Quiz — Divisions euclidiennes de polynômes, II.pdf

Question 1 sur 10 20:00

[{"id":8928,"question":"Quel est le reste de la division euclidienne de P(x) = x³ + 2x² - x + 1 par D(x) = x² + 1 ?","option_a":"A) x - 1","option_b":"B) 2x - 2","option_c":"C) -x + 2","option_d":"D) 3x - 1","option_e":"","option_f":"","bonne_reponse":"c","explication":"Le reste doit être de degré \u003C 2. En effectuant la division, on trouve P(x) = (x+2)(x²+1) + (-x+2).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A) x - 1\", \"b\": \"B) 2x - 2\", \"c\": \"C) -x + 2\", \"d\": \"D) 3x - 1\"}}","_debug_options_count":4},{"id":8929,"question":"La division euclidienne de polynômes est toujours possible dans ℝ[x].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car ℝ[x] est un anneau euclidien : la division euclidienne y est toujours définie.","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":8930,"question":"Si P(x) = (x-1)Q(x) + R, alors R est une constante. Que vaut R si P(1) = 3 ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) 3","option_d":"D) -3","option_e":"","option_f":"","bonne_reponse":"c","explication":"En remplaçant x par 1 dans P(x) = (x-1)Q(x) + R, on obtient P(1) = R. Donc R = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) 3\", \"d\": \"D) -3\"}}","_debug_options_count":4},{"id":8931,"question":"Quel est le quotient de la division de x⁴ - 1 par x² - 1 ?","option_a":"A) x² + 1","option_b":"B) x² - 1","option_c":"C) x²","option_d":"D) x² + x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"x⁴ - 1 = (x² - 1)(x² + 1) + 0. Le quotient est donc x² + 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\": \"A) x² + 1\", \"b\": \"B) x² - 1\", \"c\": \"C) x²\", \"d\": \"D) x² + x +","_debug_options_count":4},{"id":8932,"question":"Le reste de la division euclidienne de P(x) = 2x³ + x² - 5x + 3 par D(x) = x - 2 est toujours positif.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. Le reste peut être négatif. Ici, P(2) = 2(8) + 4 - 10 + 3 = 15, donc le reste est 15 (positif), mais ce n'est pas toujours le cas.","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":8933,"question":"Si P(x) = x³ + ax² + bx + c est divisible par x² + x + 1, alors a + b + c = ?","option_a":"A) 0","option_b":"B) 1","option_c":"C) -1","option_d":"D) 2","option_e":"","option_f":"","bonne_reponse":"d","explication":"Les racines de x² + x + 1 sont complexes. En utilisant P(j) = 0 (où j est une racine cubique de l'unité), on trouve a + b + c = -2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A) 0\", \"b\": \"B) 1\", \"c\": \"C) -1\", \"d\": \"D) 2\"}}","_debug_options_count":4},{"id":8934,"question":"Quel algorithme utilise-t-on pour effectuer une division euclidienne de polynômes ?","option_a":"A) Algorithme de Horner","option_b":"B) Algorithme d'Euclide","option_c":"C) Algorithme de division polynomiale","option_d":"D) Algorithme de Gauss","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'algorithme de division polynomiale est spécifique à la division euclidienne des polynômes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"A) Algorithme de Horner\", \"b\": \"B) Algorithme d'Euclide\", \"c\": \"C","_debug_options_count":4},{"id":8935,"question":"Si P(x) = (x² + 3x + 2)Q(x) + R(x), avec deg(R) \u003C 2, alors R(x) peut s'écrire sous la forme ax + b.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Vrai, car le reste doit être de degré strictement inférieur à 2, donc de la forme ax + b.","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":8936,"question":"Quel est le reste de la division de x⁵ + 1 par x² + x + 1 ?","option_a":"A) x - 1","option_b":"B) 2x + 1","option_c":"C) -x + 2","option_d":"D) x + 1","option_e":"","option_f":"","bonne_reponse":"a","explication":"En utilisant la division ou les racines de x² + x + 1, on trouve x⁵ + 1 = (x² + x + 1)(x³ - x² + 1) + (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\": \"A) x - 1\", \"b\": \"B) 2x + 1\", \"c\": \"C) -x + 2\", \"d\": \"D) x + 1\"}}","_debug_options_count":4},{"id":8937,"question":"La division euclidienne de polynômes est-elle commutative ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Faux. La division euclidienne n'est pas commutative : P(x)\/D(x) ≠ D(x)\/P(x).","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}]