Soient \(a \in \mathbb{R}\), \(b \in \mathbb{R}\), \(c \in \mathbb{R}\) et \(n \in \mathbb{N}\)

Quelques identités remarquables

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

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

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

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

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

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

De manière générale, nous avons :

Si \(n\) est pair :
7. \({a^n} - {b^n} = \) \(\left( {a + b} \right)\) \(({a^{n - 1}} - {a^{n - 2}}b + \) \(... + {( - 1)^p}{a^{n - p - 1}}{b^p}\) \( + ... - {b^{n - 1}})\)
\({a^n} - {b^n} = \) \(\left( {a - b} \right)\) \(({a^{n - 1}} + {a^{n - 2}}b\) \( + {a^{n - 3}}{b^2} + ... + \) \(a{b^{n - 2}} + {b^{n - 1}})\)

Si \(n\) est impair :
8. \({a^n} + {b^n} = \) \(\left( {a + b} \right)\) \(({a^{n - 1}} - {a^{n - 2}}b + ...\) \( + {( - 1)^p}{a^{n - p - 1}}{b^p}\) \( + ... + {b^{n - 1}})\)
\({a^n} + {b^n} = \) \(\left( {a + b} \right)\) \(({a^{n - 1}} - {a^{n - 2}}b + \) \({a^{n - 3}}{b^2} - ... - \) \(a{b^{n - 2}} + {b^{n - 1}})\)