(a+b)2{\displaystyle (a+b)^{2}} =(a+b)(a+b){\displaystyle =(a+b)(a+b)} =a(a+b)+b(a+b){\displaystyle =a(a+b)+b(a+b)} =a2+ab+ab+b2{\displaystyle =a^{2}+ab+ab+b^{2}} =a2+2ab+b2{\displaystyle =a^{2}+2ab+b^{2}}
(a−b)2{\displaystyle (a-b)^{2}} =(a−b)(a−b){\displaystyle =(a-b)(a-b)} =a(a−b)−b(a−b){\displaystyle =a(a-b)-b(a-b)} =a2−ab−ab+b2{\displaystyle =a^{2}-ab-ab+b^{2}} =a2−2ab+b2{\displaystyle =a^{2}-2ab+b^{2}}
a2−b2{\displaystyle a^{2}-b^{2}} =a2+ab−ab−b2{\displaystyle =a^{2}+ab-ab-b^{2}} =a(a+b)−b(a+b){\displaystyle =a(a+b)-b(a+b)} =(a+b)(a−b){\displaystyle =(a+b)(a-b)}
(a+b)3{\displaystyle (a+b)^{3}} =(a+b)×(a+b)2{\displaystyle =(a+b)\times (a+b)^{2}} =(a+b)(a2+2ab+b2){\displaystyle =(a+b)\left(a^{2}+2ab+b^{2}\right)} =a(a2+2ab+b2)+b(a2+2ab+b2){\displaystyle =a\left(a^{2}+2ab+b^{2}\right)+b\left(a^{2}+2ab+b^{2}\right)} =a3+2a2b+ab2+a2b+2ab2+b3{\displaystyle =a^{3}+2a^{2}b+ab^{2}+a^{2}b+2ab^{2}+b^{3}} =a3+3a2b+3ab2+b3{\displaystyle =a^{3}+3a^{2}b+3ab^{2}+b^{3}}
(a3−b3){\displaystyle (a^{3}-b^{3})} =(a−b)×(a−b)2{\displaystyle =(a-b)\times (a-b)^{2}} =(a−b)(a2−2ab+b2){\displaystyle =(a-b)\left(a^{2}-2ab+b^{2}\right)} =a(a2−2ab+b2)−b(a2−2ab+b2){\displaystyle =a\left(a^{2}-2ab+b^{2}\right)-b\left(a^{2}-2ab+b^{2}\right)} =a3−2a2b+ab2−a2b+2ab2−b3{\displaystyle =a^{3}-2a^{2}b+ab^{2}-a^{2}b+2ab^{2}-b^{3}} =a3−3a2b+3ab2−b3{\displaystyle =a^{3}-3a^{2}b+3ab^{2}-b^{3}}