和立方和立方是数学公式的一种,它属于因式分解、乘法公式及恒等式,被普遍使用。和立方是指一个数项,加上另一个数项后,总和的立方: ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 = a 3 ± b 3 ± 3 a b ( a ± b ) {\displaystyle (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!=a^{3}\pm b^{3}\pm 3ab(a\pm b)} 验证 主验证 和立方可直接计算验证: ( a + b ) 3 {\displaystyle (a+b)^{3}\,\!} = ( a + b ) ( a + b ) ( a + b ) {\displaystyle =(a+b)(a+b)(a+b)\,\!} = a ( a + b ) ( a + b ) + b ( a + b ) ( a + b ) {\displaystyle =a(a+b)(a+b)+b(a+b)(a+b)\,\!} = ( a 2 + a b ) ( a + b ) + ( a b + b 2 ) ( a + b ) {\displaystyle =(a^{2}+ab)(a+b)+(ab+b^{2})(a+b)\,\!} = a ( a 2 + a b ) + b ( a 2 + a b ) + a ( a b + b 2 ) + b ( a b + b 2 ) {\displaystyle =a(a^{2}+ab)+b(a^{2}+ab)+a(ab+b^{2})+b(ab+b^{2})\,\!} = a 3 + a 2 b + a 2 b + a b 2 + a 2 b + a b 2 + a b 2 + b 3 {\displaystyle =a^{3}+a^{2}b+a^{2}b+ab^{2}+a^{2}b+ab^{2}+ab^{2}+b^{3}\,\!} = a 3 + 3 a 2 b + 3 a b 2 + b 3 {\displaystyle =a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!} 以上计算方式便可证明: ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 {\displaystyle (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!} 运用和平方 和立方亦可运用和平方验证,首先要知道和平方的公式是: ( a + b ) 2 = a 2 + 2 a b + b 2 {\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}\,\!} 然后,利用和平方计算出和立方: ( a + b ) 3 {\displaystyle (a+b)^{3}\,\!} = ( a + b ) 2 ( a + b ) {\displaystyle =(a+b)^{2}(a+b)\,\!} = ( a 2 + 2 a b + b 2 ) ( a + b ) {\displaystyle =(a^{2}+2ab+b^{2})(a+b)\,\!} = a ( a 2 + 2 a b + b 2 ) + b ( a 2 + 2 a b + b 2 ) {\displaystyle =a(a^{2}+2ab+b^{2})+b(a^{2}+2ab+b^{2})\,\!} = a 3 + 2 a 2 b + a b 2 + a 2 b + 2 a b 2 + b 3 {\displaystyle =a^{3}+2a^{2}b+ab^{2}+a^{2}b+2ab^{2}+b^{3}\,\!} = a 3 + 3 a 2 b + 3 a b 2 + b 3 {\displaystyle =a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!} 以上计算方式便可证明: ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 {\displaystyle (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!}