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\[ \left( {x + a} \right)\left( {x + b} \right) = x^2 + \left( {a + b} \right)x + ab \] \[ \left( {a \pm b} \right)^2 = a^2 \pm 2ab + b^2 \] \[ \left( {a \pm b} \right)^3 = a^3 \pm 3a^2 b + 3ab^2 \pm b^3 \] \[ a^2 - b^2 = \left( {a - b} \right)\left( {a + b} \right) \] \[ a^3 \pm b^3 = \left( {a \pm b} \right)\left( {a^2 \mp ab + b^2 } \right) \]
\[ a^n - b^n = \left( {a - b} \right)\left( {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \cdots + ab^{n - 2} + b^{n - 1} } \right)\left( n为正整数 \right) \] \[ a^n - b^n = \left( {a + b} \right)\left( {a^{n - 1} - a^{n - 2} b + a^{n - 3} b^2 - \cdots + ab^{b - 2} - b^{n - 1} } \right)\left( n为偶数 \right) \] \[ a^n + b^n = \left( {a + b} \right)\left( {a^{n - 1} - a^{n - 2} b + a^{n - 3} b^2 - \cdots - ab^{b - 2} + b^{n - 1} } \right)\left( n为奇数 \right) \] \[ \left( {a + b + c} \right)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \] \[ a^3 + b^3 + c^3 - 3abc = \left( {a + b + c} \right)\left( {a^2 + b^2 + c^2 - ab - bc - ca} \right) \]