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。

等幂和的因式分解

$a^3+b^3\equiv (a+b)(a^2-ab+b^2)$

$a^5+b^5\equiv (a+b)(a^4-a^{3}b+a^2{b}^2-a{b}^3+b^4)$

$a^6+b^6\equiv (a^2+b^2)(a^4-a^2{b}^2+b^4)$

$a^7+b^7\equiv (a+b)(a^6-a^{5}b+a^4{b}^2-a^3{b}^3+a^2{b}^4-a{b}^5+b^6)$

$a^9+b^9\equiv (a+b)(a^2-ab+b^2)(a^6-a^3{b}^3+b^6)$

$a^{10}+b^{10}\equiv (a^2+b^2)(a^8-a^6{b}^2+a^4{b}^4-a^2{b}^6+b^8)$

$a^{11}+b^{11}\equiv (a+b)(a^{10}-a^{9}b+a^8{b}^2-a^7{b}^3+a^6{b}^4-a^5{b}^5+a^4{b}^6-a^3{b}^7+a^2{b}^8-a{b}^9+b^{10})$

$a^{12}+b^{12}\equiv (a^4+b^4)(a^8-a^4{b}^4+b^8)$

$a^{13}+b^{13}\equiv (a+b)(a^{12}-a^{11}b+a^{10}{b}^2-a^9{b}^3+a^8{b}^4-a^7{b}^5+a^6{b}^6-a^5{b}^7+a^4{b}^8-a^3{b}^9+a^2{b}^{10}-a{b}^{11}+b^{12})$

$a^{14}+b^{14}\equiv (a^2+b^2)(a^{12}-a^{10}{b}^2+a^8{b}^4-a^6{b}^6+a^4{b}^8-a^2{b}^{10}+b^{12})$

• 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。

等幂差的因式分解

$a^2-b^2\equiv (a-b)(a+b)$

$a^3-b^3\equiv (a-b)(a^2+ab+b^2)$

$a^4-b^4\equiv (a-b)(a+b)(a^2+b^2)$

$a^5-b^5\equiv (a-b)(a^4+a^{3}b+a^2{b}^2+a{b}^3+b^4)$

$a^6-b^6\equiv (a-b)(a+b)(a^2+ab+b^2)(a^2-ab+b^2)$

$a^7-b^7\equiv (a-b)(a^6+a^{5}b+a^4{b}^2+a^3{b}^3+a^2{b}^4+a{b}^5+b^6)$

$a^8-b^8\equiv (a-b)(a+b)(a^2+b^2)(a^4+b^4)$

$a^9-b^9\equiv (a-b)(a^2+ab+b^2)(a^6+a^3{b}^3+b^6)$

$a^{10}-b^{10}\equiv (a-b)(a+b)(a^4+a^{3}b+a^2{b}^2+a{b}^3+b^4)(a^4-a^{3}b+a^2{b}^2-a{b}^3+b^4)$

$a^{11}-b^{11}\equiv (a-b)(a^{10}+a^{9}b+a^8{b}^2+a^7{b}^3+a^6{b}^4+a^5{b}^5+a^4{b}^6+a^3{b}^7+a^2{b}^8+a{b}^9+b^{10})$

$a^{12}-b^{12}\equiv (a-b)(a+b)(a^2+b^2)(a^2-ab+b^2)(a^2+ab+b^2)(a^4-a^2{b}^2+b^4)$

$a^{13}-b^{13}\equiv (a-b)(a^{12}+a^{11}b+a^{10}{b}^2+a^9{b}^3+a^8{b}^4+a^7{b}^5+a^6{b}^6+a^5{b}^7+a^4{b}^8+a^3{b}^9+a^2{b}^{10}+a{b}^{11}+b^{12})$

$a^{14}-b^{14}\equiv (a+b)(a-b)(a^6+a^{5}b+a^4{b}^2+a^3{b}^3+a^2{b}^4+a{b}^5+b^6)(a^6-a^{5}b+a^4{b}^2-a^3{b}^3+a^2{b}^4-a{b}^5+b^6)$