def creerPolydegre2(nb_racines=2, rac_radical=True, rac_quotient=False):
if nb_racines == 2:
redo = True
while redo:
a = randrange(1, 4) * (-1) ** randrange(2)
alpha = randrange(1, 10) * (-1) ** randrange(2)
beta = randrange(1, 10)
gamma = [1, randrange(1, 6)][rac_radical]
if rac_quotient:
den = randrange(2, 6)
while pgcd(alpha, den) != 1 or pgcd(beta, den) != 1:
den = randrange(2, 6)
alpha = Fraction(alpha, den)
beta = Fraction(beta, den)
b = -2 * alpha * a
c = a * (alpha ** 2 - gamma * beta ** 2)
if abs(c) <= 10 and c != 0 and not factoriser(repr(Polynome([[a, 2], [b, 1], [c, 0]]))): redo = False
if c.denominator != 1:
c = 'Fraction(%s, %s)' % (c.numerator, c.denominator)
else:
c = c.numerator
if b.denominator != 1:
b = 'Fraction(%s, %s)' % (b.numerator, b.denominator)
else:
b = b.numerator
return Polynome([[a, 2], [b, 1], [c, 0]])
elif nb_racines == 1:
a, b = valeur_alea(-9, 9), valeur_alea(-9, 9)
return Polynome([[a ** 2, 2], [2 * a * b, 1], [b ** 2, 0]])
else:
pol = [[valeur_alea(-9, 9), 2 - dummy] for dummy in range(3)]
while pol[1][0] ** 2 - 4 * pol[0][0] * pol[2][0] >= 0:
pol = [[valeur_alea(-9, 9), 2 - dummy] for dummy in range(3)]
return Polynome(pol)
def creerPolydegre2(nb_racines=2, rac_radical=True, rac_quotient=False):
if nb_racines == 2:
redo = True
while redo:
a = randrange(1, 4) * (-1) ** randrange(2)
alpha = randrange(1, 10) * (-1) ** randrange(2)
beta = randrange(1, 10)
gamma = [1, randrange(1, 6)][rac_radical]
if rac_quotient:
den = randrange(2, 6)
while pgcd(alpha, den) != 1 or pgcd(beta, den) != 1:
den = randrange(2, 6)
alpha = Fraction(alpha, den)
beta = Fraction(beta, den)
b = -2 * alpha * a
c = a * (alpha ** 2 - gamma * beta ** 2)
if abs(c) <= 10 and c != 0 and not factoriser(repr(Polynome([[a, 2], [b, 1], [c, 0]]))): redo = False
if c.denominator != 1:
c = 'Fraction(%s, %s)' % (c.numerator, c.denominator)
else:
c = c.numerator
if b.denominator != 1:
b = 'Fraction(%s, %s)' % (b.numerator, b.denominator)
else:
b = b.numerator
return Polynome([[a, 2], [b, 1], [c, 0]])
elif nb_racines == 1:
a, b = valeur_alea(-9, 9), valeur_alea(-9, 9)
return Polynome([[a ** 2, 2], [2 * a * b, 1], [b ** 2, 0]])
else:
pol = [[valeur_alea(-9, 9), 2 - dummy] for dummy in range(3)]
while pol[1][0] ** 2 - 4 * pol[0][0] * pol[2][0] >= 0:
pol = [[valeur_alea(-9, 9), 2 - dummy] for dummy in range(3)]
return Polynome(pol)
def factorisation():
"""Génère un exercice de factorisation utilisant les identités remarquables ou
la distributivité
"""
l = [randrange(1, 11) for dummy in range(21)]
diff = [True, False, False]
shuffle(diff)
exo = [id_rem1, id_rem2]
lexo = [exo[randrange(2)](l[0], l[1])]
lexo.append(id_rem3(l[2], l[3]))
lexo.append(id_rem3bis(l[4], l[5], l[6]))
lexo.append(facteur_commun1(l[7:13], diff=diff.pop()))
shuffle(lexo)
exo = [facteur_commun2, facteur_commun3]
shuffle(exo)
lexo.append(exo[0](l[13:17], diff=diff.pop()))
lexo.append(exo[1](l[17:21], diff=diff.pop()))
exo = [
"\\exercice",
u"Factoriser chacune des expressions littérales suivantes :"
]
exo.append("\\begin{multicols}{2}")
cor = [
"\\exercice*",
u"Factoriser chacune des expressions littérales suivantes :"
]
cor.append("\\begin{multicols}{2}")
for i in range(len(lexo)):
p = [lexo[i]]
while True:
fact = factoriser(p[-1])
if fact:
