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salma1
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salma1

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: 09/09/2007

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MessageSujet:      EmptyMer 03 Oct 2007, 15:18

IDENTITÉS FORMELLES

Puissance de
sommes






(a
+ b)n


(a
- b)n



=



Voir [url=#Binome]Formule du binme de
Newton[/url]




Puissances
successives


x <
1






Sommes
limites



1 + x + x2 + xn =
(1 - xn+1) / (1- x)




Sommes
infinies



(1 x) (1 + x + x2 + ) = 1/ (1-x)



Voir [url=../Suite/Unsurx.htm#develop]Sommes infinie[/url]s



Autres






an - bn


=


(a
- b) (an-1 + an-2 b + ...
+
abn-2 + bn-1
)




an
- 1



=


(a
- 1) (an-1 + an-2 + ... + a +
1)














a2n+1
+ 1



=


(a
+ 1) (a2n - a2n-1 + ... - a +
1)






[url=#Top]-Ý- [/url] BINÔME DE NEWTON

On
constate






(a
+ b)2



=


a 2
+ 2ab +
b2




(a
+ b)3



=


a 3
+ 3ab + 3ab +
b3




(a
+ b)4



=


a 4
+ 4a3b + 6a2b2 + 4ab3 +
b4




(a
+ b)5



=


a 5
+ 5
a4b
+ 10
a3b2
+ 10
a2b3
+ 5
ab4
+ b5




etc.





Les coefficients sont les
nombres du [url=../Iteration/TrgPasca.htm]triangle de
Pascal[/url]








(a
- b)2



=


a 2
- 2ab +
b2




(a
- b)3



=


a 3
- 3ab + 3ab - b3




(a
- b)4



=


a 4
- 4a3b +
6a2b2 - 4ab3 +
b4




(a
- b)5



=


a 5
- 5
a4b
+ 10
a3b2
- 10
a2b3
+ 5
ab4
-
b5




etc.





Les signes
alternent




Formulation

[url=../Decompos/DivisiFe.htm#Puissance]Formule[/url] du binme de Newton






(a
+ b)n



=


an + C1n an-1b
+ Cn an-2b2 + ...


+
Cn-2n
a2bn-2 + Cn-1n
abn-1 +
bn




=







Notations






Cpn = (np)


=


n! / p! (n -
p)!




Notez

l'inversion des
indices



=


Nombre de combinaisons

de n lments pris
p p









Coefficients
binomiaux



n!


=


[url=../Compter/SixFact.htm]Factorielle[/url] n











IDENTITÉS degr n







a6
+ b6



=


(a
+ b) (a4 - a2b2 +
b4)




a6
- b6



=


(a
+ b) (
a
- b) (
a
+
ab
+ b) (
a
-
ab
+ b)




a7
+ b7



=


(a
+ b) (
a6
-
ab5
+
a2b4
-
a3b3 + a4b2
-
a5b +
b6)




a7
- b7



=


(a
- b) (
a6
+
ab5
+
a2b4
+
a3b3
+
a4b2
+
a5b
+ b6)




a8 + b8


=


aucune
factorisation




a8
- b8



=


(a
+ b) (a - b) (a + b) (a4 +
b4)




a9
+ b9



=

=

=


(a
+ b) (
a8
-
a7b
+
a6b2
- ... + b8 )


a3
+ b3) (
a6
-
a3b3
+b6)


(a
+ b) (
a2
-
ab
+ b2) (
a6
-
a3b3
+b6 )




a9
- b9



=


(a
- b) (
a2
+
ab
+ b2) (
a6
+
a3b3
+ b6 )




a10
+ b10



=


(a
+ b) (
a8
-
a6b2
+
a4b4
-
a2b6
+ b8)




a10
- b10



=


(a
+ b) (
a
- b)


(a4
+
a3b
+
ab
+
ab3
+ b4)


