差立方

差立方可直接計算驗證：

${\displaystyle (a-b)^{3}\,\!}$

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

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

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

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

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

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

以上計算方式便可證明:${\displaystyle (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}\,\!}$

差立方亦可運用差平方驗證，首先要知道差平方的公式是：

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

然後，利用差平方計算出差立方：

${\displaystyle (a-b)^{3}\,\!}$

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

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

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

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

以上計算方式便可證明:${\displaystyle (a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}\,\!}$