等冪和差公式

等幂和差，又称幂和差，指同是n次幂的a与b的和与差，所得出來的乘法公式。 a 3 + b 3 ≡ ( a + b ) ( a 2 − a b + b 2 ) {\displaystyle a^{3}+b^{3}\equiv (a+b)(a^{2}-ab+b^{2})\,\!} a 5 + b 5 ≡ ( a + b ) ( a 4 − a 3 b + a 2 b 2 − a b 3 + b 4 ) {\displaystyle a^{5}+b^{5}\equiv (a+b)(a^{4}-a^{3}b+a^{2}b^{2}-ab^{3}+b^{4})\,\!} a 6 + b 6 ≡ ( a 2 + b 2 ) ( a 4 − a 2 b 2 + b 4 ) {\displaystyle a^{6}+b^{6}\equiv (a^{2}+b^{2})(a^{4}-a^{2}b^{2}+b^{4})\,\!} a 7 + b 7 ≡ ( a + b ) ( a 6 − a 5 b + a 4 b 2 − a 3 b 3 + a 2 b 4 − a b 5 + b 6 ) {\displaystyle a^{7}+b^{7}\equiv (a+b)(a^{6}-a^{5}b+a^{4}b^{2}-a^{3}b^{3}+a^{2}b^{4}-ab^{5}+b^{6})\,\!} a 9 + b 9 ≡ ( a + b ) ( a 2 − a b + b 2 ) ( a 6 − a 3 b 3 + b 6 ) {\displaystyle a^{9}+b^{9}\equiv (a+b)(a^{2}-ab+b^{2})(a^{6}-a^{3}b^{3}+b^{6})\,\!} a 10 + b 10 ≡ ( a 2 + b 2 ) ( a 8 − a 6 b 2 + a 4 b 4 − a 2 b 6 + b 8 ) {\displaystyle a^{10}+b^{10}\equiv (a^{2}+b^{2})(a^{8}-a^{6}b^{2}+a^{4}b^{4}-a^{2}b^{6}+b^{8})\,\!} a 11 + b 11 ≡ ( a + b ) ( a 10 − a 9 b + a 8 b 2 − a 7 b 3 + a 6 b 4 − a 5 b 5 + a 4 b 6 − a 3 b 7 + a 2 b 8 − a b 9 + b 10 ) {\displaystyle a^{11}+b^{11}\equiv (a+b)(a^{10}-a^{9}b+a^{8}b^{2}-a^{7}b^{3}+a^{6}b^{4}-a^{5}b^{5}+a^{4}b^{6}-a^{3}b^{7}+a^{2}b^{8}-ab^{9}+b^{10})\,\!} a 12 + b 12 ≡ ( a 4 + b 4 ) ( a 8 − a 4 b 4 + b 8 ) {\displaystyle a^{12}+b^{12}\equiv (a^{4}+b^{4})(a^{8}-a^{4}b^{4}+b^{8})\,\!} a 13 + b 13 ≡ ( a + b ) ( a 12 − a 11 b + a 10 b 2 − a 9 b 3 + a 8 b 4 − a 7 b 5 + a 6 b 6 − a 5 b 7 + a 4 b 8 − a 3 b 9 + a 2 b 10 − a b 11 + b 12 ) {\displaystyle a^{13}+b^{13}\equiv (a+b)(a^{12}-a^{11}b+a^{10}b^{2}-a^{9}b^{3}+a^{8}b^{4}-a^{7}b^{5}+a^{6}b^{6}-a^{5}b^{7}+a^{4}b^{8}-a^{3}b^{9}+a^{2}b^{10}-ab^{11}+b^{12})\,\!} a 14 + b 14 ≡ ( a 2 + b 2 ) ( a 12 − a 10 b 2 + a 8 b 4 − a 6 b 6 + a 4 b 8 − a 2 b 10 + b 12 ) {\displaystyle a^{14}+b^{14}\equiv (a^{2}+b^{2})(a^{12}-a^{10}b^{2}+a^{8}b^{4}-a^{6}b^{6}+a^{4}b^{8}-a^{2}b^{10}+b^{12})\,\!} 注意當n是2、4、8、16、32 ......(2的幂)的時侯是無法進行因式分解的。 a 2 − b 2 ≡ ( a − b ) ( a + b ) {\displaystyle a^{2}-b^{2}\equiv (a-b)(a+b)\,\!} a 3 − b 3 ≡ ( a − b ) ( a 2 + a b + b 2 ) {\displaystyle a^{3}-b^{3}\equiv (a-b)(a^{2}+ab+b^{2})\,\!} a 4 − b 4 ≡ ( a − b ) ( a + b ) ( a 2 + b 2 ) {\displaystyle a^{4}-b^{4}\equiv (a-b)(a+b)(a^{2}+b^{2})\,\!