# Identités remarquables

## Carrés d'un binôme

• (a + b)^2 = a^2 + 2ab + b^2

• (a - b)^2 = a^2 - 2ab + b^2

## Différence de carrés

• a^2 - b^2 = (a + b)(a - b)

## Identités de degré 3

• (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

• (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

• a^3 - b^3 = (a - b)(a^2 + ab + b^2)

• a^3 - 1 = (a - 1)(a^2 + a + 1)

• a^3 + b^3 = (a + b)(a^2 - ab + b^2)

• a^3 + 1 = (a + 1)(a^2 - a + 1)

• a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)

• (a + b + c)^3 = a^3 + b^3 + c^3 + 3(a^2b + a^2c + b^2c + ab^2 + ac^2 + bc^2) + 6abc

## Identités de degré 4

• (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4

• (a - b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4

• a^4 + b^4 = (a + b)^2(a - b)^2 - 2a^2b^2

• a^4 + 1 = (a + 1)^2(a - 1)^2 - 2a^2

• a^4 - b^4 = (a^2 + b^2)(a^2 - b^2)
= (a^2 + b^2)(a + b)(a - b)
= (a + b)(a^3 - a^2b + ab^2 - b^3)
= (a - b)(a^3 + a^2b + ab^2 + b^3)

• a^4 - 1 = (a^2 + 1)(a^2 - 1)
 = (a^2 + 1)(a + 1)(a - 1)
 = (a + 1)(a^3 - a^2 + a - 1)
 = (a - 1)(a^3 + a^2 + a + 1)

• (a + b + c)^4 = a^4 + b^4 + c^4 + 4a^3b + 4a^3c + 4b^3c + 6a^2b^2 + 6a^2c^2 + 6b^2c^2 + 4ab^3 + 4ac^3 + 4bc^3 + 12ab^2c + 12abc^2 + 12a^2bc