差立方

主验证

差立方可直接计算验证：

${\displaystyle (a-b)^{3}\,\!}$ 

${\displaystyle =(a-b)(a-b)(a-b)\,\!}$ 

${\displaystyle =a(a-b)(a-b)-b(a-b)(a-b)\,\!}$ 

${\displaystyle =(a^{2}-ab)(a-b)-(ab+b^{2})(a-b)\,\!}$ 

${\displaystyle =a(a^{2}-ab)-b(a^{2}-ab)-a(ab-b^{2})+b(ab-b^{2})\,\!}$ 

${\displaystyle =a^{3}-a^{2}b-a^{2}b+ab^{2}-a^{2}b+ab^{2}+ab^{2}-b^{3}\,\!}$ 

${\displaystyle =a^{3}-3a^{2}b+3ab^{2}-b^{3}\,\!}$ 

以上计算方式便可证明:${\displaystyle (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}\,\!}$ 

运用差平方

差立方亦可运用差平方验证，首先要知道差平方的公式是：

${\displaystyle (a-b)^{2}=a^{2}-2ab+b^{2}\,\!}$ 

然后，利用差平方计算出差立方：

${\displaystyle (a-b)^{3}\,\!}$ 

${\displaystyle =(a-b)^{2}(a-b)\,\!}$ 

${\displaystyle =(a^{2}-2ab+b^{2})(a-b)\,\!}$ 

${\displaystyle =a(a^{2}-2ab+b^{2})-b(a^{2}-2ab+b^{2})\,\!}$ 

${\displaystyle =a^{3}-2a^{2}b+ab^{2}-a^{2}b+2ab^{2}-b^{3}\,\!}$ 

以上计算方式便可证明:${\displaystyle (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}\,\!}$