p.append(fact)
else:
break
p = Priorites3.texify(
[Priorites3.splitting(p[j]) for j in range(len(p))])
cor.append('\\\\\n'.join(
['$%s=%s$' % (chr(i + 65), p[j]) for j in range(len(p) - 1)]))
cor.append('\\\\')
cor.append('\\fbox{$%s=%s$}\\\\\n' % (chr(i + 65), p[-1]))
exo.append('\\\\\n'.join([
'$%s=%s$' %
(chr(i + 65), Priorites3.texify([Priorites3.splitting(lexo[i])])[0])
for i in range(len(lexo))
]))
exo.append("\\end{multicols}")
cor.append("\\end{multicols}")
return exo, cor
def factorisation():
"""Génère un exercice de factorisation utilisant les identités remarquables ou
la distributivité
"""
l = [randrange(1, 11) for dummy in range(21)]
diff = [True, False, False]
shuffle(diff)
exo = [id_rem1, id_rem2]
lexo = [exo[randrange(2)](l[0], l[1])]
lexo.append(id_rem3(l[2], l[3]))
lexo.append(id_rem3bis(l[4], l[5], l[6]))
lexo.append(facteur_commun1(l[7:13], diff=diff.pop()))
shuffle(lexo)
exo = [facteur_commun2, facteur_commun3]
shuffle(exo)
lexo.append(exo[0](l[13:17], diff=diff.pop()))
lexo.append(exo[1](l[17:21], diff=diff.pop()))
exo = ["\\exercice", u"Factoriser chacune des expressions littérales suivantes :"]
exo.append("\\begin{multicols}{2}")
cor = ["\\exercice*", u"Factoriser chacune des expressions littérales suivantes :"]
cor.append("\\begin{multicols}{2}")
for i in range(len(lexo)):
p = [lexo[i]]
while True:
fact = factoriser(p[-1])
if fact:
p.append(fact)
else: break
p = Priorites3.texify([Priorites3.splitting(p[j]) for j in range(len(p))])
cor.append('\\\\\n'.join(['$%s=%s$' % (chr(i + 65), p[j]) for j in range(len(p) - 1)]))
cor.append('\\\\')
cor.append('\\fbox{$%s=%s$}\\\\\n' % (chr(i + 65), p[-1]))
exo.append('\\\\\n'.join(['$%s=%s$' % (chr(i + 65), Priorites3.texify([Priorites3.splitting(lexo[i])])[0]) for i in range(len(lexo))]))
exo.append("\\end{multicols}")
cor.append("\\end{multicols}")
return exo, cor
def tex_answer(self):
exo = [r'\exercice*']
exo.append(_(u'Déterminer les racines des polynômes :\\par'))
noms = [r'P\,(x) &= ', r'Q\,(x) &= ', r'R\,(x) &= ']
r = ''
question = [[], [], []]
for i in range(3):
p = []
m = Polynome(list(self.exercice[i])).ordonne()
if factoriser('%r' % Polynome(m)):
p = [factoriser('%r' % Polynome(m))]
while factoriser(p[-1]):
p.append(factoriser(p[-1]))
if p and eval(Priorites3.splitting(p[-1])[0]).degre() > 0:
tmp = Priorites3.texify([Priorites3.splitting(p[j]) for j in range(len(p))])
question[i].append('{$\\! \\begin{aligned}')
question[i].append(noms[i] + str(Polynome(m, 'x')) + r'\\')
question[i].append('\\\\\n'.join(['&=%s' % (tmp[j]) for j in range(len(tmp))]))
question[i].append(r'\end{aligned}$}\par')
lp = Priorites3.splitting(p[-1])
racines = []
for e in lp:
if e[:9] == 'Polynome(':
e = eval(e)
if len(e) == 2:
racines.append(str(Fraction(-e[1][0], e[0][0]).simplifie()))
else:
racines.append('0')
if len(racines) > 1:
question[i].append(_(u'\\underline{Les racines de $%s$ sont }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), '$}\\underline{ et }\\fbox{$'.join(racines)))
elif len(racines) == 1:
question[i].append(_(u'\\underline{L\'unique racine de $%s$ est }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), racines[0]))
elif len(m) == 2 and m[0][1] == 2 and m[1][1] == 0 and m[0][0] * m[1][0] > 0:
question[i].append('$' + noms[i] + str(Polynome(m, 'x')) + r'$\par')
