Maîtrisez les racines et multiplicités des polynômes

Question 1 sur 10 10:00

[{"id":42456,"question":"Soit le polynôme P(x) = (x-2)²(x+1). Quelle est la multiplicité de la racine x=2 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"B","explication":"La racine x=2 apparaît deux fois dans la factorisation, donc sa multiplicité est 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42457,"question":"Un polynôme de degré 3 peut-il avoir 4 racines distinctes ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Un polynôme de degré n a au plus n racines distinctes. Ici, 3 \u003C 4, donc c'est impossible.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42458,"question":"Si P(a) = 0 et P'(a) ≠ 0, alors la racine a est de multiplicité :","option_a":"0","option_b":"1","option_c":"2","option_d":"3","option_e":"","option_f":"","bonne_reponse":"B","explication":"Si P(a)=0 et P'(a)≠0, la racine a est simple (multiplicité 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42459,"question":"Le polynôme P(x) = x³ - 6x² + 11x - 6 a pour racines 1, 2 et 3. Quelle est sa factorisation ?","option_a":"(x-1)(x-2)(x-3)","option_b":"(x+1)(x+2)(x+3)","option_c":"(x-1)²(x-2)","option_d":"(x-1)(x-2)²","option_e":"","option_f":"","bonne_reponse":"A","explication":"Les racines sont 1, 2 et 3, donc la factorisation est (x-1)(x-2)(x-3).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42460,"question":"Si un polynôme P(x) a une racine double en x=1, alors P(1) = 0 et P'(1) = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Une racine double signifie que P(1)=0 et P'(1)=0, car la dérivée s'annule aussi en cette racine.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42461,"question":"Quel théorème permet de relier les racines d'un polynôme à ses coefficients ?","option_a":"Théorème de Pythagore","option_b":"Théorème de Rolle","option_c":"Formules de Viète","option_d":"Théorème de Gauss","option_e":"","option_f":"","bonne_reponse":"C","explication":"Les formules de Viète relient les coefficients d'un polynôme à la somme et au produit de ses racines.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42462,"question":"Un polynôme de degré 4 peut-il avoir une racine de multiplicité 3 ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, car 3 ≤ 4. Par exemple, P(x) = (x-1)³(x-2) a une racine triple en x=1.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42463,"question":"Si P(x) = x⁴ - 5x³ + 6x², quelle est la multiplicité de la racine x=0 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"B","explication":"P(x) = x²(x² - 5x + 6), donc x=0 est une racine double (multiplicité 2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42464,"question":"Le polynôme P(x) = (x-1)(x-1)(x+2) a une racine double en x=1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, car (x-1) apparaît deux fois dans la factorisation, ce qui donne une multiplicité de 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":42465,"question":"Quelle est la somme des racines du polynôme P(x) = 2x³ - 3x² + x - 5 ?","option_a":"1.5","option_b":"-1.5","option_c":"3","option_d":"-3","option_e":"","option_f":"","bonne_reponse":"A","explication":"D'après les formules de Viète, la somme des racines est -(-3)\/2 = 1.5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]