{"id":87548,"date":"2021-01-15T11:17:56","date_gmt":"2021-01-15T10:17:56","guid":{"rendered":"https:\/\/groupe-reussite.fr\/?page_id=87548"},"modified":"2021-05-07T17:07:22","modified_gmt":"2021-05-07T16:07:22","slug":"cours-en-ligne-exercices-corriges-polynomes-terminale","status":"publish","type":"page","link":"https:\/\/groupe-reussite.fr\/ressources\/cours-en-ligne-exercices-corriges-polynomes-terminale\/","title":{"rendered":"Exercices corrig\u00e9s polynomes terminale"},"content":{"rendered":"
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\n\t\t\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Cours en ligne Maths en Terminale<\/a><\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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Chapitres Maths en Terminale G\u00e9n\u00e9rale<\/p>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tLes suites<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tContinuit\u00e9<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tAlgorithmique<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tFonctions exponentielles<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tFonctions logarithmes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tFonctions trigonom\u00e9triques<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tConditionnement et ind\u00e9pendance<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tD\u00e9rivation et convexit\u00e9<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tCalcul int\u00e9gral<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tFigures et \u00e9quation cart\u00e9sienne<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tD\u00e9nombrement<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tLoi binomiale<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tLois des grands nombres<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tLoi normale, intervalles, estimation<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tNombres complexes - alg\u00e8bre<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\u00c9quations polynomiales<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tG\u00e9om\u00e9trie et complexes<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tArithm\u00e9tique - Congruences<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tArithm\u00e9tique - PGCD PPCM<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tNombres Premiers et Fermat<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tMatrices<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tCha\u00eene de Markov<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tCONTACTEZ-NOUS<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t<\/div>\n
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Fonctions polyn\u00f4mes, exercices et corrig\u00e9s en Terminale<\/h1>\n<\/div>\n

R\u00e9sum\u00e9 de cours<\/a> Exercices et corrig\u00e9s<\/a><\/p>\n

Cours en ligne de Maths en Terminale<\/a><\/p>\n

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Exercice sur les \u00e9quations du second degr\u00e9 \u00e0 coefficients r\u00e9els<\/h2>\n

Question 1 :\u00a0<\/strong><\/p>\n

R\u00e9soudre dans \"\mathbb{C}\" l’\u00e9quation \"\qquad x^2 - 4 \, x + 13 + 4 \, \sqrt{2} = 0\".<\/p>\n

Question 2 :\u00a0<\/strong><\/p>\n

Trouver deux complexes de somme \u00e9gale \u00e0 1 et de produit \u00e9gal \u00e0 \"\dfrac 3 2\".<\/p>\n

Question 3 :\u00a0<\/strong><\/p>\n

Racines complexes de \"\qquad \qquad x ^4 + 7 \, x ^2 + 10 = 0\"<\/p>\n

Exercice sur la d\u00e9termination de fonctions polyn\u00f4mes<\/h2>\n

Question 1 :\u00a0<\/strong><\/p>\n

D\u00e9terminer les coefficients de la fonction polyn\u00f4me \"P : x \mapsto 6 \, x^4 + a \, x^3 + b \, x ^2 + c\, x + d\" admettant \"\sqrt{2}\", \"- \sqrt{2}\", \"-1/2\" et \"1/3\" pour racines.<\/p>\n

Question 2 :\u00a0<\/strong><\/p>\n

Trouver une fonction polyn\u00f4me \"P\" de degr\u00e9 3 admettant \"-1\" et \"4\" pour racines et telle que \"P(0) = 8\" et \"P(1) = 18\".
\nLe coefficient de \"x\" est \u00e9gal \u00e0 ?<\/p>\n

Question 3 :\u00a0<\/strong><\/p>\n

Soit \"P : x \mapsto x ^4 - 4 \, x^3 + 6 \, x ^2 - 4 \, x + 5\"
\n\u00c9crire \"P\" comme produit de deux polyn\u00f4mes de degr\u00e9 2 sachant que \"P( \textrm{i} ) = 0\".
\nEn d\u00e9duire les racines du polyn\u00f4me \"P\".<\/p>\n

Exercice th\u00e9orique sur les polyn\u00f4mes en Terminale Maths Expertes<\/h2>\n

Question 1 :<\/strong><\/p>\n

Il existe une unique fonction polyn\u00f4me \"P\" de degr\u00e9 3 et telle que \"P(0) = 0\" v\u00e9rifiant pour tout r\u00e9el \"x\",
\n\"\qquad \quad P(x + 1) - P(x) = x ^2\".<\/p>\n

