def resolution(self, m, pre=[], post=[]):
sgn = '+-'[m[1][0] < 0]
if isinstance(m[1][0], Fraction):
b = Priorites3.priorites(abs(m[1][0]) / 2)[-1][0]
elif m[1][0] % 2:
b = 'Fraction(%s, 2)' % abs(m[1][0])
else:
b = abs(m[1][0]) / 2
fc = ['Polynome("%sx%s%s")' % (m[0][0], sgn, b), '**', '2']
reste = ['-']
if m[2][0] > 0 or isinstance(m[2][0], Fraction):
reste.extend(Priorites3.splitting('%s**2+%r' % (b, m[2][0])))
else:
reste.extend(Priorites3.splitting('%s**2%r' % (b, m[2][0])))
etapes = list(pre)
etapes.extend(fc)
etapes.extend(reste)
etapes.extend(post)
etapes = [etapes]
for unreste in Priorites3.priorites(''.join(reste)):
calcul = list(pre)
calcul.extend(fc)
if unreste[0][0] != '-':
calcul.append('+')
else:
calcul.append('-')
unreste[0] = unreste[0][1:]
calcul.extend(unreste)
calcul.extend(post)
etapes.append(calcul)
if isinstance(b, str): # On supprime deux étapes trop détaillée UGLY
# TODO: Corriger la classe Fractions pour qu'elle gère plusieurs niveaux de détails.
etapes.pop(1)
etapes.pop(1)
if pre:
calcul = pre[0:-1]
calcul.extend(fc)
fc = list(calcul)
reste = Priorites3.priorites(''.join(reste))[-1]
reste.extend(['*', pre[0]])
for unreste in Priorites3.priorites(''.join(reste)):
calcul = list(fc)
if unreste[0][0] != '-':
calcul.append('+')
else:
calcul.append('-')
unreste[0] = unreste[0][1:]
calcul.extend(unreste)
etapes.append(calcul)
return etapes
def resolution(self, m, pre=[], post=[]):
sgn = '+-'[m[1][0] < 0]
if isinstance(m[1][0], Fraction):
b = Priorites3.priorites(abs(m[1][0]) / 2)[-1][0]
elif m[1][0] % 2:
b = 'Fraction(%s, 2)' % abs(m[1][0])
else:
b = abs(m[1][0]) / 2
fc = ['Polynome("%sx%s%s")' % (m[0][0], sgn, b), '**', '2']
reste = ['-']
if m[2][0] > 0 or isinstance(m[2][0], Fraction):
reste.extend(Priorites3.splitting('%s**2+%r' % (b, m[2][0])))
else:
reste.extend(Priorites3.splitting('%s**2%r' % (b, m[2][0])))
etapes = list(pre)
etapes.extend(fc)
etapes.extend(reste)
etapes.extend(post)
etapes = [etapes]
for unreste in Priorites3.priorites(''.join(reste)):
calcul = list(pre)
calcul.extend(fc)
if unreste[0][0] != '-':
calcul.append('+')
else:
calcul.append('-')
unreste[0] = unreste[0][1:]
calcul.extend(unreste)
calcul.extend(post)
etapes.append(calcul)
if isinstance(b, str): # On supprime deux étapes trop détaillée UGLY
# TODO: Corriger la classe Fractions pour qu'elle gère plusieurs niveaux de détails.
etapes.pop(1)
etapes.pop(1)
if pre:
calcul = pre[0:-1]
calcul.extend(fc)
fc = list(calcul)
reste = Priorites3.priorites(''.join(reste))[-1]
reste.extend(['*', pre[0]])
for unreste in Priorites3.priorites(''.join(reste)):
calcul = list(fc)
if unreste[0][0] != '-':
calcul.append('+')
else:
calcul.append('-')
unreste[0] = unreste[0][1:]
calcul.extend(unreste)
etapes.append(calcul)
return etapes
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 id_rem():
"""Génère un exercice de développement des 3 identités remarquables avec une situation piège.
Dans un premier temps, on n'utilise que des nombres entiers, puis des fractions, puis l'opposé
d'une expression littérale.
