Quiz — 19-polynomes.pdf

Question 1 sur 10 20:00

[{"id":9533,"question":"Quel est le degré du polynôme P(x) = 4x³ - 2x² + 5x - 7 ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le degré d'un polynôme est l'exposant le plus élevé de la variable x. Ici, l'exposant le plus grand est 3 (dans le terme 4x³).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":9534,"question":"La factorisation de x² - 9 est :","option_a":"(x-3)(x+3)","option_b":"(x-3)²","option_c":"(x+3)²","option_d":"x(x-9)","option_e":"","option_f":"","bonne_reponse":"a","explication":"On reconnaît une différence de carrés : x² - 9 = x² - 3² = (x-3)(x+3).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"(x-3)(x+3)\", \"b\": \"(x-3)²\", \"c\": \"(x+3)²\", \"d\": \"x(x-9)\"}}","_debug_options_count":4},{"id":9535,"question":"Vrai ou Faux ? Un polynôme de degré 2 peut avoir au maximum 2 racines réelles.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un polynôme de degré 2 (trinôme du second degré) peut avoir 0, 1 ou 2 racines réelles selon son discriminant.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9536,"question":"Quel est le résultat de (2x + 3)(x - 5) ?","option_a":"2x² - 7x - 15","option_b":"2x² - 13x - 15","option_c":"2x² + 13x - 15","option_d":"2x² - 7x + 15","option_e":"","option_f":"","bonne_reponse":"b","explication":"En développant : (2x)(x) + (2x)(-5) + 3(x) + 3(-5) = 2x² - 10x + 3x - 15 = 2x² - 7x - 15. Attention à bien regrouper les termes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"2x² - 7x - 15\", \"b\": \"2x² - 13x - 15\", \"c\": \"2x² + 13x - 15\", ","_debug_options_count":4},{"id":9537,"question":"La somme de deux polynômes de degré 3 est toujours un polynôme de degré :","option_a":"3","option_b":"2","option_c":"4","option_d":"Cela dépend des coefficients","option_e":"","option_f":"","bonne_reponse":"d","explication":"La somme de deux polynômes de degré 3 peut avoir un degré inférieur si les termes de plus haut degré s'annulent (ex : (x³ + 2x) + (-x³ + 5) = 2x + 5, degré 1).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"3\", \"b\": \"2\", \"c\": \"4\", \"d\": \"Cela dépend des coefficients\"}}","_debug_options_count":4},{"id":9538,"question":"Vrai ou Faux ? Le polynôme P(x) = x³ - 4x est divisible par (x - 2).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"En appliquant le théorème de factorisation : P(2) = 2³ - 4(2) = 8 - 8 = 0. Donc (x - 2) est un facteur de P(x).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9539,"question":"Quel est le coefficient dominant du polynôme Q(x) = -5x⁴ + 3x² - 2x + 1 ?","option_a":"-5","option_b":"3","option_c":"-2","option_d":"1","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le coefficient dominant (ou coefficient directeur) est le coefficient du terme de plus haut degré, ici -5 pour le terme -5x⁴.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-5\", \"b\": \"3\", \"c\": \"-2\", \"d\": \"1\"}}","_debug_options_count":4},{"id":9540,"question":"La factorisation de x² + 6x + 9 est :","option_a":"(x+3)²","option_b":"(x-3)²","option_c":"(x+3)(x-3)","option_d":"x(x+6) + 9","option_e":"","option_f":"","bonne_reponse":"a","explication":"On reconnaît un carré parfait : x² + 6x + 9 = (x+3)².","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"(x+3)²\", \"b\": \"(x-3)²\", \"c\": \"(x+3)(x-3)\", \"d\": \"x(x+6) + 9\"}}","_debug_options_count":4},{"id":9541,"question":"Vrai ou Faux ? Tous les polynômes de degré impair ont au moins une racine réelle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un polynôme de degré impair tend vers +∞ ou -∞ à l'infini, et par le théorème des valeurs intermédiaires, il coupe nécessairement l'axe des abscisses au moins une fois.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":9542,"question":"Quel est le résultat de (x² + 2x + 1) - (3x² - x + 4) ?","option_a":"-2x² + 3x - 3","option_b":"-2x² + x - 3","option_c":"2x² + 3x - 3","option_d":"-2x² + 3x + 3","option_e":"","option_f":"","bonne_reponse":"a","explication":"En soustrayant terme à terme : (x² - 3x²) + (2x - (-x)) + (1 - 4) = -2x² + 3x - 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-2x² + 3x - 3\", \"b\": \"-2x² + x - 3\", \"c\": \"2x² + 3x - 3\", \"d\":","_debug_options_count":4}]