Vrai ou faux ?<\/p>\n

Question 2 :<\/strong><\/p>\n

Soit \"n \in \mathbb{N}^*\".<\/p>\n

En d\u00e9duire sous forme factoris\u00e9e la valeur de \"1^2 + 2 ^2 + \cdots + n ^2\".<\/p>\n

Exercice sur l’utilisation de \"x^n - a^n\" en Terminale<\/h2>\n

Question 1 :<\/strong><\/p>\n

Soit \"n \in \mathbb{N}^*\" et \"\qquad P : x \mapsto x ^{n + 1} - x^n - x + 1\"<\/p>\n

Il existe une fonction polyn\u00f4me \"Q\" telle que pour tout r\u00e9el \"x\",<\/p>\n

\"\qquad \quad P(x) = (x - 1) ^2 \, Q(x)\"<\/p>\n

et \"Q(1) = n\".<\/p>\n

Vrai ou Faux ?<\/p>\n

Question 2 :\u00a0<\/strong><\/p>\n

Soit \"n \in \mathbb{N}^*\" et \"P : x \mapsto x ^{n + 2} - x^{n + 1}+ x ^2 -3\, x + 2\".<\/p>\n

Il existe une fonction polyn\u00f4me \"Q\" telle que pour tout r\u00e9el \"x\", \"\qquad \quad P(x) = (x - 1) ^2 \, Q(x)\"<\/p>\n

Vrai ou Faux ?<\/p>\n

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\n\t\t\t\t\tMon parcours pour<\/span>\n\t\t\t\t\n\t\t\t\t\tr\u00e9ussir en Terminale<\/span>\n\t\t\t\t<\/span>\n\t\t\t\t<\/h3>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tJe r\u00e9vise le bac en autonomie ou avec un prof\t\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/h4>\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div>\n\t\t\t\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tJ'obtiens des conseils d'orientation\t\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/h4>\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div>\n\t\t\t\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tJe m\u2019entra\u00eene sur des annales corrig\u00e9es du bac\t\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/h4>\n\t\t\t\t\n\t\t\t\t\n\t\t\t<\/div>\n\t\t\t\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tJe cherche un prof en Terminale<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Avis Google France
<\/b>\u2605\u2605\u2605\u2605\u2605<\/b><\/span> 4,8 sur 5<\/b><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\n

Correction sur les \u00e9quations du 2nd degr\u00e9 \u00e0 coefficients r\u00e9els<\/h2>\n

Question 1 :\u00a0<\/strong><\/p>\n

\"\Delta = 16 - 4 \,(13 + 4 \, \sqrt{2} = - 4\,( 9 + 4 \, \sqrt{2} )\"<\/p>\n

\"\Delta = - 4 \left ( 2^2 \, \sqrt{2} ^2 + 1 + 2 \times 2 \,\sqrt{2} \right )\"<\/p>\n

\"\Delta = - 4 \left ( 1 + 2 \, \sqrt{2} \right ) ^2\"<\/p>\n

\"\Delta = \left ( 2 \, \textrm{i} \, ( 1 + 2 \, \sqrt{2}) \right ) ^2\"<\/p>\n

L’\u00e9quation admet deux racines complexes conjugu\u00e9es :<\/p>\n

\"\displaystyle \frac {4 - 2 \, \textrm{i}\, ( 1 + 2 \, \sqrt{2}) } {2} =\boxed{ 2 - \textrm{i}\, ( 1 + 2 \, \sqrt{2})}\"<\/p>\n

et \"\boxed{\displaystyle 2 + \textrm{i}\, ( 1 + 2 \, \sqrt{2})}\".<\/p>\n

Question 2 :\u00a0<\/strong><\/p>\n

Ils sont racines de \"x^2 - S \, x + P = 0\" avec \"S = 1\" et \"P = \displaystyle \frac {3} 2\" donc de :<\/p>\n

\"\qquad \qquad x^2 - x + \dfrac 3 2 = 0\".<\/p>\n

\"\Delta = 1 - 6 = - 5 = \left ( \textrm{i} \, \sqrt{5} \right ) ^2\"<\/p>\n

Les deux racines sont<\/p>\n

\"\qquad \qquad \boxed{\dfrac {1 + \textrm{i} \, \sqrt{5} } 2}\" et \"\boxed{\dfrac {1 - \textrm{i} \, \sqrt{5} } 2}\".<\/p>\n

