Quiz — Polynômes- relations coefficients-racines.pdf

Question 1 sur 10 20:00

[{"id":31310,"question":"Soit un polynôme P(x) = x² - 5x + 6. Quelles sont ses racines ?","option_a":"x=2 et x=3","option_b":"x=1 et x=6","option_c":"x=-2 et x=-3","option_d":"x=0 et x=5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le polynôme se factorise en (x-2)(x-3), donc ses racines sont x=2 et 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=2 et x=3\", \"b\": \"x=1 et x=6\", \"c\": \"x=-2 et x=-3\", \"d\": \"x=0 et","_debug_options_count":4},{"id":31311,"question":"Pour un polynôme du second degré ax² + bx + c, la somme des racines est égale à :","option_a":"-b\/a","option_b":"b\/a","option_c":"-c\/a","option_d":"c\/a","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après les formules de Viète, la somme des racines d'un polynôme du second degré est -b\/a.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-b\/a\", \"b\": \"b\/a\", \"c\": \"-c\/a\", \"d\": \"c\/a\"}}","_debug_options_count":4},{"id":31312,"question":"Un polynôme de degré 3 a trois racines réelles. La somme de ses coefficients est toujours égale à 0.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"La somme des coefficients est égale à P(1), qui n'est pas nécessairement nulle. Par exemple, P(x) = (x-1)(x-2)(x-3) a une somme de coefficients égale à -6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":31313,"question":"Si un polynôme P(x) = x³ + px² + qx + r a pour racines 1, 2 et 3, alors p est égal à :","option_a":"6","option_b":"-6","option_c":"11","option_d":"-11","option_e":"","option_f":"","bonne_reponse":"b","explication":"D'après les formules de Viète, p = -(1 + 2 + 3) = -6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"6\", \"b\": \"-6\", \"c\": \"11\", \"d\": \"-11\"}}","_debug_options_count":4},{"id":31314,"question":"Le produit des racines d'un polynôme du second degré ax² + bx + c est égal à :","option_a":"-b\/a","option_b":"b\/a","option_c":"-c\/a","option_d":"c\/a","option_e":"","option_f":"","bonne_reponse":"d","explication":"D'après les formules de Viète, le produit des racines est c\/a.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"-b\/a\", \"b\": \"b\/a\", \"c\": \"-c\/a\", \"d\": \"c\/a\"}}","_debug_options_count":4},{"id":31315,"question":"Un polynôme de degré 4 a pour racines 1, -1, 2 et -2. Son coefficient dominant est 1. Quel est son expression développée ?","option_a":"x⁴ - 5x² + 4","option_b":"x⁴ + 5x² + 4","option_c":"x⁴ - 4x² + 5","option_d":"x⁴ + 4x² - 5","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le polynôme se factorise en (x-1)(x+1)(x-2)(x+2) = (x²-1)(x²-4) = x⁴ - 5x² + 4.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"x⁴ - 5x² + 4\", \"b\": \"x⁴ + 5x² + 4\", \"c\": \"x⁴ - 4x² + 5\",","_debug_options_count":4},{"id":31316,"question":"Si un polynôme P(x) a pour racines 0 et 1, alors il est divisible par x(x-1).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Un polynôme ayant pour racines 0 et 1 est divisible par x et (x-1), donc par x(x-1).","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":31317,"question":"Pour un polynôme P(x) = x³ + 2x² - 5x - 6, la somme des racines est égale à :","option_a":"-2","option_b":"2","option_c":"5","option_d":"-5","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après les formules de Viète, la somme des racines est -2 (coefficient de x²).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-2\", \"b\": \"2\", \"c\": \"5\", \"d\": \"-5\"}}","_debug_options_count":4},{"id":31318,"question":"Un polynôme de degré 3 a pour racines 1, 2 et k. Si la somme des coefficients est 0, alors k est égal à :","option_a":"-3","option_b":"3","option_c":"-6","option_d":"6","option_e":"","option_f":"","bonne_reponse":"a","explication":"La somme des coefficients est P(1) = 0. Le polynôme est (x-1)(x-2)(x-k). P(1) = 0 implique (1-1)(1-2)(1-k) = 0, ce qui est toujours vrai. En revanche, la somme des racines est 1+2+k = -2 (coefficient de x²), donc k = -5.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-3\", \"b\": \"3\", \"c\": \"-6\", \"d\": \"6\"}}","_debug_options_count":4},{"id":31319,"question":"Le polynôme P(x) = x⁴ - 10x³ + 35x² - 50x + 24 a pour racines 1, 2, 3 et 4. Quel est le coefficient de x³ ?","option_a":"-10","option_b":"10","option_c":"-35","option_d":"35","option_e":"","option_f":"","bonne_reponse":"a","explication":"D'après les formules de Viète, le coefficient de x³ est -(1+2+3+4) = -10.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"-10\", \"b\": \"10\", \"c\": \"-35\", \"d\": \"35\"}}","_debug_options_count":4}]