Quiz interactif : Polynômes, racines et multiplicités — Bac Sciences

Question 1 sur 10 10:00

[{"id":56171,"question":"Quelle est la racine évidente du polynôme P(x) = x³ - 2x² - x + 2 ?","option_a":"A. x = 0","option_b":"B. x = 1","option_c":"C. x = -1","option_d":"D. x = 2","option_e":"","option_f":"","bonne_reponse":"B","explication":"En testant x = 1, on obtient P(1) = 1 - 2 - 1 + 2 = 0. C'est donc une racine évidente.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56172,"question":"Si α est une racine double d'un polynôme P(x), alors P'(α) = 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Si α est une racine double, alors P(x) = (x - α)²Q(x). En dérivant, P'(x) = 2(x - α)Q(x) + (x - α)²Q'(x), donc P'(α) = 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56173,"question":"Quel est le degré minimal d'un polynôme ayant exactement 3 racines distinctes ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"C","explication":"Un polynôme de degré n peut avoir au plus n racines distinctes. Pour 3 racines distinctes, le degré minimal est 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56174,"question":"Si P(x) = (x - 2)(x² + 1), combien de racines réelles possède-t-il ?","option_a":"A. 0","option_b":"B. 1","option_c":"C. 2","option_d":"D. 3","option_e":"","option_f":"","bonne_reponse":"B","explication":"Le polynôme a une racine réelle (x = 2) et deux racines complexes (x = i et x = -i).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56175,"question":"La multiplicité d'une racine α est toujours un entier naturel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Par définition, la multiplicité d'une racine est un entier naturel (0, 1, 2, ...).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56176,"question":"Quel théorème permet de factoriser un polynôme à partir de ses racines ?","option_a":"A. Théorème de Pythagore","option_b":"B. Théorème de Rolle","option_c":"C. Théorème de factorisation","option_d":"D. Théorème des valeurs intermédiaires","option_e":"","option_f":"","bonne_reponse":"C","explication":"Le théorème de factorisation stipule que si α est une racine d'un polynôme P, alors (x - α) divise P(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56177,"question":"Si P(x) = x⁴ - 5x² + 4, combien de racines réelles distinctes possède-t-il ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"D","explication":"En factorisant, P(x) = (x² - 1)(x² - 4) = (x - 1)(x + 1)(x - 2)(x + 2). Il y a donc 4 racines réelles distinctes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56178,"question":"Si P(α) = 0 et P'(α) ≠ 0, alors α est une racine simple.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Si P(α) = 0 et P'(α) ≠ 0, alors α est une racine simple (multiplicité 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56179,"question":"Quel est le lien entre les racines d'un polynôme et son graphe ?","option_a":"A. Les racines sont les points d'intersection avec l'axe des ordonnées","option_b":"B. Les racines sont les points d'intersection avec l'axe des abscisses","option_c":"C. Les racines sont les points de maximum local","option_d":"D. Les racines sont les points de minimum local","option_e":"","option_f":"","bonne_reponse":"B","explication":"Les racines d'un polynôme sont les points où la courbe coupe l'axe des abscisses (P(x) = 0).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":56180,"question":"Si P(x) = x³ - 3x + 2, quelle est la multiplicité de la racine x = 1 ?","option_a":"A. 1","option_b":"B. 2","option_c":"C. 3","option_d":"D. 4","option_e":"","option_f":"","bonne_reponse":"B","explication":"En factorisant, P(x) = (x - 1)²(x + 2). La racine x = 1 a donc une multiplicité de 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]