Question 3 :\u00a0<\/strong><\/p>\n

En posant \"y = x ^2\", on commence par r\u00e9soudre : \"y ^2 + 7 \, y + 10 = 0\" qui a pour discriminant \"\Delta = 49 - 40 = 9\"<\/p>\n

donc deux racines r\u00e9elles distinctes \"\displaystyle \frac {- 7 - 3} 2 = - 5\" et \"\displaystyle \frac {- 7 + 3 } 2 = - 2\"<\/p>\n

On \u00e9crit donc \"\qquad y ^2 + 7 \, y + 10 = (y + 2) (y + 5)\".<\/p>\n

Puis \"P(x) = (x ^2 + 2) (x ^2 + 5)\".<\/p>\n

\"P(x) = 0\" ssi \"x ^2 = - 2\" ou \"x ^2 = - 5\"<\/p>\n

ssi \"x = \pm \, \textrm{i} \, \sqrt{2}\" ou \"x = \pm \, \textrm{i} \, \sqrt{5}\".<\/p>\n

Les 4 racines complexes de \"P(x) = 0\" sont \"\boxed{\textrm{i} \, \sqrt{2} \, ,\, - \textrm{i} \, \sqrt{2}\, ,\, \textrm{i} \, \sqrt{5} , \, - \textrm{i} \, \sqrt{5}\, }\".<\/p>\n\t\t

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COURS PARTICULIERS<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Nous avons s\u00e9lectionn\u00e9 pour vous les meilleurs professeurs particuliers.<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\u00a096%\u00a0<\/span><\/strong> de r\u00e9ussite au BAC
<\/span>\u00a044% <\/span><\/strong><\/span>\u00a0de mentions Bien et Tr\u00e8s bien<\/span>
\u00a099% <\/span><\/strong><\/span>\u00a0de recommandation \u00e0 leurs amis<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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POUR ACC\u00c9L\u00c9RER MA PROGRESSION EN TERMINALE<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tCours particuliers en terminale<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Avis Google France <\/b>\u2605\u2605\u2605\u2605\u2605<\/b><\/span> 4,9 sur 5<\/b><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t

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Correction de l’exercice sur la d\u00e9termination de fonctions polyn\u00f4mes<\/h2>\n

Question 1 :\u00a0<\/strong><\/p>\n

Comme le coefficient de \"x^4\" dans \"P\" est 6 et comme on a donn\u00e9 les 4 racines de \"P\" :<\/p>\n

\"P(x) =\displaystyle 6 \times \left (x - \sqrt{2} \right ) \left (x + \sqrt{2} \right )\" \"\qquad \qquad \qquad \times \left (x + \dfrac 1 2 \right ) \left (x - \dfrac 1 3 \right )\".<\/p>\n

\"P_1(x) = \left (x - \sqrt{2} \right ) \left (x + \sqrt{2} \right )= x^2 - 2\"<\/p>\n

\"P_2(x) = \displaystyle \left (x + \frac 1 2 \right ) \left (x - \frac 1 3 \right )\"<\/p>\n

\"P_2(x) = x ^2 + \displaystyle \left ( \frac 1 2 - \frac 1 3 \right ) x - \frac 1 6\"<\/p>\n

\"P_2(x) = x ^2 + \displaystyle \frac 1 6 \, x - \frac 1 6\"<\/p>\n

donc \"6 \, P_2(x) = 6 \, x ^2 + x - 1\"<\/p>\n

\"P(x) = (x ^2 - 2) \, (6 \, x ^2 + x - 1)\"<\/p>\n

\"P(x) = 6 \, x ^4 + x^3 - x ^2 - 12 \, x ^2 - 2\, x + 2\"<\/p>\n

\"\boxed{P(x) = 6 \, x ^4 + x^3 - 13 \, x ^2 - 2\, x + 2}\".<\/p>\n

Question 2 :\u00a0<\/strong><\/p>\n

Comme \"-1\" et \"4\" sont racines de \"P\" de degr\u00e9 3, il existe une fonction polyn\u00f4me \"Q\" de degr\u00e9 \"1\" telle que pour tout r\u00e9el \"x\", \"P(x) = (x + 1) \, (x - 4)\, Q(x)\"<\/p>\n

donc il existe des r\u00e9els \"a\" et \"b\" tels que \"\quad P(x) = (x + 1) \, (x - 4) \, (a \, x + b)\".<\/p>\n

