{"id":318,"date":"2019-08-31T13:16:09","date_gmt":"2019-08-31T11:16:09","guid":{"rendered":"http:\/\/caneri.gmollet.free.fr\/?page_id=318"},"modified":"2023-09-13T15:25:00","modified_gmt":"2023-09-13T13:25:00","slug":"activite-ii","status":"publish","type":"page","link":"https:\/\/courscaneri.ovh\/?page_id=318","title":{"rendered":"Activit\u00e9 II – Polyn\u00f4mes du second degr\u00e9 et plus"},"content":{"rendered":"

Partie I – un exemple<\/h2>\n

Soit \"P\"<\/code> le polyn\u00f4me du troisi\u00e8me degr\u00e9 d\u00e9fini par:
\n\"\forall x \in \mathbb{R}, P(x)=2x^3-14x^2+32x-24\"<\/code><\/p>\n

    \n
  1. Le nombre 2 est-il racine du polyn\u00f4me P<\/code>?<\/li>\n
  2. Trouver trois nombres r\u00e9els a<\/code>, b<\/code> et c<\/code> tels que \"\forall x \in \mathbb{R}, P(x)=(x-2)(ax^2+bx+c)\"<\/code>.<\/li>\n
  3. D\u00e9terminer le signe de\"P(x)\"<\/code> selon les valeurs de \"x\"<\/code>.<\/li>\n
  4. Factoriser \"P(x)\"<\/code> le plus possible.<\/li>\n<\/ol>\n

    Partie II – un autre exemple et une g\u00e9n\u00e9ralit\u00e9<\/h2>\n

    Soit \"Q\"<\/code> le polyn\u00f4me du troisi\u00e8me degr\u00e9 d\u00e9fini par:
    \n\"\forall x \in \mathbb{R}, Q(x)=x^3-1\"<\/code><\/p>\n

      \n
    1. Trouver une racine \u00e9vidente du polyn\u00f4me \"Q\"<\/code>.<\/li>\n
    2. Factoriser \"Q(x)\"<\/code> le plus possible.Soit \"R\"<\/code> le polyn\u00f4me du quatri\u00e8me degr\u00e9 d\u00e9fini par:
      \n\"\forall x \in \mathbb{R}, R(x)=x^4-1\"<\/code><\/li>\n
    3. Factoriser si possible \"R(x)\"<\/code>.<\/li>\n
    4. Conjecturer sur la factorisation de \"P_n\"<\/code> le polyn\u00f4me du n-i\u00e8me degr\u00e9 d\u00e9fini par:
      \n\"\forall x \in \mathbb{R}, P_n(x)=x^n-1\"<\/code><\/li>\n
    5. D\u00e9montrer cette conjecture.<\/li>\n<\/ol>\n

      Partie III – Une autre g\u00e9n\u00e9ralit\u00e9<\/h2>\n

      Soit \"a \in \mathbb{R}_+^*\"<\/code>. Soit \"Q_n\"<\/code> le polyn\u00f4me du n-i\u00e8me degr\u00e9 d\u00e9fini par:
      \n\"\forall x \in \mathbb{R}, Q_n(x)=x^n-a^n\"<\/code><\/p>\n

        \n
      1. Trouver une racine \u00e9vidente du polyn\u00f4me \"Q_n\"<\/code>.<\/li>\n
      2. Factoriser \"Q_n(x)\"<\/code>. On pourra testet les cas o\u00f9 \"n=2\"<\/code>, \"n=3\"<\/code> et \"n=4\"<\/code> et conjecturer. Ensuite on d\u00e9montrera la conjecture.<\/em><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

        Partie I – un exemple Soit le polyn\u00f4me du troisi\u00e8me degr\u00e9 d\u00e9fini par: Le nombre 2 est-il racine du polyn\u00f4me P? Trouver trois nombres r\u00e9els a, b et c tels que . D\u00e9terminer le signe de selon les valeurs de . Factoriser le plus possible. Partie II – un autre exemple et une g\u00e9n\u00e9ralit\u00e9 Soit […]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":112,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-318","page","type-page","status-publish","czr-hentry"],"_links":{"self":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/318","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=318"}],"version-history":[{"count":6,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/318\/revisions"}],"predecessor-version":[{"id":3913,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/318\/revisions\/3913"}],"up":[{"embeddable":true,"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=\/wp\/v2\/pages\/112"}],"wp:attachment":[{"href":"https:\/\/courscaneri.ovh\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=318"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}