(s<\/i>+t<\/i>)
    \n= (-p<\/i>)3<\/sup> – 3q<\/i>(-p<\/i>)
    \n= –p<\/i>3<\/sup> + 3p q<\/i>.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Symmetric Functions I This problem is from Alexander Karabegov. Given a quadratic polynomial x2 + p x + q with two real roots s and t, express the following functions of s and t as polynomials in p and q. s + t st s2 + t2 s3 + t3 Please submit all problem solutions […]<\/p>\n","protected":false},"author":130,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[181534],"tags":[],"class_list":["post-364","post","type-post","status-publish","format-standard","hentry","category-problem-of-the-week"],"_links":{"self":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/364","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/users\/130"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/comments?post=364"}],"version-history":[{"count":4,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/364\/revisions"}],"predecessor-version":[{"id":371,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/posts\/364\/revisions\/371"}],"wp:attachment":[{"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/media?parent=364"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/categories?post=364"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.acu.edu\/mu_sigma\/wp-json\/wp\/v2\/tags?post=364"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}