{"id":234367,"date":"2022-06-28T17:18:14","date_gmt":"2022-06-28T15:18:14","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=234367"},"modified":"2025-09-29T13:31:34","modified_gmt":"2025-09-29T11:31:34","slug":"degre-d-un-polynome","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/degre-d-un-polynome\/","title":{"rendered":"Comment d\u00e9terminer le degr\u00e9 d’un polyn\u00f4me ?"},"content":{"rendered":"\n

Tu te demandes comment d\u00e9terminer le degr\u00e9 d’un polyn\u00f4me ?<\/strong> Trouve la r\u00e9ponse \u00e0 ta question dans cet article, ainsi que toutes les d\u00e9finitions et les th\u00e9or\u00e8mes indispensables pour tout savoir sur le degr\u00e9 d’un polyn\u00f4me. <\/p>\n\n\n\n

Et si tu rencontres encore des difficult\u00e9s, n’h\u00e9site pas \u00e0 consulter un professeur particulier d’alg\u00e8bre<\/a> pour des explications claires et pr\u00e9cises. \ud83e\uddf6<\/p>\n\n\n\n

D\u00e9finition<\/span> : Degr\u00e9 d’un polyn\u00f4me et coefficient dominant<\/strong><\/p>\n\n\n\n\nSoit \"P = a_0 + ... + a_nX^n\" avec \"a_n \ne 0\".\n
L’entier \"n\" est le degr\u00e9 de \"P\", not\u00e9 \"deg(P)\". Par convention, le degr\u00e9 du polyn\u00f4me nul est \"-\infty\". \n
Le coefficient \"a_n\" est appel\u00e9 coefficient dominant de \"P\". Lorsque \"a_n = 1\", on dit que le polyn\u00f4me \"P\" est unitaire.\n
L’ensemble des polyn\u00f4mes \u00e0 coefficients dans \"\mathbb{K}\" de degr\u00e9 au plus \"n\" est not\u00e9 \"\mathbb{K}_n[X]\".\n\n\n\n

Th\u00e9or\u00e8me<\/span><\/strong> : <\/strong><\/p>\n\n\n\n\nSoit \"P\" et \"Q\" deux polyn\u00f4mes \u00e0 coefficients dans \"\mathbb{K}\" : \n

  • \"deg(P+Q) \leq max\{deg(P), deg(Q)\}\" ; <\/li>\n
  • \"deg(PQ) = deg(P) + deg(Q)\" ; <\/li>\n
  • \"deg(P \circ Q) = deg(P) \times deg(Q)\", si \"P \ne 0\" et \"Q \ne 0\". <\/li>\n\n\n\n

    D\u00e9monstration : <\/strong><\/p>\n\n\n\n\nNotons dans un premier temps que si \"P=0\" ou \"Q=0\", les deux premi\u00e8res relations sont v\u00e9rifi\u00e9es. Supposons que \"P\" et \"Q\" sont non nuls. On pose \"P= \sum_{k=0}^n a_kX^k\" et \"Q= \sum_{k=0}^m b_kX^k\".\n

  • Supposons que \"n \geq m\", on pose pour tout \"i \in [\![m+1;n]\!]\", \"b_i=0\". \n
    On a : \"Q = \sum_{k=0}^n b_kX^k\".\n
    Par cons\u00e9quent, \"P+Q = \sum_{k=0}^n (a_k+b_k)X^k\".\n
    On en d\u00e9duit que \"deg(P+Q) \leq n\", soit \"deg(P+Q) \leq max\{deg(P), deg(Q)\}\". <\/li>\n
    \n
  • On note \"PQ = \sum_{k=0}^{n+m} c_kX^k\" avec \"c_k = \sum_{i=0}^k a_ib_{k-i}\". Int\u00e9ressons nous au coefficient dominant du polyn\u00f4me \"PQ\" :\n
    \n
    \n

      <\/span>   <\/span>\"\[<\/p>\n
    \nOn en d\u00e9duit que \"deg(PQ) = deg(P) + deg(Q)\".<\/li>\n
    \n

  • On a \"deg(P \circ Q) = deg(Q^n) = n deg(Q)\" en utilisant la formule du degr\u00e9 d’un produit. On a bien : \"deg(P \circ Q) = deg(P) \times deg(Q)\". <\/li>\n\n\n\n

    Proposition<\/span> :<\/strong> <\/p>\n\n\n\n\nSoient \"P\" et \"Q\" deux polyn\u00f4mes de \"\mathbb{K}[X]\". \n

      <\/span>   <\/span>\"\[<\/p>\nOn dit que l’anneau \"(\mathbb{K}[X],+,\times)\" est int\u00e8gre.\n\n\n\n

    D\u00e9monstration<\/strong> :<\/p>\n\n\n\n\nOn suppose que \"PQ=0\", alors \"deg(PQ)=deg(P)+deg(Q) = -\infty\". Cette \u00e9galit\u00e9 n’est possible uniquement si on a \"deg(P)= -\infty\" ou \"deg(Q)= -\infty\", ce qui prouve le r\u00e9sultat.\n\n\n\n

    Proposition<\/span> :<\/strong><\/p>\n\n\n\n\nLes \u00e9l\u00e9ments inversibles de \"\mathbb{K}[X]\" sont les polyn\u00f4mes de degr\u00e9 0, appel\u00e9s polyn\u00f4mes constants.\n\n\n\n

    D\u00e9monstration :<\/strong><\/p>\n\n\n\n\n(\"\Rightarrow\") Soit \"P\" un polyn\u00f4me inversible de \"\mathbb{K}[X]\", alors il existe \"Q \in \mathbb{K}[X]\" tel que \"PQ = 1\". On en d\u00e9duit que \"deg(P) + deg(Q) = 0\" ce qui est possible uniquement si \"deg(P)=0\" et \"deg(Q)=0\". On en d\u00e9duit bien le r\u00e9sultat.\n
    (\"\Leftarrow\") Si \"P\" est un polyn\u00f4me de degr\u00e9 0, alors il existe \"a\in \mathbb{K}^*\" tel que \"P=a\". En prenant \"Q= \frac {1}{a}\", on a bien \"PQ=1\", \"P\" est bien inversible.\n\n\n\n

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    Cet article est extrait de l’ouvrage Maths MPSI-MP2I. Tout-en-un : cours, m\u00e9thodes, entra\u00eenement et corrig\u00e9s <\/em>(\u00e9ditions Vuibert, juin 2021) <\/em>\u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n<\/div>\n<\/div>\n\n\n

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    \n 5\/5 - (2 votes) <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

    Tu te demandes comment d\u00e9terminer le degr\u00e9 d’un polyn\u00f4me ? Trouve la r\u00e9ponse \u00e0 ta question dans cet (…)<\/p>\n","protected":false},"author":158,"featured_media":244631,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":true,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-234367","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/234367","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/158"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=234367"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/234367\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/244631"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=234367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=234367"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=234367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}