} a 5 − b 5 ≡ ( a − b ) ( a 4 + a 3 b + a 2 b 2 + a b 3 + b 4 ) {\displaystyle a^{5}-b^{5}\equiv (a-b)(a^{4}+a^{3}b+a^{2}b^{2}+ab^{3}+b^{4})\,\!} a 6 − b 6 ≡ ( a − b ) ( a + b ) ( a 2 + a b + b 2 ) ( a 2 − a b + b 2 ) {\displaystyle a^{6}-b^{6}\equiv (a-b)(a+b)(a^{2}+ab+b^{2})(a^{2}-ab+b^{2})\,\!} a 7 − b 7 ≡ ( a − b ) ( a 6 + a 5 b + a 4 b 2 + a 3 b 3 + a 2 b 4 + a b 5 + b 6 ) {\displaystyle a^{7}-b^{7}\equiv (a-b)(a^{6}+a^{5}b+a^{4}b^{2}+a^{3}b^{3}+a^{2}b^{4}+ab^{5}+b^{6})\,\!} a 8 − b 8 ≡ ( a − b ) ( a + b ) ( a 2 + b 2 ) ( a 4 + b 4 ) {\displaystyle a^{8}-b^{8}\equiv (a-b)(a+b)(a^{2}+b^{2})(a^{4}+b^{4})\,\!} a 9 − b 9 ≡ ( a − b ) ( a 2 + a b + b 2 ) ( a 6 + a 3 b 3 + b 6 ) {\displaystyle a^{9}-b^{9}\equiv (a-b)(a^{2}+ab+b^{2})(a^{6}+a^{3}b^{3}+b^{6})\,\!} a 10 − b 10 ≡ ( a − b ) ( a + b ) ( a 4 + a 3 b + a 2 b 2 + a b 3 + b 4 ) ( a 4 − a 3 b + a 2 b 2 − a b 3 + b 4 ) {\displaystyle a^{10}-b^{10}\equiv (a-b)(a+b)(a^{4}+a^{3}b+a^{2}b^{2}+ab^{3}+b^{4})(a^{4}-a^{3}b+a^{2}b^{2}-ab^{3}+b^{4})\,\!} a 11 − b 11 ≡ ( a − b ) ( a 10 + a 9 b + a 8 b 2 + a 7 b 3 + a 6 b 4 + a 5 b 5 + a 4 b 6 + a 3 b 7 + a 2 b 8 + a b 9 + b 10 ) {\displaystyle a^{11}-b^{11}\equiv (a-b)(a^{10}+a^{9}b+a^{8}b^{2}+a^{7}b^{3}+a^{6}b^{4}+a^{5}b^{5}+a^{4}b^{6}+a^{3}b^{7}+a^{2}b^{8}+ab^{9}+b^{10})\,\!} a 12 − b 12 ≡ ( a − b ) ( a + b ) ( a 2 + b 2 ) ( a 2 − a b + b 2 ) ( a 2 + a b + b 2 ) ( a 4 − a 2 b 2 + b 4 ) {\displaystyle a^{12}-b^{12}\equiv (a-b)(a+b)(a^{2}+b^{2})(a^{2}-ab+b^{2})(a^{2}+ab+b^{2})(a^{4}-a^{2}b^{2}+b^{4})\,\!} a 13 − b 13 ≡ ( a − b ) ( a 12 + a 11 b + a 10 b 2 + a 9 b 3 + a 8 b 4 + a 7 b 5 + a 6 b 6 + a 5 b 7 + a 4 b 8 + a 3 b 9 + a 2 b 10 + a b 11 + b 12 ) {\displaystyle a^{13}-b^{13}\equiv (a-b)(a^{12}+a^{11}b+a^{10}b^{2}+a^{9}b^{3}+a^{8}b^{4}+a^{7}b^{5}+a^{6}b^{6}+a^{5}b^{7}+a^{4}b^{8}+a^{3}b^{9}+a^{2}b^{10}+ab^{11}+b^{12})\,\!} a 14 − b 14 ≡ ( a + b ) ( a − b ) ( a 6 + a 5 b + a 4 b 2 + a 3 b 3 + a 2 b 4 + a b 5 + b 6 ) ( a 6 − a 5 b + a 4 b 2 − a 3 b 3 + a 2 b 4 − a b 5 + b 6 ) {\displaystyle a^{14}-b^{14}\equiv (a+b)(a-b)(a^{6}+a^{5}b+a^{4}b^{2}+a^{3}b^{3}+a^{2}b^{4}+ab^{5}+b^{6})(a^{6}-a^{5}b+a^{4}b^{2}-a^{3}b^{3}+a^{2}b^{4}-ab^{5}+b^{6})\,\!}