\"P(0) = -4 \, b = 8\"<\/p>\n

et \"P(1) = 2 (-3) (a + b) = 18\"<\/p>\n

ssi \"b = - 2\" et \"a + b = - 3\"<\/p>\n

ssi \"b = - 2\" et \"a = - 1\".<\/p>\n

donc \"P(x) = -(x + 1) (x - 4) (x + 2)\"<\/p>\n

\"P(x) = - (x^2 - 3 \, x - 4)(x + 2)\"<\/p>\n

\"P(x) = - x^3 + 3 \, x ^2 + 4 \, x - 2 \, x^2 + 6 \, x + 8\"<\/p>\n

\"\boxed{P(x) = - x^3 + x^2 + 10\, x + 8}\".<\/p>\n

Question 3 :\u00a0<\/strong><\/p>\n

Comme \"P(\textrm{i} ) = 0\", \"\overline{P(\textrm{i}) } = 0\" soit \"P( - \textrm{i}) = 0\" car \"P\" est \u00e0 coefficients r\u00e9els,<\/p>\n

donc \"P(x) = (x - \textrm{i}) (x + \textrm{i}) \, (a \, x ^2 + b \, x + c )\"<\/p>\n

soit \"P(x) = (x ^2 + 1) \, (a \, x ^2 + b \, x + c )\"<\/p>\n

en d\u00e9veloppant<\/p>\n

\"P(x) = a \, x ^4 + b \, x^3 + (c + a) \, x ^2 + b\, x + c\"<\/p>\n

On obtient le syst\u00e8me<\/p>\n

\"\left \{ \begin{matrix} a = 1 \\ b = - 4 \\ c + a = 6 \\ b = - 4 \\ c = 5 \end{matrix} \right.\" ssi \"\left \{ \begin{matrix} a = 1 \\ b = - 4 \\ c= 5 \end{matrix} \right.\"<\/p>\n

\"\boxed{P(x) = (x ^2 + 1) \, ( x ^2 - 4 \, x + 5 ) }\".<\/p>\n

On cherche les racines de \"\qquad Q(x) = x ^2 - 4 \, x + 5 = 0\"<\/p>\n

\"\Delta = 16 - 20 = - 4 = (2 \, \textrm{i} ) ^2\"<\/p>\n

Les racines de \"Q(x) = 0\" sont donc \"\displaystyle \frac {4 + 2 \, \textrm{i} } 2 = 2 + \textrm{i}\" et \"2 - \textrm{i}\"<\/p>\n

Les racines de \"P\" sont \"\boxed{\textrm{i},\, - \textrm{i} ,\, 2 + \textrm{i},\, 2 - \textrm{i}}\".<\/p>\n\t\t

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STAGE REVISION BAC<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Profite de tes vacances pour gagner des points au bac.<\/h4>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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96%<\/span><\/strong> \u00a0de r\u00e9ussite au BAC<\/span>
44%<\/strong><\/span> de mentions Bien et Tr\u00e8s bien<\/span>
99%<\/span><\/strong> de recommandation \u00e0 leurs amis<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tStage terminale<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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Avis Google France <\/b>\u2605\u2605\u2605\u2605\u2605<\/b><\/span> 4,9 sur 5<\/b><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\n

Correction de l’exercice th\u00e9orique sur les polyn\u00f4mes en Terminale<\/h2>\n

Question 1 :\u00a0<\/strong><\/p>\n

Vrai<\/p>\n

On cherche donc des r\u00e9els \"a\", \"b\" et \"c\" tels que \"P(x) = a\, x ^3 + b \, x^2 + c\, x\".<\/p>\n

\"P(x + 1) =\" \"\; \; a\, (x + 1) ^3 + b (x + 1) ^2 + c(x + 1)\"<\/p>\n

On rappelle que \"(x + 1) ^3 = x^3 + 3 \, x ^2 + 3 \, x + 1\"<\/p>\n

\"P(x + 1) = a \, x^3 + (3 \, a + b) \, x ^2\" \"\qquad +\, (3 \, a + 2 \, b + c) \, x + a + b + c\"<\/p>\n

\"P(x + 1) - P(x) =\" \"\qquad \quad 3 \, a \, x ^2 + (3 \, a + 2 \, b)\, x + a + b + c\"<\/p>\n

Pour tout \"x\", \"P(x + 1) - P(x) = x^2\" ssi<\/p>\n

\"\left \{ \begin{matrix} 3 \, a = 1 \\ 3 \, a + 2 \, b = 0 \\ a + b + c = 0 \end{matrix} \right.\" ssi \"\left \{ \begin{matrix} a = 1/3 \\ 2 \, b = - 1 \\ c = -<\/p>\n

donc \"\boxed{P(x) = \displaystyle\frac 1 3 \, x ^3 - \frac 1 2 \, x^2 + \frac 1 6 \, x}\".<\/p>\n

