Soient $a \in \mathbb{R}$, $b \in \mathbb{R}$, $c \in \mathbb{R}$, $n \in \mathbb{N}$ et $k \in \mathbb{N}$.

1. ${\left( {a - b} \right)^2} =$ ${a^2} - 2ab + {b^2}$

2. ${\left( {a + b} \right)^2} =$ ${a^2} + 2ab + {b^2}$

3. ${\left( {a - b} \right)^3} =$ ${a^3} - 3{a^2}b +$ $3a{b^2} - {b^3}$

4. ${\left( {a + b} \right)^3} =$ ${a^3} + 3{a^2}b +$ $3a{b^2} + {b^3}$

5. ${\left( {a - b} \right)^4} = {a^4}$ $- 4{a^3}b + 6{a^2}{b^2}$ $- 4a{b^3} + {b^4}$

6. ${\left( {a + b} \right)^4} = {a^4}$ $+ 4{a^3}b + 6{a^2}{b^2}$ $+ 4a{b^3} + {b^4}$

Formule du binôme
7. ${\left( {a + b} \right)^n} =$ $C_n^0{a^n} + C_n^1{a^{n - 1}}b +$ $C_n^2{a^{n - 2}}{b^2} + ...$ $+ C_n^n{b^n}$
Avec $C_n^k = \frac{{n!}}{{k!\left( {n - k} \right)!}}$ les coefficients binomials
• Si $a = b = 1$, on a :
$1 + C_n^1 +$ $C_n^2 + ... + C_n^n$ $= {2^n}$

8. ${\left( {a + b + c} \right)^2} =$ ${a^2} + {b^2} + {c^2} +$ $2ab + 2ac + 2bc$

9. $(a + b + c + ...$ $+ u + v{)^2} =$ ${a^2} + {b^2} + {c^2} + ...$ $+ {u^2} + {v^2} +$ $2(ab + ac + ...$ $+ bu + bv + uv)$