question[i][-1] = question[i][-1].replace('&', '')
if m[1][0] > 0: question[i].append('$' + noms[i][:7] + ' \\geqslant %r$' % m[1][0])
else: question[i].append('$' + noms[i][:7] + ' \\leqslant %r$' % m[1][0])
question[i].append(_(u'car un carré est toujours positif.\\par\n\\underline{$%s$ n\'a donc pas de racine.}') % (noms[i].rstrip(r' &= ')))
else:
question[i].append('$' + noms[i] + str(Polynome(m, 'x')) + r'\quad$')
question[i][-1] = question[i][-1].replace('&', '')
question[i].append(_(u'On calcule le discriminant de $%s$ avec $a=%s$, $b=%s$ et $c=%s$ :\\par\\medskip') % (noms[i].rstrip(r' &= '), m[0][0], m[1][0], m[2][0]))
question[i].append(r'\begin{tabularx}{\linewidth}[t]{XXX}')
question[i].append(r'{$\! \begin{aligned}')
if m[1][0]>0:
sol = [[str(m[1][0]), '**', '2', '-', '4', '*', str(m[0][0]), '*', str(m[2][0])]]
sol.extend(Priorites3.priorites('%s**2-4*%s*%s' % (m[1][0], m[0][0], m[2][0])))
else:
sol = [['(', str(m[1][0]), ')', '**', '2', '-', '4', '*', str(m[0][0]), '*', str(m[2][0])]]
sol.extend(Priorites3.priorites('(%s)**2-4*%s*%s' % (m[1][0], m[0][0], m[2][0])))
solTeX = Priorites3.texify(sol)
for s in solTeX:
question[i].append(u'\\Delta &= %s\\\\' % s)
question[i].append(r'\end{aligned}$}')
question[i].append(r'&')
question[i].append(r'{$\! \begin{aligned}')
delta = sol[-1][0]
print(sol)
sol = [['Fraction(SquareRoot([[%s, None], [-1, %s]]),\'2*%s\')' % (-m[1][0], delta, m[0][0])]]
sol.extend(Priorites3.priorites(sol[0][0]))
sol = Priorites3.texify(sol)
for s in sol:
question[i].append(u'x_1 &= %s\\\\' % s)
racines = [sol[-1]]
question[i].append(r'\end{aligned}$}')
question[i].append(r'&')
question[i].append(r'{$\! \begin{aligned}')
sol = [['Fraction(SquareRoot([[%s, None], [1, %s]]),\'2*%s\')' % (-m[1][0], delta, m[0][0])]]
sol.extend(Priorites3.priorites(sol[0][0]))
sol = Priorites3.texify(sol)
for s in sol:
question[i].append(u'x_2 &= %s\\\\' % s)
racines.append(sol[-1])
question[i].append(r'\end{aligned}$}')
question[i].append(r'\end{tabularx}\par')
question[i].append(_(u'\\underline{Les racines de $%s$ sont }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), _('$}\\underline{ et }\\fbox{$').join(racines)))
if i == 1: question.append(question[1])
if len(question) == 4:
question.pop(1)
if question[0][0][-6:] == r'\quad$':
question[1].insert(0, r'\par\medskip\begin{tabularx}{\linewidth}[t]{XX}')
question[2].insert(0, r'&')
question[2].append(r'\end{tabularx}\par\medskip')
else:
question[0].insert(0, r'\begin{tabularx}{\linewidth}[t]{XX}')
question[1].insert(0, r'&')
question[1].append(r'\end{tabularx}\par\medskip')
for i in range(3): exo.extend(question[i])
return exo
def tex_answer(self):
exo = [r'\exercice*']
exo.append(_(u'Déterminer les racines des polynômes :\\par'))
noms = [r'P\,(x) &= ', r'Q\,(x) &= ', r'R\,(x) &= ']
r = ''
question = [[], [], []]
for i in range(3):
p = []
m = Polynome(list(self.exercice[i])).ordonne()
if factoriser('%r' % Polynome(m)):
p = [factoriser('%r' % Polynome(m))]
while factoriser(p[-1]):
p.append(factoriser(p[-1]))
if p and eval(Priorites3.splitting(p[-1])[0]).degre() > 0:
tmp = Priorites3.texify([Priorites3.splitting(p[j]) for j in range(len(p))])
question[i].append('{$\\! \\begin{aligned}')