Question 2 :\u00a0<\/strong><\/p>\n

Soit \"S _ n = 1^2 + 2 ^2 + \cdots + n ^2\".<\/p>\n

On \u00e9crit la relation \"P(x + 1) - P(x) = x ^2\" en prenant comme valeurs successives de \"x\" : \"0 , 1 ,\, 2 , \cdots ,\, n\"<\/p>\n

\"P(1) - P(0) = 0^2\"<\/p>\n

\"P(2) - P(1) = 1^2\"<\/p>\n

\"P(3) - P(2) = 2^2\"<\/p>\n

\"\vdots\"<\/p>\n

\"P(n ) - P(n - 2) = (n - 1) ^2\"<\/p>\n

\"P(n+ 1 ) - P(n)^2 = n ^2\"<\/p>\n

\"-----------------\"<\/p>\n

Puis en sommant ces relations, apr\u00e8s simplifications, il ne reste que<\/p>\n

\"P(n + 1) - P(0) = S_n\" avec \"P(0) = 0\"<\/p>\n

\"S_n = \displaystyle \frac 1 3 (n + 1) ^3 - \frac 1 2\, (n + 1)^2 + \frac 1 6 \, (n + 1)\"<\/p>\n

On factorise \"\displaystyle \frac {n + 1} 6\"<\/p>\n

\"S_n = \displaystyle \frac {n + 1} 6 \left ( 2 \, (n + 1) ^2 -3\, ( n + 1) + 1 \right )\"<\/p>\n

\"S_n = \displaystyle \frac {n + 1} 6 \left ( 2\, n ^2 + 4 \, n + 2 - 3 \, n - 2 \right )\"<\/p>\n

\"S_n = \displaystyle \frac {n + 1} 6 \left ( 2\, n ^2 + n \right )\"<\/p>\n

\"\boxed {S_n = \displaystyle \frac {n(n + 1) (2 \, n + 1)} 6 }\".<\/p>\n

Correction d’exercice sur l’utilisation de \"x^n - a^n\" en Terminale<\/h2>\n

Question 1 :\u00a0<\/strong><\/p>\n

Vrai<\/p>\n

\"P(x) = x^n (x - 1) - (x - 1)\"<\/p>\n

\"P(x) = (x - 1) \, (x ^n - 1)\"<\/p>\n

Comme \"x ^n - 1 = (x - 1) \, Q(x)\" avec \"\boxed{Q(x) = x ^{n - 1} + x ^ {n - 2} + \cdots + x + 1}\"<\/p>\n

\"P(x) = (x - 1) ^2 \, Q(x)\".<\/p>\n

Il est imm\u00e9diat que \"Q(1) = n\".<\/p>\n

Question 2 :\u00a0<\/strong><\/p>\n

Vrai<\/p>\n

\"P(x) = x ^{n + 2} - x^{n + 1}+ x ^2 -3\, x + 2\"<\/p>\n

\"\bullet\" 1 est racine \u00e9vidente de \"\qquad \qquad x ^2 -3\, x + 2 = 0\",<\/p>\n

l’autre racine est \u00e9gale au produit \"2\" des racines donc<\/p>\n

\"\qquad x^2 -3\, x + 2 = (x - 1) (x - 2)\".<\/p>\n

\"\bullet\" Puis \"P(x) = x^ {n + 1} \, (x - 1) + (x - 1) (x - 2)\"<\/p>\n

\"P(x) = (x - 1) (x ^{n + 1} + x - 2)\"<\/p>\n

\"\bullet\" \"R(x) = x ^{n + 1} + x - 2\"<\/p>\n

\"R(1) = 0\", donc on peut factoriser \"x - 1\"<\/p>\n

\"R(x) = x ^{n + 1 } - 1 + (x - 1)\"<\/p>\n

comme \"x ^{n + 1 } - 1 = (x - 1) \, R_1(x)\" avec \"R_1(x) = x ^n + x ^{n - 1} + \cdots + x + 1\"<\/p>\n

\"R(x) = (x - 1) (R_1(x) + 1)\"<\/p>\n

donc \"\boxed{P(x) = (x - 1) ^2 Q(x)}\" avec \"Q(x) = R_1(x) + 1\"<\/p>\n

\"\boxed{Q(x) = x ^n + x ^{n - 1} + \cdots + x + 2}\".<\/p>\n

Profitez aussi des autres cours en ligne avec exercices corrig\u00e9s pour vous entra\u00eener sur les notions fondamentales de maths au programme de maths expertes en Terminale :<\/p>\n