"""
l = [randrange(1, 11) for dummy in range(14)]
while pgcd(l[8], l[9]) != 1 or pgcd(l[10], l[11]) != 1 or (l[9] == 1 and l[11] == 1):
# On crée deux rationnels irréductibles non tous deux entiers.
l = [randrange(1, 11) for dummy in range(14)]
lpoly = [id_rem1(l[0], l[1]), id_rem2(l[2], l[3]), id_rem3(l[4], l[5]), id_rem4(l[6], l[7])]
shuffle(lpoly)
lid = [id_rem1, id_rem2, id_rem3, id_rem4]
lpoly2 = [lid.pop(randrange(4))(Fraction(l[8], l[9]), Fraction(l[10], l[11]))]
lpoly2.append('-' + lid.pop(randrange(3))(l[12], l[13]))
shuffle(lpoly2)
lpoly.extend(lpoly2)
expr = [Priorites3.texify([Priorites3.splitting(lpoly[i])]) for i in range(6)]
exo = ["\\exercice", u"Développer chacune des expressions littérales suivantes :"]
exo.append("\\begin{multicols}{2}")
exo.append('\\\\\n'.join(['$%s=%s$' % (chr(i + 65), expr[i][0]) for i in range(6)]))
exo.append("\\end{multicols}")
cor = ["\\exercice*", u"Développer chacune des expressions littérales suivantes :"]
cor.append("\\begin{multicols}{2}")
for i in range(6):
dev = Priorites3.texify(Priorites3.priorites(lpoly[i]))
dev.insert(0, expr[i][0])
cor.append('\\\\\n'.join(['$%s=%s$' % (chr(i + 65), dev[j]) for j in range(len(dev) - 1)]))
cor.append('\\\\')
cor.append('\\fbox{$%s=%s$}\\\\\n' % (chr(i + 65), dev[-1]))
cor.append("\\end{multicols}")
return exo, cor
def corrige(nom, fct, ant):
from pyromaths.outils import Priorites3
sol = []
calc = fct(ant)
res = Priorites3.priorites(calc)
res = Priorites3.texify(res)
sol.append(r"\par $%s\,(%s)=%s$\par" % (nom, ant, Priorites3.texify([Priorites3.splitting(calc)])[0]))
sol.append('\\par\n'.join(['$%s\\,(%s)=%s$' % (nom, ant, res[j]) for j in range(len(res) - 1)]))
sol.append(r'\par')
sol.append('\\fbox{$%s\\,(%s)=%s$}\\\\\n' % (nom, ant, res[-1]))
return sol
def corrige(nom, fct, ant):
from pyromaths.outils import Priorites3
sol = []
calc = fct(ant)
res = Priorites3.priorites(calc)
res = Priorites3.texify(res)
sol.append(r"\par $%s\,(%s)=%s$\par" %
(nom, ant, Priorites3.texify([Priorites3.splitting(calc)])[0]))
sol.append('\\par\n'.join(
['$%s\\,(%s)=%s$' % (nom, ant, res[j]) for j in range(len(res) - 1)]))
sol.append(r'\par')
sol.append('\\fbox{$%s\\,(%s)=%s$}\\\\\n' % (nom, ant, res[-1]))
return sol
def print_coef(coef):
"""Gère le format du coef
"""
if isinstance(coef, (float, int)):
if coef > 0: return "+" + decimaux(coef)
else: return decimaux(coef)
if isinstance(coef, Fraction):
if isinstance(coef.n, int) and isinstance(coef.d, int) and coef.n < 0 and coef.d > 0:
return "-" + str(Fraction(-coef.n, coef.d, coef.code))
return "+" + str(coef)
if isinstance(coef, str):
texte = "(" + "".join(Priorites3.texify([Priorites3.splitting(coef)])) + ")"
if texte[0] != "-": return "+" + texte
else: return texte
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 print_coef(coef):
"""Gère le format du coef
"""
if isinstance(coef, (float, int)):
if coef > 0: return "+" + decimaux(coef)
else: return decimaux(coef)
if isinstance(coef, Fraction):
if isinstance(coef.n, int) and isinstance(
coef.d, int) and coef.n < 0 and coef.d > 0:
return "-" + str(Fraction(-coef.n, coef.d, coef.code))
return "+" + str(coef)
if isinstance(coef, str):
texte = "(" + "".join(
Priorites3.texify([Priorites3.splitting(coef)])) + ")"
if texte[0] != "-": return "+" + texte
else: return texte
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
