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