IDENTITÉS FORMELLESPuissance de
sommes
(a
+ b)n
(a
- b)n
=
§
Voir [url=#Binome]Formule du binôme de
Newton[/url]
Puissances
successivesx <
1
Sommes
limitées
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 NEWTONOn
constate
(a
+ b)2
=
a 2
+ 2ab +
b2
(a
+ b)3
=
a 3
+ 3a²b + 3ab² +
b3
(a
+ b)4
=
a 4
+ 4a3b + 6a2b2 + 4ab3 +
b4
(a
+ b)5
=
a 5
+ 5a4b
+ 10a3b2
+ 10a2b3
+ 5ab4
+ 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
- 3a²b + 3ab² - b3
(a
- b)4
=
a 4
- 4a3b +
6a2b2 - 4ab3 +
b4
(a
- b)5
=
a 5
- 5a4b
+ 10a3b2
- 10a2b3
+ 5ab4
-
b5
etc.
Les signes
alternent
Formulation[url=../Decompos/DivisiFe.htm#Puissance]Formule[/url] du binôme de Newton
(a
+ b)n
=
an + C1n an-1b
+ C²n an-2b2 + ...
+
Cn-2n
a2bn-2 + Cn-1n
abn-1 +
bn
=
NotationsCpn = (np)=
n! / p! (n -
p)!Notez l'inversion des
indices=
Nombre de
combinaisons de
n éléments pris
p à p Coefficients
binomiaux
n!=
[url=../Compter/SixFact.htm]Factorielle[/url]
n IDENTITÉS degré n
a6
+ b6
=
(a²
+ b²) (a4 - a2b2 +
b4)
a6
- b6
=
(a
+ b) (a
- b) (a²
+ ab
+ b²) (a²
- ab
+ b²)
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) (a² + b²) (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
=
(a²
+ b²) (a8
- a6b2
+a4b4
- a2b6
+ b8)
a10
- b10
=
(a
+ b) (a
- b)
(a4
+ a3b
+ a²b²
+ ab3
+ b4)
(a4
- a3b + a²b²
- 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) (a²
+ b²)
(a²
+ ab
+ b²) (a²
- ab
+ b²)
(a4
- a²b²
+ b4)
IDENTITÉS degré 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
+ 5a4b
+ 10a3b2
+ 10a2b3
+ 5ab4
+ b5
(a
- b)5
=
a 5
-
5a4b
+ 10a3b2
-
10a2b3
+ 5ab4
-
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 degré 4
(a
+ b)4
=
a 4
+ 4a3b + 6a2b2 + 4ab3 +
b4
(a
- b)4
=
a 4
- 4a3b + 6a2b2 - 4ab3 +
b4
a4
+ b4
=
=
(a + b)² (a - b)² - 2a²b²
aucune
factorisation
a4 - b4
=
=
=
=
(a²
+ b²) (a²
- b²)
(a²
+ b²) (a
+ b) (a
- b)
(a
+ b) (a3
- a2b
+ ab2
- b3 )
(a
- b) (a3
+ a2b
+ ab2
+ b3 )
a4
+ 4b4
=
=
(a²
+ 2ab
+ 2b²)( a²
- 2ab
+ 2b²)
[(a
+ b)² + b²]
[(a
- b)² + b²]
a4
- 4b4
=
(a²
+ 2b²) (a² - 2b²) évident
(a
+ b + c)4
=
a4 + b4
+ c4
+
4a3b +
4a3c +
4b3c
+
6a²b²+
6a²c²+
6b²c²
+ 4ab3 + 4ac3 + 4bc3
12ab²c + 12abc² + 12a²bc
(a
+ b)4 + (a - b)4
=
2a4 + 12a²b² + 2b4
(a
+ b)4 - (a - b)4
=
8ab (a² +
b²)
(a
+ b)4 (a -
b)4
=
a8 +
b8 + 6a4b4 - 4a²b² (a4 +
b4)
(a
+ b)4 / (a - b)4
pas
intéressant
a3 + a² - a - 1=(a + 1)² (a - 1)(a
+ b)3=a 3
+ 3a²b + 3ab² +
b3(a
- b)3=a 3
- 3a²b + 3ab² -
b3a3
- b3=(a
- b) (a² + ab + b²
)a3
+ b3=(a
+ b) (a² - ab + b²
)a3
+ b3 + c3 pas
intéressanta3 +
1=(a
+ 1) (a² - a + 1 )(a
+ b + c)3=a3 +
b3 +c3 + 3 ( a²b + a²c + b²c + ab² + ac² + bc²
)+ 6 abc(a
+ b)3 + (a - b)3=2a3 + 6ab²(a
+ b)3 - (a - b)3=2b3 + 6a²b(a
+ b)3 * (a - b)3=a6 - 3a4b2 + 3a2b4 -
b6(a
+ b)3 / (a - b)3 pas
intéressant(a-b)3 + (b-c)3 + (c-a)3=3(a
- b)(b - c)(c - a)