(a4
-
a3b + ab
-
ab3 +
b4)




a11
+ b11



=


(a
+ b) ( a10 - a9b + a8b2 - ... + b10
)




a11
- b11



=


(a
- b) ( a10 + a9b + a8b2 + ... + b10
)




a12
+ b12



=


(a4
+ b4) (a8 - a4b4 +
b8)




a12
- b12



=


(a
+ b) (
a
- b) (
a
+ b)


(a
+
ab
+ b) (
a
-
ab
+ b)


(a4
-
ab
+ b4)




IDENTITÉS degr 5


utres






n6
-
1



=
(n+1) (n-1)
(n4+n2+1)




n7
-
n



=
(n+1) n (n-1) (n4+n2+1)






(a
+ b)5



=


a 5
+ 5
a4b
+ 10
a3b2
+ 10
a2b3
+ 5
ab4
+ b5




(a
- b)5



=


a 5
-
5
a4b
+ 10
a3b2
-
10
a2b3
+ 5
ab4
-
b5




a5
+ b5



=


(a
+ b) (
a4
-
a3b
+
a2b2
-
ab3
+ b4 )




a5
- b5



=


(a
- b) (
a4
+
a3b
+
a2b2
+
ab3
+ b4 )




a5
+ 1



=


(a
+ 1) (a4 - a3 +
a2 - a + 1
)




a5
- 1



=


(a - 1) (a4 + a3 +
a2 + a + 1
)








n5
n





= (n 2) (n 1)
n (n + 1)
(n + 2)


+
5 (n 1) n (n +
1)



[url=#Top]-[/url] IDENTITÉS degr 4







(a
+ b)4



=


a 4
+ 4a3b + 6a2b2 + 4ab3 +
b4




(a
- b)4



=


a 4
- 4a3b + 6a2b2 - 4ab3 +
b4




a4
+ b4



=

=


(a + b) (a - b) - 2ab

aucune
factorisation




a4 - b4


=

=

=

=


(a
+ b) (
a
- b)


(a
+ b) (
a
+ b) (
a
- b)


(a
+ b) (
a3
-
a2b
+
ab2
- b3 )


(a
- b) (
a3
+
a2b
+
ab2
+ b3 )




a4
+ 4b4



=

=


(a
+ 2
ab
+ 2b)(
a
- 2
ab
+ 2b)


[(a
+ b) + b]
[(
a
- b) + b]




a4
- 4b4



=


(a
+ 2b) (a - 2b)
vident



(a
+ b + c)4



=


a4 + b4
+ c4


+
4
a3b +
4
a3c +
4b3c


+
6
ab+
6
ac+
6bc


+ 4ab3 + 4ac3 + 4bc3

12abc + 12abc + 12abc



(a
+ b)4 + (a - b)4



=


2a4 + 12ab + 2b4



(a
+ b)4 - (a - b)4



=


8ab (a +
b)




(a
+ b)4 (a -
b)4



=


a8 +
b8 + 6a4b4 - 4ab (a4 +
b4)




(a
+ b)4 / (a - b)4






pas
intressant





a3 + a - a - 1


=


(a + 1) (a - 1)





(a
+ b)3



=


a 3
+ 3ab + 3ab +
b3




(a
- b)3



=


a 3
- 3ab + 3ab -
b3




a3
- b3



=


(a
- b) (a + ab + b
)




a3
+ b3



=


(a
+ b) (a - ab + b
)




a3
+ b3 + c3






pas
intressant




a3 +
1



=


(a
+ 1) (a - a + 1 )




(a
+ b + c)3



=


a3 +
b3 +c3


+ 3 ( ab + ac + bc + ab + ac + bc
)


+ 6 abc



(a
+ b)3 + (a - b)3



=


2a3 + 6ab



(a
+ b)3 - (a - b)3



=


2b3 + 6ab



(a
+ b)3 * (a - b)3



=


a6 - 3a4b2 + 3a2b4 -
b6




(a
+ b)3 / (a - b)3






pas
intressant




(a-b)3 + (b-c)3 + (c-a)3


=


3(a
- b)(b - c)(c - a)
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