question[i].append(noms[i] + str(Polynome(m, 'x')) + r'\\')
question[i].append('\\\\\n'.join(['&=%s' % (tmp[j]) for j in range(len(tmp))]))
question[i].append(r'\end{aligned}$}\par')
lp = Priorites3.splitting(p[-1])
racines = []
for e in lp:
if e[:9] == 'Polynome(':
e = eval(e)
if len(e) == 2:
racines.append(str(Fraction(-e[1][0], e[0][0]).simplifie()))
else:
racines.append('0')
if len(racines) > 1:
question[i].append(_(u'\\underline{Les racines de $%s$ sont }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), '$}\\underline{ et }\\fbox{$'.join(racines)))
elif len(racines) == 1:
question[i].append(_(u'\\underline{L\'unique racine de $%s$ est }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), racines[0]))
elif len(m) == 2 and m[0][1] == 2 and m[1][1] == 0 and m[0][0] * m[1][0] > 0:
question[i].append('$' + noms[i] + str(Polynome(m, 'x')) + r'$\par')
question[i][-1] = question[i][-1].replace('&', '')
if m[1][0] > 0: question[i].append('$' + noms[i][:7] + ' \\ge %r$' % m[1][0])
else: question[i].append('$' + noms[i][:7] + ' \\le %r$' % m[1][0])
question[i].append(_(u'car un carré est toujours positif.\\par\n\\underline{$%s$ n\'a donc pas de racine.}') % (noms[i].rstrip(r' &= ')))
else:
question[i].append('$' + noms[i] + str(Polynome(m, 'x')) + r'\quad$')
question[i][-1] = question[i][-1].replace('&', '')
question[i].append(_(u'On calcule le discriminant de $%s$ avec $a=%s$, $b=%s$ et $c=%s$ :\\par\\medskip') % (noms[i].rstrip(r' &= '), m[0][0], m[1][0], m[2][0]))
question[i].append(r'\begin{tabularx}{\linewidth}[t]{XXX}')
question[i].append(r'{$\! \begin{aligned}')
sol = [[str(m[1][0]), '**', '2', '-', '4', '*', str(m[0][0]), '*', str(m[2][0])]]
sol.extend(Priorites3.priorites('%s**2-4*%s*%s' % (m[1][0], m[0][0], m[2][0])))
solTeX = Priorites3.texify(sol)
for s in solTeX:
question[i].append(u'\\Delta &= %s\\\\' % s)
question[i].append(r'\end{aligned}$}')
question[i].append(r'&')
question[i].append(r'{$\! \begin{aligned}')
delta = sol[-1][0]
sol = [['Fraction(SquareRoot([[%s, None], [-1, %s]]),\'2*%s\')' % (-m[1][0], delta, m[0][0])]]
sol.extend(Priorites3.priorites(sol[0][0]))
sol = Priorites3.texify(sol)
for s in sol:
question[i].append(u'x_1 &= %s\\\\' % s)
racines = [sol[-1]]
question[i].append(r'\end{aligned}$}')
question[i].append(r'&')
question[i].append(r'{$\! \begin{aligned}')
sol = [['Fraction(SquareRoot([[%s, None], [1, %s]]),\'2*%s\')' % (-m[1][0], delta, m[0][0])]]
sol.extend(Priorites3.priorites(sol[0][0]))
sol = Priorites3.texify(sol)
for s in sol:
question[i].append(u'x_2 &= %s\\\\' % s)
racines.append(sol[-1])
question[i].append(r'\end{aligned}$}')
question[i].append(r'\end{tabularx}\par')
question[i].append(_(u'\\underline{Les racines de $%s$ sont }\\fbox{$%s$}') % (noms[i].rstrip(r' &= '), _('$}\\underline{ et }\\fbox{$').join(racines)))
if i == 1: question.append(question[1])
if len(question) == 4:
question.pop(1)
if question[0][0][-6:] == r'\quad$':
question[1].insert(0, r'\par\medskip\begin{tabularx}{\linewidth}[t]{XX}')
question[2].insert(0, r'&')
question[2].append(r'\end{tabularx}\par\medskip')
else:
question[0].insert(0, r'\begin{tabularx}{\linewidth}[t]{XX}')
question[1].insert(0, r'&')
question[1].append(r'\end{tabularx}\par\medskip')
for i in range(3): exo.extend(question[i])
return exo
